What is a p-value, explained for high school researchers

TL;DR: This post answers a single precise question: what is a p-value, and how should high school researchers interpret and report it correctly? It is written for students in grades 9 through 12 who are conducting original quantitative research and need to understand statistical significance without a university statistics course. After reading, you will be able to calculate, interpret, and report a p-value accurately in a research paper. Students whose work is ready for peer review can submit original research to the Princeton Journal of Pre-Collegiate Research.

Introduction

Peer reviewers reject more student research papers for misinterpreting p-values than for almost any other statistical error. The mistake is rarely a calculation error. It is a conceptual one: students report a p-value below 0.05 and then write that their hypothesis has been "proven" or that the result is "significant" in a general sense. Those two words mean different things, and conflating them is a substantive flaw that experienced reviewers identify immediately. Understanding what is a p-value, explained for high school researchers, requires understanding what the number actually measures and what it does not. This post provides that explanation precisely, with worked examples and common error patterns drawn from real research contexts.

What is a p-value and what does it measure?

A p-value is the probability of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is true. It does not measure the probability that your hypothesis is correct. A p-value below 0.05 means there is less than a 5 percent chance of seeing your result if no real effect exists, not that your conclusion is 95 percent likely to be right.

To understand this precisely, start with the null hypothesis. In any experiment, the null hypothesis states that there is no effect, no difference, and no relationship between the variables being tested. The p-value tells you how surprising your data would be if the null hypothesis were true.

For example: a student tests whether listening to classical music improves short-term memory scores in a sample of 40 peers. The null hypothesis is that music has no effect on scores. After running a two-sample t-test, the student obtains a p-value of 0.03. This means that if music truly had no effect, there would be only a 3 percent probability of observing a difference this large by chance alone. Because 0.03 is below the conventional threshold of 0.05, the result is described as statistically significant.

Statistical significance does not mean the effect is large, important, or practically meaningful. It means the result is unlikely to be a product of random variation, given the sample size and the data collected. A very large sample can produce a statistically significant result for a difference so small it has no real-world relevance. This distinction matters in every discipline, from psychology to biology to economics.

The threshold of 0.05 is a convention, not a law. It was proposed by statistician Ronald Fisher in 1925 and became the default in many fields. Some disciplines, including particle physics, require p-values below 0.000001 before claiming a discovery. Medical research often uses 0.01. High school researchers should state the threshold they are using and explain why it is appropriate for their field.

For further context on how statistical significance is applied in student research, the post on statistical significance in high school research provides a connected explanation of how these concepts interact in practice.

What does a p-value not tell you?

A p-value does not measure effect size, practical importance, or the probability that your hypothesis is true. These are the three most consequential things students assume a p-value confirms, and none of them are correct.

Effect size is a separate calculation that tells you how large the observed difference actually is. Common effect size measures include Cohen's d for comparing two means and Pearson's r for correlations. A study with 500 participants might find a statistically significant result with a Cohen's d of 0.08, which is considered negligible by most standards. A study with 20 participants might find a non-significant result but show a Cohen's d of 0.6, which represents a medium effect that could be meaningful with a larger sample.

The American Statistical Association issued a formal statement in 2016 warning against using p-values as the sole basis for scientific conclusions. The statement explicitly notes that a p-value does not measure the probability that the studied hypothesis is true. High school researchers who cite this distinction in their methodology sections demonstrate a level of statistical literacy that reviewers notice.

Practical importance is a judgment that requires domain knowledge, not a calculation. A drug that reduces blood pressure by 1 mmHg with a p-value of 0.001 is statistically significant but clinically irrelevant. A student researcher studying the effect of sleep duration on GPA should report both the p-value and the actual difference in GPA points observed, so readers can judge whether the finding matters in context.

What are the most common p-value mistakes high school researchers make?

The most common p-value error in student research is interpreting a significant result as proof that the hypothesis is true. A p-value below 0.05 rules out random chance as the most likely explanation for the data. It does not rule out confounding variables, measurement error, or sampling bias. Students who write "the results prove that" after reporting a p-value are making a logical error that reviewers will flag in revision requests.

