## P-Value And Statistical Significance: What It Is & Why It Matters

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Editor-in-Chief for Simply Psychology

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Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

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The p-value in statistics quantifies the evidence against a null hypothesis. A low p-value suggests data is inconsistent with the null, potentially favoring an alternative hypothesis. Common significance thresholds are 0.05 or 0.01.

## Hypothesis testing

When you perform a statistical test, a p-value helps you determine the significance of your results in relation to the null hypothesis.

The null hypothesis (H0) states no relationship exists between the two variables being studied (one variable does not affect the other). It states the results are due to chance and are not significant in supporting the idea being investigated. Thus, the null hypothesis assumes that whatever you try to prove did not happen.

The alternative hypothesis (Ha or H1) is the one you would believe if the null hypothesis is concluded to be untrue.

The alternative hypothesis states that the independent variable affected the dependent variable, and the results are significant in supporting the theory being investigated (i.e., the results are not due to random chance).

## What a p-value tells you

A p-value, or probability value, is a number describing how likely it is that your data would have occurred by random chance (i.e., that the null hypothesis is true).

The level of statistical significance is often expressed as a p-value between 0 and 1.

The smaller the p -value, the less likely the results occurred by random chance, and the stronger the evidence that you should reject the null hypothesis.

Remember, a p-value doesn’t tell you if the null hypothesis is true or false. It just tells you how likely you’d see the data you observed (or more extreme data) if the null hypothesis was true. It’s a piece of evidence, not a definitive proof.

## Example: Test Statistic and p-Value

Suppose you’re conducting a study to determine whether a new drug has an effect on pain relief compared to a placebo. If the new drug has no impact, your test statistic will be close to the one predicted by the null hypothesis (no difference between the drug and placebo groups), and the resulting p-value will be close to 1. It may not be precisely 1 because real-world variations may exist. Conversely, if the new drug indeed reduces pain significantly, your test statistic will diverge further from what’s expected under the null hypothesis, and the p-value will decrease. The p-value will never reach zero because there’s always a slim possibility, though highly improbable, that the observed results occurred by random chance.

## P-value interpretation

The significance level (alpha) is a set probability threshold (often 0.05), while the p-value is the probability you calculate based on your study or analysis.

## A p-value less than or equal to your significance level (typically ≤ 0.05) is statistically significant.

A p-value less than or equal to a predetermined significance level (often 0.05 or 0.01) indicates a statistically significant result, meaning the observed data provide strong evidence against the null hypothesis.

This suggests the effect under study likely represents a real relationship rather than just random chance.

For instance, if you set α = 0.05, you would reject the null hypothesis if your p -value ≤ 0.05.

It indicates strong evidence against the null hypothesis, as there is less than a 5% probability the null is correct (and the results are random).

Therefore, we reject the null hypothesis and accept the alternative hypothesis.

## Example: Statistical Significance

Upon analyzing the pain relief effects of the new drug compared to the placebo, the computed p-value is less than 0.01, which falls well below the predetermined alpha value of 0.05. Consequently, you conclude that there is a statistically significant difference in pain relief between the new drug and the placebo.

## What does a p-value of 0.001 mean?

A p-value of 0.001 is highly statistically significant beyond the commonly used 0.05 threshold. It indicates strong evidence of a real effect or difference, rather than just random variation.

Specifically, a p-value of 0.001 means there is only a 0.1% chance of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is correct.

Such a small p-value provides strong evidence against the null hypothesis, leading to rejecting the null in favor of the alternative hypothesis.

## A p-value more than the significance level (typically p > 0.05) is not statistically significant and indicates strong evidence for the null hypothesis.

This means we retain the null hypothesis and reject the alternative hypothesis. You should note that you cannot accept the null hypothesis; we can only reject it or fail to reject it.

Note : when the p-value is above your threshold of significance, it does not mean that there is a 95% probability that the alternative hypothesis is true.

## One-Tailed Test

## Two-Tailed Test

## How do you calculate the p-value ?

Most statistical software packages like R, SPSS, and others automatically calculate your p-value. This is the easiest and most common way.

Online resources and tables are available to estimate the p-value based on your test statistic and degrees of freedom.

These tables help you understand how often you would expect to see your test statistic under the null hypothesis.

Understanding the Statistical Test:

Different statistical tests are designed to answer specific research questions or hypotheses. Each test has its own underlying assumptions and characteristics.

For example, you might use a t-test to compare means, a chi-squared test for categorical data, or a correlation test to measure the strength of a relationship between variables.

Be aware that the number of independent variables you include in your analysis can influence the magnitude of the test statistic needed to produce the same p-value.

This factor is particularly important to consider when comparing results across different analyses.

## Example: Choosing a Statistical Test

If you’re comparing the effectiveness of just two different drugs in pain relief, a two-sample t-test is a suitable choice for comparing these two groups. However, when you’re examining the impact of three or more drugs, it’s more appropriate to employ an Analysis of Variance ( ANOVA) . Utilizing multiple pairwise comparisons in such cases can lead to artificially low p-values and an overestimation of the significance of differences between the drug groups.

## How to report

A statistically significant result cannot prove that a research hypothesis is correct (which implies 100% certainty).

Instead, we may state our results “provide support for” or “give evidence for” our research hypothesis (as there is still a slight probability that the results occurred by chance and the null hypothesis was correct – e.g., less than 5%).

## Example: Reporting the results

In our comparison of the pain relief effects of the new drug and the placebo, we observed that participants in the drug group experienced a significant reduction in pain ( M = 3.5; SD = 0.8) compared to those in the placebo group ( M = 5.2; SD = 0.7), resulting in an average difference of 1.7 points on the pain scale (t(98) = -9.36; p < 0.001).

The 6th edition of the APA style manual (American Psychological Association, 2010) states the following on the topic of reporting p-values:

“When reporting p values, report exact p values (e.g., p = .031) to two or three decimal places. However, report p values less than .001 as p < .001.

The tradition of reporting p values in the form p < .10, p < .05, p < .01, and so forth, was appropriate in a time when only limited tables of critical values were available.” (p. 114)

- Do not use 0 before the decimal point for the statistical value p as it cannot equal 1. In other words, write p = .001 instead of p = 0.001.
- Please pay attention to issues of italics ( p is always italicized) and spacing (either side of the = sign).
- p = .000 (as outputted by some statistical packages such as SPSS) is impossible and should be written as p < .001.
- The opposite of significant is “nonsignificant,” not “insignificant.”

## Why is the p -value not enough?

A lower p-value is sometimes interpreted as meaning there is a stronger relationship between two variables.

However, statistical significance means that it is unlikely that the null hypothesis is true (less than 5%).

To understand the strength of the difference between the two groups (control vs. experimental) a researcher needs to calculate the effect size .

## When do you reject the null hypothesis?

In statistical hypothesis testing, you reject the null hypothesis when the p-value is less than or equal to the significance level (α) you set before conducting your test. The significance level is the probability of rejecting the null hypothesis when it is true. Commonly used significance levels are 0.01, 0.05, and 0.10.

Remember, rejecting the null hypothesis doesn’t prove the alternative hypothesis; it just suggests that the alternative hypothesis may be plausible given the observed data.

The p -value is conditional upon the null hypothesis being true but is unrelated to the truth or falsity of the alternative hypothesis.

## What does p-value of 0.05 mean?

If your p-value is less than or equal to 0.05 (the significance level), you would conclude that your result is statistically significant. This means the evidence is strong enough to reject the null hypothesis in favor of the alternative hypothesis.

## Are all p-values below 0.05 considered statistically significant?

No, not all p-values below 0.05 are considered statistically significant. The threshold of 0.05 is commonly used, but it’s just a convention. Statistical significance depends on factors like the study design, sample size, and the magnitude of the observed effect.

A p-value below 0.05 means there is evidence against the null hypothesis, suggesting a real effect. However, it’s essential to consider the context and other factors when interpreting results.

Researchers also look at effect size and confidence intervals to determine the practical significance and reliability of findings.

## How does sample size affect the interpretation of p-values?

Sample size can impact the interpretation of p-values. A larger sample size provides more reliable and precise estimates of the population, leading to narrower confidence intervals.

With a larger sample, even small differences between groups or effects can become statistically significant, yielding lower p-values. In contrast, smaller sample sizes may not have enough statistical power to detect smaller effects, resulting in higher p-values.

Therefore, a larger sample size increases the chances of finding statistically significant results when there is a genuine effect, making the findings more trustworthy and robust.

## Can a non-significant p-value indicate that there is no effect or difference in the data?

No, a non-significant p-value does not necessarily indicate that there is no effect or difference in the data. It means that the observed data do not provide strong enough evidence to reject the null hypothesis.

There could still be a real effect or difference, but it might be smaller or more variable than the study was able to detect.

Other factors like sample size, study design, and measurement precision can influence the p-value. It’s important to consider the entire body of evidence and not rely solely on p-values when interpreting research findings.

## Can P values be exactly zero?

While a p-value can be extremely small, it cannot technically be absolute zero. When a p-value is reported as p = 0.000, the actual p-value is too small for the software to display. This is often interpreted as strong evidence against the null hypothesis. For p values less than 0.001, report as p < .001

## Further Information

- P-values and significance tests (Kahn Academy)
- Hypothesis testing and p-values (Kahn Academy)
- Wasserstein, R. L., Schirm, A. L., & Lazar, N. A. (2019). Moving to a world beyond “ p “< 0.05”.
- Criticism of using the “ p “< 0.05”.
- Publication manual of the American Psychological Association
- Statistics for Psychology Book Download

Bland, J. M., & Altman, D. G. (1994). One and two sided tests of significance: Authors’ reply. BMJ: British Medical Journal , 309 (6958), 874.

Goodman, S. N., & Royall, R. (1988). Evidence and scientific research. American Journal of Public Health , 78 (12), 1568-1574.

Goodman, S. (2008, July). A dirty dozen: twelve p-value misconceptions . In Seminars in hematology (Vol. 45, No. 3, pp. 135-140). WB Saunders.

Lang, J. M., Rothman, K. J., & Cann, C. I. (1998). That confounded P-value. Epidemiology (Cambridge, Mass.) , 9 (1), 7-8.

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## What Is P-Value?

Understanding p-value.

- P-Value in Hypothesis Testing

## The Bottom Line

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## P-Value: What It Is, How to Calculate It, and Why It Matters

Yarilet Perez is an experienced multimedia journalist and fact-checker with a Master of Science in Journalism. She has worked in multiple cities covering breaking news, politics, education, and more. Her expertise is in personal finance and investing, and real estate.

In statistics, a p-value is defined as a number that indicates how likely you are to obtain a value that is at least equal to or more than the actual observation if the null hypothesis is correct.

The p-value serves as an alternative to rejection points to provide the smallest level of significance at which the null hypothesis would be rejected. A smaller p-value means stronger evidence in favor of the alternative hypothesis.

