How to Interpret P-Values: Examples for Students

Learning how to interpret p-values is one of the biggest turning points in statistics homework. Many students can run a t-test, ANOVA, chi-square test, or regression in software, but then get stuck when the output says something like p = .032 or Sig. = .487.

This guide explains what a p-value means, what it does not mean, how to compare it with alpha, how one-tailed and two-tailed p-values differ, and how to write a clear conclusion for statistics assignments.

Method references: NIST on critical values and p-values, Penn State STAT 500 hypothesis testing lesson, and the ASA Statement on p-values and statistical significance.

What Is a P-Value?

A p-value is a probability calculated under the assumption that the null hypothesis is true. It tells you how unusual your observed test statistic would be if the null hypothesis were really true.

NIST describes a p-value as the probability of the test statistic being at least as extreme as the observed one, given that the null hypothesis is true. Penn State’s hypothesis testing lesson uses the same logic: calculate the test statistic, find the p-value, compare it to alpha, then make a decision about the null hypothesis.

A smaller p-value means the observed result would be more unusual under the null hypothesis. That is why small p-values are often treated as evidence against the null hypothesis. But a p-value does not prove the alternative hypothesis, and it does not measure how important the effect is.

P-Value vs Alpha

Alpha, often written as α, is the significance level chosen before the test. In many introductory statistics classes, alpha is set to 0.05. The p-value is the result calculated from the data. The decision rule compares these two numbers.

This is exactly why your instructor may ask you to state alpha before running the test. Choosing alpha after seeing the p-value can turn hypothesis testing into a moving target.

P-Value Decision Table

Be careful with borderline p-values. The ASA statement warns against treating p < .05 as a magic line where a claim instantly becomes true and p > .05 as a line where a claim instantly becomes false. Your conclusion should still consider the study design, sample size, assumptions, effect size, and practical context.

P-Value Examples for Common Tests

The p-value appears in many different statistical tests. The meaning is always tied to the null hypothesis and the alternative hypothesis for that specific test.

T-Test P-Value Example

Suppose an independent samples t-test compares average exam scores for online and in-person students. The output gives t(58) = 2.41, p = .019, with alpha = .05.

Use the t-test calculator if you need to check a t statistic, degrees of freedom, p-value, and decision.

ANOVA P-Value Example

Suppose a one-way ANOVA compares exam scores across online, hybrid, and in-person sections. The output gives F(2, 87) = 5.62, p = .005, with alpha = .05.

Use the ANOVA calculator for one-way ANOVA calculations, or read T-Test vs ANOVA if you are unsure which test fits your assignment.

Chi-Square P-Value Example

Suppose a chi-square test of independence examines whether tutoring preference is related to class level. The output gives χ2(3) = 8.12, p = .044, with alpha = .05.

For this type of calculation, use the chi-square calculator.

Regression P-Value Example

In regression output, p-values can appear for the overall model and for individual predictors. For example, a regression coefficient might show B = 4.20, p = .018.

For regression interpretation, see Regression and ANOVA Help or how to interpret Excel regression output.

One-Tailed vs Two-Tailed P-Values

The alternative hypothesis determines whether a p-value is one-tailed or two-tailed.

Penn State’s hypothesis testing material explains that the p-value is calculated differently depending on whether the alternative hypothesis is left-tailed, right-tailed, or two-tailed. Many homework problems use two-tailed tests unless the prompt clearly says the researcher predicted a specific direction before collecting data.

For z, t, chi-square, and F p-values with tail choices, use the p-value calculator or z-test calculator.

How to Write a P-Value Conclusion

Most assignments expect more than a sentence like “p is less than .05.” A good conclusion should mention the test, p-value, alpha, decision about the null hypothesis, and what the result means in the context of the question.

Template for a Significant Result

Example: Because p = .019 is less than alpha = .05, we reject the null hypothesis. There is statistically significant evidence that average exam scores differ between online and in-person students.

Template for a Non-Significant Result

Example: Because p = .284 is greater than alpha = .05, we fail to reject the null hypothesis. There is not enough statistical evidence to conclude that average exam scores differ between online and in-person students.

APA-Style P-Value Wording

In APA-style reporting, include the test statistic and p-value. For example:

If your course avoids the phrase “statistically significant,” you can write more cautiously: “The result provides evidence of a difference in average exam scores between the groups.” Always follow your instructor’s rubric.

What a P-Value Does Not Mean

This is where many students lose marks. A p-value is useful, but it is easy to overstate what it proves.

• A p-value is not the probability that the null hypothesis is true. A p-value is calculated assuming the null hypothesis is true; it does not assign a probability to the hypothesis itself.
• A p-value is not the probability that the result happened by chance alone. The ASA statement specifically warns against that interpretation.
• A p-value does not measure effect size. A very small p-value can come from a tiny effect in a large sample. A larger p-value can occur with an important effect in a small sample.
• A p-value does not prove practical importance. Statistical evidence and real-world importance are related but not the same thing.
• A non-significant p-value does not prove no effect exists. It means the data did not provide enough evidence to reject the null hypothesis under the chosen test and assumptions.

If your assignment asks whether the result is important, use an effect size, confidence interval, context, and study design. Statskan has an effect size calculator and a confidence interval calculator for related interpretation.

Common P-Value Mistakes

• Writing “accept the null hypothesis”: most introductory courses prefer “fail to reject the null hypothesis.”
• Reporting p = 0.000: software may display .000, but report it as p < .001 instead.
• Ignoring alpha: the p-value must be compared to the significance level selected for the test.
• Using the wrong tail: one-tailed and two-tailed tests answer different alternative hypotheses.
• Confusing p-value with effect size: p-values help with evidence against H0; effect size helps with magnitude.
• Interpreting p-values without assumptions: p-values depend on the test being appropriate for the data.
• Cherry-picking only significant results: selective reporting makes p-values much less meaningful.

Frequently Asked Questions