P-value is often used to promote credibility for studies or reports by government agencies. For example, the U.S. Census Bureau stipulates that any analysis with a p-value greater than 0.10 must be accompanied by a statement that the difference is not statistically different from zero. The Census Bureau also has standards in place stipulating which p-values are acceptable for various publications.

## Key Takeaways

- A p-value is a statistical measurement used to validate a hypothesis against observed data.
- A p-value measures the probability of obtaining the observed results, assuming that the null hypothesis is true.
- The lower the p-value, the greater the statistical significance of the observed difference.
- A p-value of 0.05 or lower is generally considered statistically significant.
- P-value can serve as an alternative to—or in addition to—preselected confidence levels for hypothesis testing.

Jessica Olah / Investopedia

P-values are usually found using p-value tables or spreadsheets/statistical software. These calculations are based on the assumed or known probability distribution of the specific statistic tested. The sample size, which determines the reliability of the observed data, directly influences the accuracy of the p-value calculation. he p-value approach to hypothesis testing uses the calculated he p-value approach to hypothesis testing uses the calculated P-values are calculated from the deviation between the observed value and a chosen reference value, given the probability distribution of the statistic, with a greater difference between the two values corresponding to a lower p-value.

Mathematically, the p-value is calculated using integral calculus from the area under the probability distribution curve for all values of statistics that are at least as far from the reference value as the observed value is, relative to the total area under the probability distribution curve. Standard deviations, which quantify the dispersion of data points from the mean, are instrumental in this calculation.

The calculation for a p-value varies based on the type of test performed. The three test types describe the location on the probability distribution curve: lower-tailed test, upper-tailed test, or two-tailed test . In each case, the degrees of freedom play a crucial role in determining the shape of the distribution and thus, the calculation of the p-value.

In a nutshell, the greater the difference between two observed values, the less likely it is that the difference is due to simple random chance, and this is reflected by a lower p-value.

## The P-Value Approach to Hypothesis Testing

The p-value approach to hypothesis testing uses the calculated probability to determine whether there is evidence to reject the null hypothesis. This determination relies heavily on the test statistic, which summarizes the information from the sample relevant to the hypothesis being tested. The null hypothesis, also known as the conjecture, is the initial claim about a population (or data-generating process). The alternative hypothesis states whether the population parameter differs from the value of the population parameter stated in the conjecture.

In practice, the significance level is stated in advance to determine how small the p-value must be to reject the null hypothesis. Because different researchers use different levels of significance when examining a question, a reader may sometimes have difficulty comparing results from two different tests. P-values provide a solution to this problem.

Even a low p-value is not necessarily proof of statistical significance, since there is still a possibility that the observed data are the result of chance. Only repeated experiments or studies can confirm if a relationship is statistically significant.

For example, suppose a study comparing returns from two particular assets was undertaken by different researchers who used the same data but different significance levels. The researchers might come to opposite conclusions regarding whether the assets differ.

If one researcher used a confidence level of 90% and the other required a confidence level of 95% to reject the null hypothesis, and if the p-value of the observed difference between the two returns was 0.08 (corresponding to a confidence level of 92%), then the first researcher would find that the two assets have a difference that is statistically significant , while the second would find no statistically significant difference between the returns.

To avoid this problem, the researchers could report the p-value of the hypothesis test and allow readers to interpret the statistical significance themselves. This is called a p-value approach to hypothesis testing. Independent observers could note the p-value and decide for themselves whether that represents a statistically significant difference or not.

## Example of P-Value

An investor claims that their investment portfolio’s performance is equivalent to that of the Standard & Poor’s (S&P) 500 Index . To determine this, the investor conducts a two-tailed test.

The null hypothesis states that the portfolio’s returns are equivalent to the S&P 500’s returns over a specified period, while the alternative hypothesis states that the portfolio’s returns and the S&P 500’s returns are not equivalent—if the investor conducted a one-tailed test , the alternative hypothesis would state that the portfolio’s returns are either less than or greater than the S&P 500’s returns.

The p-value hypothesis test does not necessarily make use of a preselected confidence level at which the investor should reset the null hypothesis that the returns are equivalent. Instead, it provides a measure of how much evidence there is to reject the null hypothesis. The smaller the p-value, the greater the evidence against the null hypothesis.

Thus, if the investor finds that the p-value is 0.001, there is strong evidence against the null hypothesis, and the investor can confidently conclude that the portfolio’s returns and the S&P 500’s returns are not equivalent.

Although this does not provide an exact threshold as to when the investor should accept or reject the null hypothesis, it does have another very practical advantage. P-value hypothesis testing offers a direct way to compare the relative confidence that the investor can have when choosing among multiple different types of investments or portfolios relative to a benchmark such as the S&P 500.

For example, for two portfolios, A and B, whose performance differs from the S&P 500 with p-values of 0.10 and 0.01, respectively, the investor can be much more confident that portfolio B, with a lower p-value, will actually show consistently different results.

## Is a 0.05 P-Value Significant?

A p-value less than 0.05 is typically considered to be statistically significant, in which case the null hypothesis should be rejected. A p-value greater than 0.05 means that deviation from the null hypothesis is not statistically significant, and the null hypothesis is not rejected.

## What Does a P-Value of 0.001 Mean?

A p-value of 0.001 indicates that if the null hypothesis tested were indeed true, then there would be a one-in-1,000 chance of observing results at least as extreme. This leads the observer to reject the null hypothesis because either a highly rare data result has been observed or the null hypothesis is incorrect.

## How Can You Use P-Value to Compare 2 Different Results of a Hypothesis Test?

If you have two different results, one with a p-value of 0.04 and one with a p-value of 0.06, the result with a p-value of 0.04 will be considered more statistically significant than the p-value of 0.06. Beyond this simplified example, you could compare a 0.04 p-value to a 0.001 p-value. Both are statistically significant, but the 0.001 example provides an even stronger case against the null hypothesis than the 0.04.

The p-value is used to measure the significance of observational data. When researchers identify an apparent relationship between two variables, there is always a possibility that this correlation might be a coincidence. A p-value calculation helps determine if the observed relationship could arise as a result of chance.

U.S. Census Bureau. “ Statistical Quality Standard E1: Analyzing Data .”

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S.3.2 hypothesis testing (p-value approach).

The P -value approach involves determining "likely" or "unlikely" by determining the probability — assuming the null hypothesis was true — of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed. If the P -value is small, say less than (or equal to) \(\alpha\), then it is "unlikely." And, if the P -value is large, say more than \(\alpha\), then it is "likely."

If the P -value is less than (or equal to) \(\alpha\), then the null hypothesis is rejected in favor of the alternative hypothesis. And, if the P -value is greater than \(\alpha\), then the null hypothesis is not rejected.

Specifically, the four steps involved in using the P -value approach to conducting any hypothesis test are:

- Specify the null and alternative hypotheses.
- Using the sample data and assuming the null hypothesis is true, calculate the value of the test statistic. Again, to conduct the hypothesis test for the population mean μ , we use the t -statistic \(t^*=\frac{\bar{x}-\mu}{s/\sqrt{n}}\) which follows a t -distribution with n - 1 degrees of freedom.
- Using the known distribution of the test statistic, calculate the P -value : "If the null hypothesis is true, what is the probability that we'd observe a more extreme test statistic in the direction of the alternative hypothesis than we did?" (Note how this question is equivalent to the question answered in criminal trials: "If the defendant is innocent, what is the chance that we'd observe such extreme criminal evidence?")
- Set the significance level, \(\alpha\), the probability of making a Type I error to be small — 0.01, 0.05, or 0.10. Compare the P -value to \(\alpha\). If the P -value is less than (or equal to) \(\alpha\), reject the null hypothesis in favor of the alternative hypothesis. If the P -value is greater than \(\alpha\), do not reject the null hypothesis.

## Example S.3.2.1

Mean gpa section .

In our example concerning the mean grade point average, suppose that our random sample of n = 15 students majoring in mathematics yields a test statistic t * equaling 2.5. Since n = 15, our test statistic t * has n - 1 = 14 degrees of freedom. Also, suppose we set our significance level α at 0.05 so that we have only a 5% chance of making a Type I error.

## Right Tailed

The P -value for conducting the right-tailed test H 0 : μ = 3 versus H A : μ > 3 is the probability that we would observe a test statistic greater than t * = 2.5 if the population mean \(\mu\) really were 3. Recall that probability equals the area under the probability curve. The P -value is therefore the area under a t n - 1 = t 14 curve and to the right of the test statistic t * = 2.5. It can be shown using statistical software that the P -value is 0.0127. The graph depicts this visually.

The P -value, 0.0127, tells us it is "unlikely" that we would observe such an extreme test statistic t * in the direction of H A if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the P -value, 0.0127, is less than \(\alpha\) = 0.05, we reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ > 3.

Note that we would not reject H 0 : μ = 3 in favor of H A : μ > 3 if we lowered our willingness to make a Type I error to \(\alpha\) = 0.01 instead, as the P -value, 0.0127, is then greater than \(\alpha\) = 0.01.

## Left Tailed

In our example concerning the mean grade point average, suppose that our random sample of n = 15 students majoring in mathematics yields a test statistic t * instead of equaling -2.5. The P -value for conducting the left-tailed test H 0 : μ = 3 versus H A : μ < 3 is the probability that we would observe a test statistic less than t * = -2.5 if the population mean μ really were 3. The P -value is therefore the area under a t n - 1 = t 14 curve and to the left of the test statistic t* = -2.5. It can be shown using statistical software that the P -value is 0.0127. The graph depicts this visually.

The P -value, 0.0127, tells us it is "unlikely" that we would observe such an extreme test statistic t * in the direction of H A if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the P -value, 0.0127, is less than α = 0.05, we reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ < 3.

Note that we would not reject H 0 : μ = 3 in favor of H A : μ < 3 if we lowered our willingness to make a Type I error to α = 0.01 instead, as the P -value, 0.0127, is then greater than \(\alpha\) = 0.01.

In our example concerning the mean grade point average, suppose again that our random sample of n = 15 students majoring in mathematics yields a test statistic t * instead of equaling -2.5. The P -value for conducting the two-tailed test H 0 : μ = 3 versus H A : μ ≠ 3 is the probability that we would observe a test statistic less than -2.5 or greater than 2.5 if the population mean μ really was 3. That is, the two-tailed test requires taking into account the possibility that the test statistic could fall into either tail (hence the name "two-tailed" test). The P -value is, therefore, the area under a t n - 1 = t 14 curve to the left of -2.5 and to the right of 2.5. It can be shown using statistical software that the P -value is 0.0127 + 0.0127, or 0.0254. The graph depicts this visually.

Note that the P -value for a two-tailed test is always two times the P -value for either of the one-tailed tests. The P -value, 0.0254, tells us it is "unlikely" that we would observe such an extreme test statistic t * in the direction of H A if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the P -value, 0.0254, is less than α = 0.05, we reject the null hypothesis H 0 : μ = 3 in favor of the alternative hypothesis H A : μ ≠ 3.

Note that we would not reject H 0 : μ = 3 in favor of H A : μ ≠ 3 if we lowered our willingness to make a Type I error to α = 0.01 instead, as the P -value, 0.0254, is then greater than \(\alpha\) = 0.01.

Now that we have reviewed the critical value and P -value approach procedures for each of the three possible hypotheses, let's look at three new examples — one of a right-tailed test, one of a left-tailed test, and one of a two-tailed test.

The good news is that, whenever possible, we will take advantage of the test statistics and P -values reported in statistical software, such as Minitab, to conduct our hypothesis tests in this course.

Statistics Made Easy

## An Explanation of P-Values and Statistical Significance

In statistics, p-values are commonly used in hypothesis testing for t-tests, chi-square tests, regression analysis, ANOVAs, and a variety of other statistical methods.

Despite being so common, people often interpret p-values incorrectly, which can lead to errors when interpreting the findings from an analysis or a study.

This post explains how to understand and interpret p-values in a clear, practical way.

## Hypothesis Testing

To understand p-values, we first need to understand the concept of hypothesis testing .

A hypothesis test is a formal statistical test we use to reject or fail to reject some hypothesis. For example, we may hypothesize that a new drug, method, or procedure provides some benefit over a current drug, method, or procedure.

To test this, we can conduct a hypothesis test where we use a null and alternative hypothesis:

Null hypothesis – There is no effect or difference between the new method and the old method.

Alternative hypothesis – There does exist some effect or difference between the new method and the old method.

A p-value indicates how believable the null hypothesis is, given the sample data. Specifically, assuming the null hypothesis is true, the p-value tells us the probability of obtaining an effect at least as large as the one we actually observed in the sample data.

If the p-value of a hypothesis test is sufficiently low, we can reject the null hypothesis. Specifically, when we conduct a hypothesis test, we must choose a significance level at the outset. Common choices for significance levels are 0.01, 0.05, and 0.10.

If the p-values is less than our significance level, then we can reject the null hypothesis.

Otherwise, if the p-value is equal to or greater than our significance level, then we fail to reject the null hypothesis.

## How to Interpret a P-Value

The textbook definition of a p-value is:

A p-value is the probability of observing a sample statistic that is at least as extreme as your sample statistic, given that the null hypothesis is true.

For example, suppose a factory claims that they produce tires that have a mean weight of 200 pounds. An auditor hypothesizes that the true mean weight of tires produced at this factory is different from 200 pounds so he runs a hypothesis test and finds that the p-value of the test is 0.04. Here is how to interpret this p-value:

If the factory does indeed produce tires that have a mean weight of 200 pounds, then 4% of all audits will obtain the effect observed in the sample, or larger, because of random sample error. This tells us that obtaining the sample data that the auditor did would be pretty rare if indeed the factory produced tires that have a mean weight of 200 pounds.

Depending on the significance level used in this hypothesis test, the auditor would likely reject the null hypothesis that the true mean weight of tires produced at this factory is indeed 200 pounds. The sample data that he obtained from the audit is not very consistent with the null hypothesis.

## How Not to Interpret a P-Value

The biggest misconception about p-values is that they are equivalent to the probability of making a mistake by rejecting a true null hypothesis (known as a Type I error).

There are two primary reasons that p-values can’t be the error rate:

1. P-values are calculated based on the assumption that the null hypothesis is true and that the difference between the sample data and the null hypothesis is simple caused by random chance. Thus, p-values can’t tell you the probability that the null is true or false since it is 100% true based on the perspective of the calculations.

2. Although a low p-value indicates that your sample data are unlikely assuming the null is true, a p-value still can’t tell you which of the following cases is more likely:

- The null is false
- The null is true but you obtained an odd sample

In regards to the previous example, here is a correct and incorrect way to interpret the p-value:

- Correct Interpretation: Assuming the factory does produce tires with a mean weight of 200 pounds, you would obtain the observed difference that you did obtain in your sample or a more extreme difference in 4% of audits due to random sampling error.
- Incorrect Interpretation: If you reject the null hypothesis, there is a 4% chance that you are making a mistake.

## Examples of Interpreting P-Values

The following examples illustrate correct ways to interpret p-values in the context of hypothesis testing.

A phone company claims that 90% of its customers are satisfied with their service. To test this claim, an independent researcher gathered a simple random sample of 200 customers and asked them if they are satisfied with their service, to which 85% responded yes. The p-value associated with this sample data turned out to be 0.018.

Correct interpretation of p-value: Assuming that 90% of the customers actually are satisfied with their service, the researcher would obtain the observed difference that he did obtain in his sample or a more extreme difference in 1.8% of audits due to random sampling error.

A company invents a new battery for phones. The company claims that this new battery will work for at least 10 minutes longer than the old battery. To test this claim, a researcher takes a simple random sample of 80 new batteries and 80 old batteries. The new batteries run for an average of 120 minutes with a standard deviation of 12 minutes and the old batteries run for an average of 115 minutes with a standard deviation of 15 minutes. The p-value that results from the test for a difference in population means is 0.011.

Correct interpretation of p-value: Assuming that the new battery works for the same amount of time or less than the old battery, the researcher would obtain the observed difference or a more extreme difference in 1.1% of studies due to random sampling error.

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## StatPearls [Internet].

Hypothesis testing, p values, confidence intervals, and significance.

Jacob Shreffler ; Martin R. Huecker .

## Affiliations

Last Update: March 13, 2023 .

- Definition/Introduction

Medical providers often rely on evidence-based medicine to guide decision-making in practice. Often a research hypothesis is tested with results provided, typically with p values, confidence intervals, or both. Additionally, statistical or research significance is estimated or determined by the investigators. Unfortunately, healthcare providers may have different comfort levels in interpreting these findings, which may affect the adequate application of the data.

- Issues of Concern

Without a foundational understanding of hypothesis testing, p values, confidence intervals, and the difference between statistical and clinical significance, it may affect healthcare providers' ability to make clinical decisions without relying purely on the research investigators deemed level of significance. Therefore, an overview of these concepts is provided to allow medical professionals to use their expertise to determine if results are reported sufficiently and if the study outcomes are clinically appropriate to be applied in healthcare practice.

Hypothesis Testing

Investigators conducting studies need research questions and hypotheses to guide analyses. Starting with broad research questions (RQs), investigators then identify a gap in current clinical practice or research. Any research problem or statement is grounded in a better understanding of relationships between two or more variables. For this article, we will use the following research question example:

Research Question: Is Drug 23 an effective treatment for Disease A?

Research questions do not directly imply specific guesses or predictions; we must formulate research hypotheses. A hypothesis is a predetermined declaration regarding the research question in which the investigator(s) makes a precise, educated guess about a study outcome. This is sometimes called the alternative hypothesis and ultimately allows the researcher to take a stance based on experience or insight from medical literature. An example of a hypothesis is below.

Research Hypothesis: Drug 23 will significantly reduce symptoms associated with Disease A compared to Drug 22.

The null hypothesis states that there is no statistical difference between groups based on the stated research hypothesis.

Researchers should be aware of journal recommendations when considering how to report p values, and manuscripts should remain internally consistent.

Regarding p values, as the number of individuals enrolled in a study (the sample size) increases, the likelihood of finding a statistically significant effect increases. With very large sample sizes, the p-value can be very low significant differences in the reduction of symptoms for Disease A between Drug 23 and Drug 22. The null hypothesis is deemed true until a study presents significant data to support rejecting the null hypothesis. Based on the results, the investigators will either reject the null hypothesis (if they found significant differences or associations) or fail to reject the null hypothesis (they could not provide proof that there were significant differences or associations).

To test a hypothesis, researchers obtain data on a representative sample to determine whether to reject or fail to reject a null hypothesis. In most research studies, it is not feasible to obtain data for an entire population. Using a sampling procedure allows for statistical inference, though this involves a certain possibility of error. [1] When determining whether to reject or fail to reject the null hypothesis, mistakes can be made: Type I and Type II errors. Though it is impossible to ensure that these errors have not occurred, researchers should limit the possibilities of these faults. [2]

Significance

Significance is a term to describe the substantive importance of medical research. Statistical significance is the likelihood of results due to chance. [3] Healthcare providers should always delineate statistical significance from clinical significance, a common error when reviewing biomedical research. [4] When conceptualizing findings reported as either significant or not significant, healthcare providers should not simply accept researchers' results or conclusions without considering the clinical significance. Healthcare professionals should consider the clinical importance of findings and understand both p values and confidence intervals so they do not have to rely on the researchers to determine the level of significance. [5] One criterion often used to determine statistical significance is the utilization of p values.

P values are used in research to determine whether the sample estimate is significantly different from a hypothesized value. The p-value is the probability that the observed effect within the study would have occurred by chance if, in reality, there was no true effect. Conventionally, data yielding a p<0.05 or p<0.01 is considered statistically significant. While some have debated that the 0.05 level should be lowered, it is still universally practiced. [6] Hypothesis testing allows us to determine the size of the effect.

An example of findings reported with p values are below:

Statement: Drug 23 reduced patients' symptoms compared to Drug 22. Patients who received Drug 23 (n=100) were 2.1 times less likely than patients who received Drug 22 (n = 100) to experience symptoms of Disease A, p<0.05.

Statement:Individuals who were prescribed Drug 23 experienced fewer symptoms (M = 1.3, SD = 0.7) compared to individuals who were prescribed Drug 22 (M = 5.3, SD = 1.9). This finding was statistically significant, p= 0.02.

For either statement, if the threshold had been set at 0.05, the null hypothesis (that there was no relationship) should be rejected, and we should conclude significant differences. Noticeably, as can be seen in the two statements above, some researchers will report findings with < or > and others will provide an exact p-value (0.000001) but never zero [6] . When examining research, readers should understand how p values are reported. The best practice is to report all p values for all variables within a study design, rather than only providing p values for variables with significant findings. [7] The inclusion of all p values provides evidence for study validity and limits suspicion for selective reporting/data mining.

While researchers have historically used p values, experts who find p values problematic encourage the use of confidence intervals. [8] . P-values alone do not allow us to understand the size or the extent of the differences or associations. [3] In March 2016, the American Statistical Association (ASA) released a statement on p values, noting that scientific decision-making and conclusions should not be based on a fixed p-value threshold (e.g., 0.05). They recommend focusing on the significance of results in the context of study design, quality of measurements, and validity of data. Ultimately, the ASA statement noted that in isolation, a p-value does not provide strong evidence. [9]

When conceptualizing clinical work, healthcare professionals should consider p values with a concurrent appraisal study design validity. For example, a p-value from a double-blinded randomized clinical trial (designed to minimize bias) should be weighted higher than one from a retrospective observational study [7] . The p-value debate has smoldered since the 1950s [10] , and replacement with confidence intervals has been suggested since the 1980s. [11]

Confidence Intervals

A confidence interval provides a range of values within given confidence (e.g., 95%), including the accurate value of the statistical constraint within a targeted population. [12] Most research uses a 95% CI, but investigators can set any level (e.g., 90% CI, 99% CI). [13] A CI provides a range with the lower bound and upper bound limits of a difference or association that would be plausible for a population. [14] Therefore, a CI of 95% indicates that if a study were to be carried out 100 times, the range would contain the true value in 95, [15] confidence intervals provide more evidence regarding the precision of an estimate compared to p-values. [6]

In consideration of the similar research example provided above, one could make the following statement with 95% CI:

Statement: Individuals who were prescribed Drug 23 had no symptoms after three days, which was significantly faster than those prescribed Drug 22; there was a mean difference between the two groups of days to the recovery of 4.2 days (95% CI: 1.9 – 7.8).

It is important to note that the width of the CI is affected by the standard error and the sample size; reducing a study sample number will result in less precision of the CI (increase the width). [14] A larger width indicates a smaller sample size or a larger variability. [16] A researcher would want to increase the precision of the CI. For example, a 95% CI of 1.43 – 1.47 is much more precise than the one provided in the example above. In research and clinical practice, CIs provide valuable information on whether the interval includes or excludes any clinically significant values. [14]

Null values are sometimes used for differences with CI (zero for differential comparisons and 1 for ratios). However, CIs provide more information than that. [15] Consider this example: A hospital implements a new protocol that reduced wait time for patients in the emergency department by an average of 25 minutes (95% CI: -2.5 – 41 minutes). Because the range crosses zero, implementing this protocol in different populations could result in longer wait times; however, the range is much higher on the positive side. Thus, while the p-value used to detect statistical significance for this may result in "not significant" findings, individuals should examine this range, consider the study design, and weigh whether or not it is still worth piloting in their workplace.

Similarly to p-values, 95% CIs cannot control for researchers' errors (e.g., study bias or improper data analysis). [14] In consideration of whether to report p-values or CIs, researchers should examine journal preferences. When in doubt, reporting both may be beneficial. [13] An example is below:

Reporting both: Individuals who were prescribed Drug 23 had no symptoms after three days, which was significantly faster than those prescribed Drug 22, p = 0.009. There was a mean difference between the two groups of days to the recovery of 4.2 days (95% CI: 1.9 – 7.8).

- Clinical Significance

Recall that clinical significance and statistical significance are two different concepts. Healthcare providers should remember that a study with statistically significant differences and large sample size may be of no interest to clinicians, whereas a study with smaller sample size and statistically non-significant results could impact clinical practice. [14] Additionally, as previously mentioned, a non-significant finding may reflect the study design itself rather than relationships between variables.

Healthcare providers using evidence-based medicine to inform practice should use clinical judgment to determine the practical importance of studies through careful evaluation of the design, sample size, power, likelihood of type I and type II errors, data analysis, and reporting of statistical findings (p values, 95% CI or both). [4] Interestingly, some experts have called for "statistically significant" or "not significant" to be excluded from work as statistical significance never has and will never be equivalent to clinical significance. [17]

The decision on what is clinically significant can be challenging, depending on the providers' experience and especially the severity of the disease. Providers should use their knowledge and experiences to determine the meaningfulness of study results and make inferences based not only on significant or insignificant results by researchers but through their understanding of study limitations and practical implications.

- Nursing, Allied Health, and Interprofessional Team Interventions

All physicians, nurses, pharmacists, and other healthcare professionals should strive to understand the concepts in this chapter. These individuals should maintain the ability to review and incorporate new literature for evidence-based and safe care.

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Disclosure: Jacob Shreffler declares no relevant financial relationships with ineligible companies.

Disclosure: Martin Huecker declares no relevant financial relationships with ineligible companies.

This book is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ), which permits others to distribute the work, provided that the article is not altered or used commercially. You are not required to obtain permission to distribute this article, provided that you credit the author and journal.

- Cite this Page Shreffler J, Huecker MR. Hypothesis Testing, P Values, Confidence Intervals, and Significance. [Updated 2023 Mar 13]. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2024 Jan-.

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- Review Clinical versus statistical significance: interpreting P values and confidence intervals related to measures of association to guide decision making. [J Pharm Pract. 2010] Review Clinical versus statistical significance: interpreting P values and confidence intervals related to measures of association to guide decision making. Ferrill MJ, Brown DA, Kyle JA. J Pharm Pract. 2010 Aug; 23(4):344-51. Epub 2010 Apr 13.
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## p-value Calculator

What is p-value, how do i calculate p-value from test statistic, how to interpret p-value, how to use the p-value calculator to find p-value from test statistic, how do i find p-value from z-score, how do i find p-value from t, p-value from chi-square score (χ² score), p-value from f-score.

Welcome to our p-value calculator! You will never again have to wonder how to find the p-value, as here you can determine the one-sided and two-sided p-values from test statistics, following all the most popular distributions: normal, t-Student, chi-squared, and Snedecor's F.

P-values appear all over science, yet many people find the concept a bit intimidating. Don't worry – in this article, we will explain not only what the p-value is but also how to interpret p-values correctly . Have you ever been curious about how to calculate the p-value by hand? We provide you with all the necessary formulae as well!

🙋 If you want to revise some basics from statistics, our normal distribution calculator is an excellent place to start.

Formally, the p-value is the probability that the test statistic will produce values at least as extreme as the value it produced for your sample . It is crucial to remember that this probability is calculated under the assumption that the null hypothesis H 0 is true !

More intuitively, p-value answers the question:

Assuming that I live in a world where the null hypothesis holds, how probable is it that, for another sample, the test I'm performing will generate a value at least as extreme as the one I observed for the sample I already have?

It is the alternative hypothesis that determines what "extreme" actually means , so the p-value depends on the alternative hypothesis that you state: left-tailed, right-tailed, or two-tailed. In the formulas below, S stands for a test statistic, x for the value it produced for a given sample, and Pr(event | H 0 ) is the probability of an event, calculated under the assumption that H 0 is true:

Left-tailed test: p-value = Pr(S ≤ x | H 0 )

Right-tailed test: p-value = Pr(S ≥ x | H 0 )

Two-tailed test:

p-value = 2 × min{Pr(S ≤ x | H 0 ), Pr(S ≥ x | H 0 )}

(By min{a,b} , we denote the smaller number out of a and b .)

If the distribution of the test statistic under H 0 is symmetric about 0 , then: p-value = 2 × Pr(S ≥ |x| | H 0 )

or, equivalently: p-value = 2 × Pr(S ≤ -|x| | H 0 )

As a picture is worth a thousand words, let us illustrate these definitions. Here, we use the fact that the probability can be neatly depicted as the area under the density curve for a given distribution. We give two sets of pictures: one for a symmetric distribution and the other for a skewed (non-symmetric) distribution.

- Symmetric case: normal distribution:

- Non-symmetric case: chi-squared distribution:

In the last picture (two-tailed p-value for skewed distribution), the area of the left-hand side is equal to the area of the right-hand side.

To determine the p-value, you need to know the distribution of your test statistic under the assumption that the null hypothesis is true . Then, with the help of the cumulative distribution function ( cdf ) of this distribution, we can express the probability of the test statistics being at least as extreme as its value x for the sample:

Left-tailed test:

p-value = cdf(x) .

Right-tailed test:

p-value = 1 - cdf(x) .

p-value = 2 × min{cdf(x) , 1 - cdf(x)} .

If the distribution of the test statistic under H 0 is symmetric about 0 , then a two-sided p-value can be simplified to p-value = 2 × cdf(-|x|) , or, equivalently, as p-value = 2 - 2 × cdf(|x|) .

The probability distributions that are most widespread in hypothesis testing tend to have complicated cdf formulae, and finding the p-value by hand may not be possible. You'll likely need to resort to a computer or to a statistical table, where people have gathered approximate cdf values.

Well, you now know how to calculate the p-value, but… why do you need to calculate this number in the first place? In hypothesis testing, the p-value approach is an alternative to the critical value approach . Recall that the latter requires researchers to pre-set the significance level, α, which is the probability of rejecting the null hypothesis when it is true (so of type I error ). Once you have your p-value, you just need to compare it with any given α to quickly decide whether or not to reject the null hypothesis at that significance level, α. For details, check the next section, where we explain how to interpret p-values.

As we have mentioned above, the p-value is the answer to the following question:

What does that mean for you? Well, you've got two options:

- A high p-value means that your data is highly compatible with the null hypothesis; and
- A small p-value provides evidence against the null hypothesis , as it means that your result would be very improbable if the null hypothesis were true.

However, it may happen that the null hypothesis is true, but your sample is highly unusual! For example, imagine we studied the effect of a new drug and got a p-value of 0.03 . This means that in 3% of similar studies, random chance alone would still be able to produce the value of the test statistic that we obtained, or a value even more extreme, even if the drug had no effect at all!

The question "what is p-value" can also be answered as follows: p-value is the smallest level of significance at which the null hypothesis would be rejected. So, if you now want to make a decision on the null hypothesis at some significance level α , just compare your p-value with α :

- If p-value ≤ α , then you reject the null hypothesis and accept the alternative hypothesis; and
- If p-value ≥ α , then you don't have enough evidence to reject the null hypothesis.

Obviously, the fate of the null hypothesis depends on α . For instance, if the p-value was 0.03 , we would reject the null hypothesis at a significance level of 0.05 , but not at a level of 0.01 . That's why the significance level should be stated in advance and not adapted conveniently after the p-value has been established! A significance level of 0.05 is the most common value, but there's nothing magical about it. Here, you can see what too strong a faith in the 0.05 threshold can lead to. It's always best to report the p-value, and allow the reader to make their own conclusions.

Also, bear in mind that subject area expertise (and common reason) is crucial. Otherwise, mindlessly applying statistical principles, you can easily arrive at statistically significant, despite the conclusion being 100% untrue.

As our p-value calculator is here at your service, you no longer need to wonder how to find p-value from all those complicated test statistics! Here are the steps you need to follow:

Pick the alternative hypothesis : two-tailed, right-tailed, or left-tailed.

Tell us the distribution of your test statistic under the null hypothesis: is it N(0,1), t-Student, chi-squared, or Snedecor's F? If you are unsure, check the sections below, as they are devoted to these distributions.

If needed, specify the degrees of freedom of the test statistic's distribution.

Enter the value of test statistic computed for your data sample.

Our calculator determines the p-value from the test statistic and provides the decision to be made about the null hypothesis. The standard significance level is 0.05 by default.

Go to the advanced mode if you need to increase the precision with which the calculations are performed or change the significance level .

In terms of the cumulative distribution function (cdf) of the standard normal distribution, which is traditionally denoted by Φ , the p-value is given by the following formulae:

Left-tailed z-test:

p-value = Φ(Z score )

Right-tailed z-test:

p-value = 1 - Φ(Z score )

Two-tailed z-test:

p-value = 2 × Φ(−|Z score |)

p-value = 2 - 2 × Φ(|Z score |)

🙋 To learn more about Z-tests, head to Omni's Z-test calculator .

We use the Z-score if the test statistic approximately follows the standard normal distribution N(0,1) . Thanks to the central limit theorem, you can count on the approximation if you have a large sample (say at least 50 data points) and treat your distribution as normal.

A Z-test most often refers to testing the population mean , or the difference between two population means, in particular between two proportions. You can also find Z-tests in maximum likelihood estimations.

The p-value from the t-score is given by the following formulae, in which cdf t,d stands for the cumulative distribution function of the t-Student distribution with d degrees of freedom:

Left-tailed t-test:

p-value = cdf t,d (t score )

Right-tailed t-test:

p-value = 1 - cdf t,d (t score )

Two-tailed t-test:

p-value = 2 × cdf t,d (−|t score |)

p-value = 2 - 2 × cdf t,d (|t score |)

Use the t-score option if your test statistic follows the t-Student distribution . This distribution has a shape similar to N(0,1) (bell-shaped and symmetric) but has heavier tails – the exact shape depends on the parameter called the degrees of freedom . If the number of degrees of freedom is large (>30), which generically happens for large samples, the t-Student distribution is practically indistinguishable from the normal distribution N(0,1).

The most common t-tests are those for population means with an unknown population standard deviation, or for the difference between means of two populations , with either equal or unequal yet unknown population standard deviations. There's also a t-test for paired (dependent) samples .

🙋 To get more insights into t-statistics, we recommend using our t-test calculator .

Use the χ²-score option when performing a test in which the test statistic follows the χ²-distribution .

This distribution arises if, for example, you take the sum of squared variables, each following the normal distribution N(0,1). Remember to check the number of degrees of freedom of the χ²-distribution of your test statistic!

How to find the p-value from chi-square-score ? You can do it with the help of the following formulae, in which cdf χ²,d denotes the cumulative distribution function of the χ²-distribution with d degrees of freedom:

Left-tailed χ²-test:

p-value = cdf χ²,d (χ² score )

Right-tailed χ²-test:

p-value = 1 - cdf χ²,d (χ² score )

Remember that χ²-tests for goodness-of-fit and independence are right-tailed tests! (see below)

Two-tailed χ²-test:

p-value = 2 × min{cdf χ²,d (χ² score ), 1 - cdf χ²,d (χ² score )}

(By min{a,b} , we denote the smaller of the numbers a and b .)

The most popular tests which lead to a χ²-score are the following:

Testing whether the variance of normally distributed data has some pre-determined value. In this case, the test statistic has the χ²-distribution with n - 1 degrees of freedom, where n is the sample size. This can be a one-tailed or two-tailed test .

Goodness-of-fit test checks whether the empirical (sample) distribution agrees with some expected probability distribution. In this case, the test statistic follows the χ²-distribution with k - 1 degrees of freedom, where k is the number of classes into which the sample is divided. This is a right-tailed test .

Independence test is used to determine if there is a statistically significant relationship between two variables. In this case, its test statistic is based on the contingency table and follows the χ²-distribution with (r - 1)(c - 1) degrees of freedom, where r is the number of rows, and c is the number of columns in this contingency table. This also is a right-tailed test .

Finally, the F-score option should be used when you perform a test in which the test statistic follows the F-distribution , also known as the Fisher–Snedecor distribution. The exact shape of an F-distribution depends on two degrees of freedom .

To see where those degrees of freedom come from, consider the independent random variables X and Y , which both follow the χ²-distributions with d 1 and d 2 degrees of freedom, respectively. In that case, the ratio (X/d 1 )/(Y/d 2 ) follows the F-distribution, with (d 1 , d 2 ) -degrees of freedom. For this reason, the two parameters d 1 and d 2 are also called the numerator and denominator degrees of freedom .

The p-value from F-score is given by the following formulae, where we let cdf F,d1,d2 denote the cumulative distribution function of the F-distribution, with (d 1 , d 2 ) -degrees of freedom:

Left-tailed F-test:

p-value = cdf F,d1,d2 (F score )

Right-tailed F-test:

p-value = 1 - cdf F,d1,d2 (F score )

Two-tailed F-test:

p-value = 2 × min{cdf F,d1,d2 (F score ), 1 - cdf F,d1,d2 (F score )}

Below we list the most important tests that produce F-scores. All of them are right-tailed tests .

A test for the equality of variances in two normally distributed populations . Its test statistic follows the F-distribution with (n - 1, m - 1) -degrees of freedom, where n and m are the respective sample sizes.

ANOVA is used to test the equality of means in three or more groups that come from normally distributed populations with equal variances. We arrive at the F-distribution with (k - 1, n - k) -degrees of freedom, where k is the number of groups, and n is the total sample size (in all groups together).

A test for overall significance of regression analysis . The test statistic has an F-distribution with (k - 1, n - k) -degrees of freedom, where n is the sample size, and k is the number of variables (including the intercept).

With the presence of the linear relationship having been established in your data sample with the above test, you can calculate the coefficient of determination, R 2 , which indicates the strength of this relationship . You can do it by hand or use our coefficient of determination calculator .

A test to compare two nested regression models . The test statistic follows the F-distribution with (k 2 - k 1 , n - k 2 ) -degrees of freedom, where k 1 and k 2 are the numbers of variables in the smaller and bigger models, respectively, and n is the sample size.

You may notice that the F-test of an overall significance is a particular form of the F-test for comparing two nested models: it tests whether our model does significantly better than the model with no predictors (i.e., the intercept-only model).

## Can p-value be negative?

No, the p-value cannot be negative. This is because probabilities cannot be negative, and the p-value is the probability of the test statistic satisfying certain conditions.

## What does a high p-value mean?

A high p-value means that under the null hypothesis, there's a high probability that for another sample, the test statistic will generate a value at least as extreme as the one observed in the sample you already have. A high p-value doesn't allow you to reject the null hypothesis.

## What does a low p-value mean?

A low p-value means that under the null hypothesis, there's little probability that for another sample, the test statistic will generate a value at least as extreme as the one observed for the sample you already have. A low p-value is evidence in favor of the alternative hypothesis – it allows you to reject the null hypothesis.

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## Table of Contents

What is p-value , p value vs alpha level, p values and critical values, how is p-value calculated, p-value in hypothesis testing, p-values and statistical significance, reporting p-values, our learners also ask, what is p-value in statistical hypothesis.

Few statistical estimates are as significant as the p-value. The p-value or probability value is a number, calculated from a statistical test , that describes how likely your results would have occurred if the null hypothesis were true. A P-value less than 0.5 is statistically significant, while a value higher than 0.5 indicates the null hypothesis is true; hence it is not statistically significant. So, what is P-Value exactly, and why is it so important?

In statistical hypothesis testing , P-Value or probability value can be defined as the measure of the probability that a real-valued test statistic is at least as extreme as the value actually obtained. P-value shows how likely it is that your set of observations could have occurred under the null hypothesis. P-Values are used in statistical hypothesis testing to determine whether to reject the null hypothesis. The smaller the p-value, the stronger the likelihood that you should reject the null hypothesis.

## Your Data Analytics Career is Around The Corner!

P-values are expressed as decimals and can be converted into percentage. For example, a p-value of 0.0237 is 2.37%, which means there's a 2.37% chance of your results being random or having happened by chance. The smaller the P-value, the more significant your results are.

In a hypothesis test, you can compare the p value from your test with the alpha level selected while running the test. Now, let’s try to understand what is P-Value vs Alpha level.

A P-value indicates the probability of getting an effect no less than that actually observed in the sample data.

An alpha level will tell you the probability of wrongly rejecting a true null hypothesis. The level is selected by the researcher and obtained by subtracting your confidence level from 100%. For instance, if you are 95% confident in your research, the alpha level will be 5% (0.05).

When you run the hypothesis test, if you get:

- A small p value (<=0.05), you should reject the null hypothesis
- A large p value (>0.05), you should not reject the null hypothesis

In addition to the P-value, you can use other values given by your test to determine if your null hypothesis is true.

For example, if you run an F-test to compare two variances in Excel, you will obtain a p-value, an f-critical value, and a f-value. Compare the f-value with f-critical value. If f-critical value is lower, you should reject the null hypothesis.

P-Values are usually calculated using p-value tables or spreadsheets, or calculated automatically using statistical software like R, SPSS, etc.

Depending on the test statistic and degrees of freedom (subtracting no. of independent variables from no. of observations) of your test, you can find out from the tables how frequently you can expect the test statistic to be under the null hypothesis.

How to calculate P-value depends on which statistical test you’re using to test your hypothesis.

- Every statistical test uses different assumptions and generates different statistics. Select the test method that best suits your data and matches the effect or relationship being tested.
- The number of independent variables included in your test determines how big or small the test statistic should be in order to generate the same p-value.

Regardless of what statistical test you are using, the p-value will always denote the same thing – how frequently you can expect to get a test statistic as extreme or even more extreme than the one given by your test.

In the P-Value approach to hypothesis testing, a calculated probability is used to decide if there’s evidence to reject the null hypothesis, also known as the conjecture. The conjecture is the initial claim about a data population, while the alternative hypothesis ascertains if the observed population parameter differs from the population parameter value according to the conjecture.

Effectively, the significance level is declared in advance to determine how small the P-value needs to be such that the null hypothesis is rejected. The levels of significance vary from one researcher to another; so it can get difficult for readers to compare results from two different tests. That is when P-value makes things easier.

Readers could interpret the statistical significance by referring to the reported P-value of the hypothesis test. This is known as the P-value approach to hypothesis testing. Using this, readers could decide for themselves whether the p value represents a statistically significant difference.

The level of statistical significance is usually represented as a P-value between 0 and 1. The smaller the p-value, the more likely it is that you would reject the null hypothesis.

- A P-Value < or = 0.05 is considered statistically significant. It denotes strong evidence against the null hypothesis, since there is below 5% probability of the null being correct. So, we reject the null hypothesis and accept the alternative hypothesis.
- But if P-Value is lower than your threshold of significance, though the null hypothesis can be rejected, it does not mean that there is 95% probability of the alternative hypothesis being true.
- A P-Value >0.05 is not statistically significant. It denotes strong evidence for the null hypothesis being true. Thus, we retain the null hypothesis and reject the alternative hypothesis. We cannot accept null hypothesis; we can only reject or not reject it.

A statistically significant result does not prove a research hypothesis to be correct. Instead, it provides support for or provides evidence for the hypothesis.

- You should report exact P-Values upto two or three decimal places.
- For P-values less than .001, report as p < .001.
- Do not use 0 before the decimal point as it cannot equal1. Write p = .001, and not p = 0.001
- Make sure p is always italicized and there is space on either side of the = sign.
- It is impossible to get P = .000, and should be written as p < .001

An investor says that the performance of their investment portfolio is equivalent to that of the Standard & Poor’s (S&P) 500 Index. He performs a two-tailed test to determine this.

The null hypothesis here says that the portfolio’s returns are equivalent to the returns of S&P 500, while the alternative hypothesis says that the returns of the portfolio and the returns of the S&P 500 are not equivalent.

The p-value hypothesis test gives a measure of how much evidence is present to reject the null hypothesis. The smaller the p value, the higher the evidence against null hypothesis.

Therefore, if the investor gets a P value of .001, it indicates strong evidence against null hypothesis. So he confidently deduces that the portfolio’s returns and the S&P 500’s returns are not equivalent.

## 1. What does P-value mean?

P-Value or probability value is a number that denotes the likelihood of your data having occurred under the null hypothesis of your statistical test.

## 2. What does p 0.05 mean?

A P-value less than 0.05 is deemed to be statistically significant, meaning the null hypothesis should be rejected in such a case. A P-Value greater than 0.05 is not considered to be statistically significant, meaning the null hypothesis should not be rejected.

## 3. What is P-value and how is it calculated?

The p-value or probability value is a number, calculated from a statistical test, that tells how likely it is that your results would have occurred under the null hypothesis of the test.

P-values are usually automatically calculated using statistical software. They can also be calculated using p-value tables for the relevant statistical test. P values are calculated based on the null distribution of the test statistic. In case the test statistic is far from the mean of the null distribution, the p-value obtained is small. It indicates that the test statistic is unlikely to have occurred under the null hypothesis.

## 4. What is p-value in research?

P values are used in hypothesis testing to help determine whether the null hypothesis should be rejected. It plays a major role when results of research are discussed. Hypothesis testing is a statistical methodology frequently used in medical and clinical research studies.

## 5. Why is the p-value significant?

Statistical significance is a term that researchers use to say that it is not likely that their observations could have occurred if the null hypothesis were true. The level of statistical significance is usually represented as a P-value or probability value between 0 and 1. The smaller the p-value, the more likely it is that you would reject the null hypothesis.

## 6. What is null hypothesis and what is p-value?

A null hypothesis is a kind of statistical hypothesis that suggests that there is no statistical significance in a set of given observations. It says there is no relationship between your variables.

P-value or probability value is a number, calculated from a statistical test, that tells how likely it is that your results would have occurred under the null hypothesis of the test.

P-Value is used to determine the significance of observational data. Whenever researchers notice an apparent relation between two variables, a P-Value calculation helps ascertain if the observed relationship happened as a result of chance. Learn more about statistical analysis and data analytics and fast track your career with our Professional Certificate Program In Data Analytics .

## Data Science & Business Analytics Courses Duration and Fees

Data Science & Business Analytics programs typically range from a few weeks to several months, with fees varying based on program and institution.

## Recommended Reads

Free eBook: Top Programming Languages For A Data Scientist

Inferential Statistics Explained: From Basics to Advanced!

A Comprehensive Look at Percentile in Statistics

Machine Learning Career Guide: A Playbook to Becoming a Machine Learning Engineer

The Difference Between Data Mining and Statistics

A Complete Guide on Hypothesis Testing in Statistics

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## AP®︎/College Statistics

Course: ap®︎/college statistics > unit 10.

- Idea behind hypothesis testing
- Examples of null and alternative hypotheses
- Writing null and alternative hypotheses

## P-values and significance tests

- Comparing P-values to different significance levels
- Estimating a P-value from a simulation
- Estimating P-values from simulations
- Using P-values to make conclusions

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## Video transcript

- Quality Improvement
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## Understanding Hypothesis Tests: Significance Levels (Alpha) and P values in Statistics

Topics: Hypothesis Testing , Statistics

What do significance levels and P values mean in hypothesis tests? What is statistical significance anyway? In this post, I’ll continue to focus on concepts and graphs to help you gain a more intuitive understanding of how hypothesis tests work in statistics.

To bring it to life, I’ll add the significance level and P value to the graph in my previous post in order to perform a graphical version of the 1 sample t-test. It’s easier to understand when you can see what statistical significance truly means!

Here’s where we left off in my last post . We want to determine whether our sample mean (330.6) indicates that this year's average energy cost is significantly different from last year’s average energy cost of $260.

The probability distribution plot above shows the distribution of sample means we’d obtain under the assumption that the null hypothesis is true (population mean = 260) and we repeatedly drew a large number of random samples.

I left you with a question: where do we draw the line for statistical significance on the graph? Now we'll add in the significance level and the P value, which are the decision-making tools we'll need.

We'll use these tools to test the following hypotheses:

- Null hypothesis: The population mean equals the hypothesized mean (260).
- Alternative hypothesis: The population mean differs from the hypothesized mean (260).

## What Is the Significance Level (Alpha)?

The significance level, also denoted as alpha or α, is the probability of rejecting the null hypothesis when it is true. For example, a significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference.

These types of definitions can be hard to understand because of their technical nature. A picture makes the concepts much easier to comprehend!

The significance level determines how far out from the null hypothesis value we'll draw that line on the graph. To graph a significance level of 0.05, we need to shade the 5% of the distribution that is furthest away from the null hypothesis.

In the graph above, the two shaded areas are equidistant from the null hypothesis value and each area has a probability of 0.025, for a total of 0.05. In statistics, we call these shaded areas the critical region for a two-tailed test. If the population mean is 260, we’d expect to obtain a sample mean that falls in the critical region 5% of the time. The critical region defines how far away our sample statistic must be from the null hypothesis value before we can say it is unusual enough to reject the null hypothesis.

Our sample mean (330.6) falls within the critical region, which indicates it is statistically significant at the 0.05 level.

We can also see if it is statistically significant using the other common significance level of 0.01.

The two shaded areas each have a probability of 0.005, which adds up to a total probability of 0.01. This time our sample mean does not fall within the critical region and we fail to reject the null hypothesis. This comparison shows why you need to choose your significance level before you begin your study. It protects you from choosing a significance level because it conveniently gives you significant results!

Thanks to the graph, we were able to determine that our results are statistically significant at the 0.05 level without using a P value. However, when you use the numeric output produced by statistical software , you’ll need to compare the P value to your significance level to make this determination.

## Ready for a demo of Minitab Statistical Software? Just ask!

## What Are P values?

P-values are the probability of obtaining an effect at least as extreme as the one in your sample data, assuming the truth of the null hypothesis.

This definition of P values, while technically correct, is a bit convoluted. It’s easier to understand with a graph!

To graph the P value for our example data set, we need to determine the distance between the sample mean and the null hypothesis value (330.6 - 260 = 70.6). Next, we can graph the probability of obtaining a sample mean that is at least as extreme in both tails of the distribution (260 +/- 70.6).

In the graph above, the two shaded areas each have a probability of 0.01556, for a total probability 0.03112. This probability represents the likelihood of obtaining a sample mean that is at least as extreme as our sample mean in both tails of the distribution if the population mean is 260. That’s our P value!

When a P value is less than or equal to the significance level, you reject the null hypothesis. If we take the P value for our example and compare it to the common significance levels, it matches the previous graphical results. The P value of 0.03112 is statistically significant at an alpha level of 0.05, but not at the 0.01 level.

If we stick to a significance level of 0.05, we can conclude that the average energy cost for the population is greater than 260.

A common mistake is to interpret the P-value as the probability that the null hypothesis is true. To understand why this interpretation is incorrect, please read my blog post How to Correctly Interpret P Values .

## Discussion about Statistically Significant Results

A hypothesis test evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data. A test result is statistically significant when the sample statistic is unusual enough relative to the null hypothesis that we can reject the null hypothesis for the entire population. “Unusual enough” in a hypothesis test is defined by:

- The assumption that the null hypothesis is true—the graphs are centered on the null hypothesis value.
- The significance level—how far out do we draw the line for the critical region?
- Our sample statistic—does it fall in the critical region?

Keep in mind that there is no magic significance level that distinguishes between the studies that have a true effect and those that don’t with 100% accuracy. The common alpha values of 0.05 and 0.01 are simply based on tradition. For a significance level of 0.05, expect to obtain sample means in the critical region 5% of the time when the null hypothesis is true . In these cases, you won’t know that the null hypothesis is true but you’ll reject it because the sample mean falls in the critical region. That’s why the significance level is also referred to as an error rate!

This type of error doesn’t imply that the experimenter did anything wrong or require any other unusual explanation. The graphs show that when the null hypothesis is true, it is possible to obtain these unusual sample means for no reason other than random sampling error. It’s just luck of the draw.

Significance levels and P values are important tools that help you quantify and control this type of error in a hypothesis test. Using these tools to decide when to reject the null hypothesis increases your chance of making the correct decision.

If you like this post, you might want to read the other posts in this series that use the same graphical framework:

- Previous: Why We Need to Use Hypothesis Tests
- Next: Confidence Intervals and Confidence Levels

If you'd like to see how I made these graphs, please read: How to Create a Graphical Version of the 1-sample t-Test .

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In Statistics, the researcher checks the significance of the observed result, which is known as test static . For this test, a hypothesis test is also utilized. The P-value or probability value concept is used everywhere in statistical analysis. It determines the statistical significance and the measure of significance testing. In this article, let us discuss its definition, formula, table, interpretation and how to use P-value to find the significance level etc. in detail.

Table of Contents:

## P-value Definition

The P-value is known as the probability value. It is defined as the probability of getting a result that is either the same or more extreme than the actual observations. The P-value is known as the level of marginal significance within the hypothesis testing that represents the probability of occurrence of the given event. The P-value is used as an alternative to the rejection point to provide the least significance at which the null hypothesis would be rejected. If the P-value is small, then there is stronger evidence in favour of the alternative hypothesis.

## P-value Table

The P-value table shows the hypothesis interpretations:

Generally, the level of statistical significance is often expressed in p-value and the range between 0 and 1. The smaller the p-value, the stronger the evidence and hence, the result should be statistically significant. Hence, the rejection of the null hypothesis is highly possible, as the p-value becomes smaller.

Let us look at an example to better comprehend the concept of P-value.

Let’s say a researcher flips a coin ten times with the null hypothesis that it is fair. The total number of heads is the test statistic, which is two-tailed. Assume the researcher notices alternating heads and tails on each flip (HTHTHTHTHT). As this is the predicted number of heads, the test statistic is 5 and the p-value is 1 (totally unexceptional).

Assume that the test statistic for this research was the “number of alternations” (i.e., the number of times H followed T or T followed H), which is two-tailed once again. This would result in a test statistic of 9, which is extremely high and has a p-value of 1/2 8 = 1/256, or roughly 0.0039. This would be regarded as extremely significant, much beyond the 0.05 level. These findings suggest that the data set is exceedingly improbable to have happened by random in terms of one test statistic, yet they do not imply that the coin is biased towards heads or tails.

The data have a high p-value according to the first test statistic, indicating that the number of heads observed is not impossible. The data have a low p-value according to the second test statistic, indicating that the pattern of flips observed is extremely unlikely. There is no “alternative hypothesis,” (therefore only the null hypothesis can be rejected), and such evidence could have a variety of explanations – the data could be falsified, or the coin could have been flipped by a magician who purposefully swapped outcomes.

This example shows that the p-value is entirely dependent on the test statistic used and that p-values can only be used to reject a null hypothesis, not to explore an alternate hypothesis.

## P-value Formula

We Know that P-value is a statistical measure, that helps to determine whether the hypothesis is correct or not. P-value is a number that lies between 0 and 1. The level of significance(α) is a predefined threshold that should be set by the researcher. It is generally fixed as 0.05. The formula for the calculation for P-value is

Step 1: Find out the test static Z is

P0 = assumed population proportion in the null hypothesis

N = sample size

Step 2: Look at the Z-table to find the corresponding level of P from the z value obtained.

## P-Value Example

An example to find the P-value is given here.

Question: A statistician wants to test the hypothesis H 0 : μ = 120 using the alternative hypothesis Hα: μ > 120 and assuming that α = 0.05. For that, he took the sample values as

n =40, σ = 32.17 and x̄ = 105.37. Determine the conclusion for this hypothesis?

We know that,

Now substitute the given values

Now, using the test static formula, we get

t = (105.37 – 120) / 5.0865

Therefore, t = -2.8762

Using the Z-Score table , we can find the value of P(t>-2.8762)

From the table, we get

P (t<-2.8762) = P(t>2.8762) = 0.003

If P(t>-2.8762) =1- 0.003 =0.997

P- value =0.997 > 0.05

Therefore, from the conclusion, if p>0.05, the null hypothesis is accepted or fails to reject.

Hence, the conclusion is “fails to reject H 0. ”

## Frequently Asked Questions on P-Value

What is meant by p-value.

The p-value is defined as the probability of obtaining the result at least as extreme as the observed result of a statistical hypothesis test, assuming that the null hypothesis is true.

## What does a smaller P-value represent?

The smaller the p-value, the greater the statistical significance of the observed difference, which results in the rejection of the null hypothesis in favour of alternative hypotheses.

## What does the p-value greater than 0.05 represent?

If the p-value is greater than 0.05, then the result is not statistically significant.

## Can the p-value be greater than 1?

P-value means probability value, which tells you the probability of achieving the result under a certain hypothesis. Since it is a probability, its value ranges between 0 and 1, and it cannot exceed 1.

## What does the p-value less than 0.05 represent?

If the p-value is less than 0.05, then the result is statistically significant, and hence we can reject the null hypothesis in favour of the alternative hypothesis.

## Introduction to Data Analysis

- What is Data Analysis?
- Data Analytics and its type
- How to Install Numpy on Windows?
- How to Install Pandas in Python?
- How to Install Matplotlib on python?
- How to Install Python Tensorflow in Windows?

## Data Analysis Libraries

- Pandas Tutorial
- NumPy Tutorial - Python Library
- Data Analysis with SciPy
- Introduction to TensorFlow

## Data Visulization Libraries

- Matplotlib Tutorial
- Python Seaborn Tutorial
- Plotly tutorial
- Introduction to Bokeh in Python

## Exploratory Data Analysis (EDA)

- Univariate, Bivariate and Multivariate data and its analysis
- Measures of Central Tendency in Statistics
- Measures of spread - Range, Variance, and Standard Deviation
- Interquartile Range and Quartile Deviation using NumPy and SciPy
- Anova Formula
- Skewness of Statistical Data
- How to Calculate Skewness and Kurtosis in Python?
- Difference Between Skewness and Kurtosis
- Histogram | Meaning, Example, Types and Steps to Draw
- Interpretations of Histogram
- Quantile Quantile plots
- What is Univariate, Bivariate & Multivariate Analysis in Data Visualisation?
- Using pandas crosstab to create a bar plot
- Exploring Correlation in Python
- Mathematics | Covariance and Correlation
- Factor Analysis | Data Analysis
- Data Mining - Cluster Analysis
- MANOVA Test in R Programming
- Python - Central Limit Theorem
- Probability Distribution Function
- Probability Density Estimation & Maximum Likelihood Estimation
- Exponential Distribution in R Programming - dexp(), pexp(), qexp(), and rexp() Functions
- Mathematics | Probability Distributions Set 4 (Binomial Distribution)
- Poisson Distribution | Definition, Formula, Table and Examples
- P-Value: Comprehensive Guide to Understand, Apply, and Interpret
- Z-Score in Statistics
- How to Calculate Point Estimates in R?
- Confidence Interval
- Chi-square test in Machine Learning

## Understanding Hypothesis Testing

Data preprocessing.

- ML | Data Preprocessing in Python
- ML | Overview of Data Cleaning
- ML | Handling Missing Values
- Detect and Remove the Outliers using Python

## Data Transformation

- Data Normalization Machine Learning
- Sampling distribution Using Python

## Time Series Data Analysis

- Data Mining - Time-Series, Symbolic and Biological Sequences Data
- Basic DateTime Operations in Python
- Time Series Analysis & Visualization in Python
- How to deal with missing values in a Timeseries in Python?
- How to calculate MOVING AVERAGE in a Pandas DataFrame?
- What is a trend in time series?
- How to Perform an Augmented Dickey-Fuller Test in R
- AutoCorrelation

## Case Studies and Projects

- Top 8 Free Dataset Sources to Use for Data Science Projects
- Step by Step Predictive Analysis - Machine Learning
- 6 Tips for Creating Effective Data Visualizations

Hypothesis testing involves formulating assumptions about population parameters based on sample statistics and rigorously evaluating these assumptions against empirical evidence. This article sheds light on the significance of hypothesis testing and the critical steps involved in the process.

## What is Hypothesis Testing?

Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. It evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data.

Example: You say an average height in the class is 30 or a boy is taller than a girl. All of these is an assumption that we are assuming, and we need some statistical way to prove these. We need some mathematical conclusion whatever we are assuming is true.

## Defining Hypotheses

## Key Terms of Hypothesis Testing

- P-value: The P value , or calculated probability, is the probability of finding the observed/extreme results when the null hypothesis(H0) of a study-given problem is true. If your P-value is less than the chosen significance level then you reject the null hypothesis i.e. accept that your sample claims to support the alternative hypothesis.
- Test Statistic: The test statistic is a numerical value calculated from sample data during a hypothesis test, used to determine whether to reject the null hypothesis. It is compared to a critical value or p-value to make decisions about the statistical significance of the observed results.
- Critical value : The critical value in statistics is a threshold or cutoff point used to determine whether to reject the null hypothesis in a hypothesis test.
- Degrees of freedom: Degrees of freedom are associated with the variability or freedom one has in estimating a parameter. The degrees of freedom are related to the sample size and determine the shape.

## Why do we use Hypothesis Testing?

Hypothesis testing is an important procedure in statistics. Hypothesis testing evaluates two mutually exclusive population statements to determine which statement is most supported by sample data. When we say that the findings are statistically significant, thanks to hypothesis testing.

## One-Tailed and Two-Tailed Test

One tailed test focuses on one direction, either greater than or less than a specified value. We use a one-tailed test when there is a clear directional expectation based on prior knowledge or theory. The critical region is located on only one side of the distribution curve. If the sample falls into this critical region, the null hypothesis is rejected in favor of the alternative hypothesis.

## One-Tailed Test

There are two types of one-tailed test:

## Two-Tailed Test

A two-tailed test considers both directions, greater than and less than a specified value.We use a two-tailed test when there is no specific directional expectation, and want to detect any significant difference.

## What are Type 1 and Type 2 errors in Hypothesis Testing?

In hypothesis testing, Type I and Type II errors are two possible errors that researchers can make when drawing conclusions about a population based on a sample of data. These errors are associated with the decisions made regarding the null hypothesis and the alternative hypothesis.

## How does Hypothesis Testing work?

Step 1: define null and alternative hypothesis.

We first identify the problem about which we want to make an assumption keeping in mind that our assumption should be contradictory to one another, assuming Normally distributed data.

## Step 2 – Choose significance level

## Step 3 – Collect and Analyze data.

Gather relevant data through observation or experimentation. Analyze the data using appropriate statistical methods to obtain a test statistic.

## Step 4-Calculate Test Statistic

The data for the tests are evaluated in this step we look for various scores based on the characteristics of data. The choice of the test statistic depends on the type of hypothesis test being conducted.

There are various hypothesis tests, each appropriate for various goal to calculate our test. This could be a Z-test , Chi-square , T-test , and so on.

- Z-test : If population means and standard deviations are known. Z-statistic is commonly used.
- t-test : If population standard deviations are unknown. and sample size is small than t-test statistic is more appropriate.
- Chi-square test : Chi-square test is used for categorical data or for testing independence in contingency tables
- F-test : F-test is often used in analysis of variance (ANOVA) to compare variances or test the equality of means across multiple groups.

We have a smaller dataset, So, T-test is more appropriate to test our hypothesis.

T-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.

## Step 5 – Comparing Test Statistic:

In this stage, we decide where we should accept the null hypothesis or reject the null hypothesis. There are two ways to decide where we should accept or reject the null hypothesis.

## Method A: Using Crtical values

Comparing the test statistic and tabulated critical value we have,

- If Test Statistic>Critical Value: Reject the null hypothesis.
- If Test Statistic≤Critical Value: Fail to reject the null hypothesis.

Note: Critical values are predetermined threshold values that are used to make a decision in hypothesis testing. To determine critical values for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

## Method B: Using P-values

We can also come to an conclusion using the p-value,

Note : The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed in the sample, assuming the null hypothesis is true. To determine p-value for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

## Step 7- Interpret the Results

At last, we can conclude our experiment using method A or B.

## Calculating test statistic

To validate our hypothesis about a population parameter we use statistical functions . We use the z-score, p-value, and level of significance(alpha) to make evidence for our hypothesis for normally distributed data .

## 1. Z-statistics:

When population means and standard deviations are known.

- μ represents the population mean,
- σ is the standard deviation
- and n is the size of the sample.

## 2. T-Statistics

T test is used when n<30,

t-statistic calculation is given by:

- t = t-score,
- x̄ = sample mean
- μ = population mean,
- s = standard deviation of the sample,
- n = sample size

## 3. Chi-Square Test

Chi-Square Test for Independence categorical Data (Non-normally distributed) using:

- i,j are the rows and columns index respectively.

## Real life Hypothesis Testing example

Let’s examine hypothesis testing using two real life situations,

## Case A: D oes a New Drug Affect Blood Pressure?

Imagine a pharmaceutical company has developed a new drug that they believe can effectively lower blood pressure in patients with hypertension. Before bringing the drug to market, they need to conduct a study to assess its impact on blood pressure.

- Before Treatment: 120, 122, 118, 130, 125, 128, 115, 121, 123, 119
- After Treatment: 115, 120, 112, 128, 122, 125, 110, 117, 119, 114

## Step 1 : Define the Hypothesis

- Null Hypothesis : (H 0 )The new drug has no effect on blood pressure.
- Alternate Hypothesis : (H 1 )The new drug has an effect on blood pressure.

## Step 2: Define the Significance level

Let’s consider the Significance level at 0.05, indicating rejection of the null hypothesis.

If the evidence suggests less than a 5% chance of observing the results due to random variation.

## Step 3 : Compute the test statistic

Using paired T-test analyze the data to obtain a test statistic and a p-value.

The test statistic (e.g., T-statistic) is calculated based on the differences between blood pressure measurements before and after treatment.

t = m/(s/√n)

- m = mean of the difference i.e X after, X before
- s = standard deviation of the difference (d) i.e d i = X after, i − X before,
- n = sample size,

then, m= -3.9, s= 1.8 and n= 10

we, calculate the , T-statistic = -9 based on the formula for paired t test

## Step 4: Find the p-value

The calculated t-statistic is -9 and degrees of freedom df = 9, you can find the p-value using statistical software or a t-distribution table.

thus, p-value = 8.538051223166285e-06

Step 5: Result

- If the p-value is less than or equal to 0.05, the researchers reject the null hypothesis.
- If the p-value is greater than 0.05, they fail to reject the null hypothesis.

Conclusion: Since the p-value (8.538051223166285e-06) is less than the significance level (0.05), the researchers reject the null hypothesis. There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.

## Python Implementation of Hypothesis Testing

Let’s create hypothesis testing with python, where we are testing whether a new drug affects blood pressure. For this example, we will use a paired T-test. We’ll use the scipy.stats library for the T-test.

Scipy is a mathematical library in Python that is mostly used for mathematical equations and computations.

We will implement our first real life problem via python,

In the above example, given the T-statistic of approximately -9 and an extremely small p-value, the results indicate a strong case to reject the null hypothesis at a significance level of 0.05.

- The results suggest that the new drug, treatment, or intervention has a significant effect on lowering blood pressure.
- The negative T-statistic indicates that the mean blood pressure after treatment is significantly lower than the assumed population mean before treatment.

## Case B : Cholesterol level in a population

Data: A sample of 25 individuals is taken, and their cholesterol levels are measured.

Cholesterol Levels (mg/dL): 205, 198, 210, 190, 215, 205, 200, 192, 198, 205, 198, 202, 208, 200, 205, 198, 205, 210, 192, 205, 198, 205, 210, 192, 205.

Populations Mean = 200

Population Standard Deviation (σ): 5 mg/dL(given for this problem)

## Step 1: Define the Hypothesis

- Null Hypothesis (H 0 ): The average cholesterol level in a population is 200 mg/dL.
- Alternate Hypothesis (H 1 ): The average cholesterol level in a population is different from 200 mg/dL.

As the direction of deviation is not given , we assume a two-tailed test, and based on a normal distribution table, the critical values for a significance level of 0.05 (two-tailed) can be calculated through the z-table and are approximately -1.96 and 1.96.

Step 4: Result

Since the absolute value of the test statistic (2.04) is greater than the critical value (1.96), we reject the null hypothesis. And conclude that, there is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL

## Limitations of Hypothesis Testing

- Although a useful technique, hypothesis testing does not offer a comprehensive grasp of the topic being studied. Without fully reflecting the intricacy or whole context of the phenomena, it concentrates on certain hypotheses and statistical significance.
- The accuracy of hypothesis testing results is contingent on the quality of available data and the appropriateness of statistical methods used. Inaccurate data or poorly formulated hypotheses can lead to incorrect conclusions.
- Relying solely on hypothesis testing may cause analysts to overlook significant patterns or relationships in the data that are not captured by the specific hypotheses being tested. This limitation underscores the importance of complimenting hypothesis testing with other analytical approaches.

Hypothesis testing stands as a cornerstone in statistical analysis, enabling data scientists to navigate uncertainties and draw credible inferences from sample data. By systematically defining null and alternative hypotheses, choosing significance levels, and leveraging statistical tests, researchers can assess the validity of their assumptions. The article also elucidates the critical distinction between Type I and Type II errors, providing a comprehensive understanding of the nuanced decision-making process inherent in hypothesis testing. The real-life example of testing a new drug’s effect on blood pressure using a paired T-test showcases the practical application of these principles, underscoring the importance of statistical rigor in data-driven decision-making.

## Frequently Asked Questions (FAQs)

1. what are the 3 types of hypothesis test.

There are three types of hypothesis tests: right-tailed, left-tailed, and two-tailed. Right-tailed tests assess if a parameter is greater, left-tailed if lesser. Two-tailed tests check for non-directional differences, greater or lesser.

## 2.What are the 4 components of hypothesis testing?

Null Hypothesis ( ): No effect or difference exists. Alternative Hypothesis ( ): An effect or difference exists. Significance Level ( ): Risk of rejecting null hypothesis when it’s true (Type I error). Test Statistic: Numerical value representing observed evidence against null hypothesis.

## 3.What is hypothesis testing in ML?

Statistical method to evaluate the performance and validity of machine learning models. Tests specific hypotheses about model behavior, like whether features influence predictions or if a model generalizes well to unseen data.

## 4.What is the difference between Pytest and hypothesis in Python?

Pytest purposes general testing framework for Python code while Hypothesis is a Property-based testing framework for Python, focusing on generating test cases based on specified properties of the code.

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The p value is a number, calculated from a statistical test, that describes how likely you are to have found a particular set of observations if the null hypothesis were true. P values are used in hypothesis testing to help decide whether to reject the null hypothesis. The smaller the p value, the more likely you are to reject the null hypothesis.

P Value Definition. A p value is used in hypothesis testing to help you support or reject the null hypothesis. The p value is the evidence against a null hypothesis. The smaller the p-value, the stronger the evidence that you should reject the null hypothesis. P values are expressed as decimals although it may be easier to understand what they ...

To find the p value for your sample, do the following: Identify the correct test statistic. Calculate the test statistic using the relevant properties of your sample. Specify the characteristics of the test statistic's sampling distribution. Place your test statistic in the sampling distribution to find the p value.

A p-value, or probability value, is a number describing how likely it is that your data would have occurred by random chance (i.e., that the null hypothesis is true). The level of statistical significance is often expressed as a p-value between 0 and 1. The smaller the p -value, the less likely the results occurred by random chance, and the ...

P-Value: The p-value is the level of marginal significance within a statistical hypothesis test representing the probability of the occurrence of a given event. The p-value is used as an ...

The P -value is, therefore, the area under a tn - 1 = t14 curve to the left of -2.5 and to the right of 2.5. It can be shown using statistical software that the P -value is 0.0127 + 0.0127, or 0.0254. The graph depicts this visually. Note that the P -value for a two-tailed test is always two times the P -value for either of the one-tailed tests.

Interpreting P values. By Jim Frost 98 Comments. P values determine whether your hypothesis test results are statistically significant. Statistics use them all over the place. You'll find P values in t-tests, distribution tests, ANOVA, and regression analysis. P values have become so important that they've taken on a life of their own.

The p -value is used in the context of null hypothesis testing in order to quantify the statistical significance of a result, the result being the observed value of the chosen statistic . [note 2] The lower the p -value is, the lower the probability of getting that result if the null hypothesis were true. A result is said to be statistically ...

Hypothesis Testing. To understand p-values, we first need to understand the concept of hypothesis testing. A hypothesis test is a formal statistical test we use to reject or fail to reject some hypothesis. For example, we may hypothesize that a new drug, method, or procedure provides some benefit over a current drug, method, or procedure.

Medical providers often rely on evidence-based medicine to guide decision-making in practice. Often a research hypothesis is tested with results provided, typically with p values, confidence intervals, or both. Additionally, statistical or research significance is estimated or determined by the investigators. Unfortunately, healthcare providers may have different comfort levels in interpreting ...

To determine the p-value, you need to know the distribution of your test statistic under the assumption that the null hypothesis is true.Then, with the help of the cumulative distribution function (cdf) of this distribution, we can express the probability of the test statistics being at least as extreme as its value x for the sample:Left-tailed test:

Then, if the null hypothesis is wrong, then the data will tend to group at a point that is not the value in the null hypothesis (1.2), and then our p-value will wind up being very small. If the null hypothesis is correct, or close to being correct, then the p-value will be larger, because the data values will group around the value we hypothesized.

Onward! We use p -values to make conclusions in significance testing. More specifically, we compare the p -value to a significance level α to make conclusions about our hypotheses. If the p -value is lower than the significance level we chose, then we reject the null hypothesis H 0 in favor of the alternative hypothesis H a .

The p-value or probability value is a number, calculated from a statistical test, that tells how likely it is that your results would have occurred under the null hypothesis of the test. P-values are usually automatically calculated using statistical software. They can also be calculated using p-value tables for the relevant statistical test.

Hypothesis testing is a vital process in inferential statistics where the goal is to use sample data to draw conclusions about an entire population. In the testing process, you use significance levels and p-values to determine whether the test results are statistically significant.

What is the P-value method in Hypothesis Testing? The P-value method is used in Hypothesis Testing to check the significance of the given Null Hypothesis. Then, deciding to reject or support it is based upon the specified significance level or threshold. ... Hypothesis testing is a statistical method used to make inferences about population ...

About. Transcript. We compare a P-value to a significance level to make a conclusion in a significance test. Given the null hypothesis is true, a p-value is the probability of getting a result as or more extreme than the sample result by random chance alone. If a p-value is lower than our significance level, we reject the null hypothesis.

When a P value is less than or equal to the significance level, you reject the null hypothesis. If we take the P value for our example and compare it to the common significance levels, it matches the previous graphical results. The P value of 0.03112 is statistically significant at an alpha level of 0.05, but not at the 0.01 level.

The P-value is known as the level of marginal significance within the hypothesis testing that represents the probability of occurrence of the given event. The P-value is used as an alternative to the rejection point to provide the least significance at which the null hypothesis would be rejected. If the P-value is small, then there is stronger ...

Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test. Significance is usually denoted by a p-value, or probability value. Statistical significance is arbitrary - it depends on the threshold, or alpha value, chosen by the ...

The p-value, or probability value, is a measure in statistics that quantifies the strength of evidence against a null hypothesis. It indicates the likelihood of observing a test statistic as extreme as, or more extreme than, the one obtained from the data, assuming the null hypothesis is true.

Note: The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed in the sample, assuming the null hypothesis is true. To determine p-value for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

P-values are the backbone of many statistical tests, providing a metric to gauge whether observed data deviates from what is expected under the null hypothesis. Essentially, a p-value is the ...