The second common mistake is p-hacking, which occurs when a student runs multiple statistical tests on the same dataset and reports only the tests that produced significant results. If you run 20 tests at the 0.05 threshold, you would expect one significant result by chance alone even if no real effect exists. The correction for this is the Bonferroni adjustment, which divides the significance threshold by the number of tests performed. A student running five tests should use 0.01 as the threshold for each, not 0.05.

The third mistake is reporting a p-value without reporting the test statistic and degrees of freedom that produced it. Writing "p = 0.04" without also writing "t(38) = 2.11" gives reviewers no way to verify the calculation. Complete reporting is required in peer-reviewed work. The format varies by statistical test but always includes the test statistic, the degrees of freedom or sample size, and the p-value.

The fourth mistake is treating a non-significant result as a failed experiment. A p-value above 0.05 means the data did not provide sufficient evidence to reject the null hypothesis. It does not mean the null hypothesis is true or that the study was worthless. Non-significant results are publishable when the methodology is sound and the question is well-posed. Reporting a null result honestly is a contribution to the literature.

How to calculate and report a p-value correctly, step by step

1. State the null and alternative hypotheses before collecting data. The null hypothesis must be stated in advance, not chosen after seeing the results. Example: "There is no difference in mean test scores between students who studied with background music and those who studied in silence."

2. Choose the appropriate statistical test for your data type. Use a two-sample t-test for comparing two group means with continuous data. Use chi-square for comparing proportions or categorical outcomes. Use Pearson's r for measuring linear correlation between two continuous variables. The test choice must match the data structure.

3. Set your significance threshold before running the test. State whether you are using 0.05, 0.01, or another threshold, and explain why that threshold is appropriate for your field.

4. Run the test using a reliable tool. Free tools including Google Sheets, R, and Python's SciPy library all produce accurate p-values. The post on free tools for high school researchers covers several accessible options in detail.

5. Report the full test result, not just the p-value. Include the test statistic, degrees of freedom, and p-value in the format required by your target journal. Example: "A two-sample t-test revealed a statistically significant difference in mean scores between groups, t(38) = 2.43, p = 0.02."

6. Report effect size alongside the p-value. Calculate Cohen's d or another appropriate effect size measure and include it in the results section. This gives readers the information they need to judge practical significance.

7. Interpret the result accurately in the discussion section. Write that the result is consistent with the alternative hypothesis, not that it proves it. Acknowledge limitations including sample size, sampling method, and potential confounds.

Students whose quantitative research is complete and correctly reported can review the peer review process for high school journals to understand what reviewers assess before submission.

The Princeton Journal of Pre-Collegiate Research publishes original quantitative and qualitative research across all academic disciplines. If your statistical analysis is complete and your paper is ready for review, examine the submission guidelines at princeton-jpcr.org before preparing your manuscript.

Frequently asked questions about p-values in high school research

What is a p-value in simple terms for a high school student?

A p-value is the probability of seeing your results by chance if there is actually no real effect. A p-value of 0.03 means there is a 3 percent chance your data would look this way if the null hypothesis were true. It does not confirm your hypothesis; it measures how surprising your data is under the assumption that nothing is happening. The conventional threshold for calling a result statistically significant is 0.05, though this varies by discipline.

How long does it take to get a research paper with statistical analysis peer reviewed?

Peer review for quantitative research typically involves close scrutiny of methodology, statistical reporting, and interpretation, which adds time to the review process. At PJPCR, the standard review and publication timeline is 2 to 3 months from submission to a final decision. A fast-track option is available for students who need a quicker turnaround. The review process is thorough regardless of timeline, and statistical methods receive the same level of assessment in both tracks.

Do I need a university mentor to submit research with statistical analysis to a journal?

No. A university mentor is not required to submit to most student research journals, including PJPCR. What is required is that the methodology is sound, the statistical reporting is complete, and the conclusions are proportionate to the evidence. Many high school students conduct rigorous quantitative research independently or with a high school teacher as a faculty advisor. The quality of the work determines eligibility for review, not the institutional affiliation of the supervisor.

What makes a high school research paper with statistical analysis publishable?

A publishable quantitative paper reports a clearly stated hypothesis, an appropriate statistical test, complete results including test statistics and effect sizes, and a discussion that accurately interprets the findings without overstating them. Reviewers assess whether the conclusions are supported by the data, not whether the results are positive or significant. A well-reported null result from a sound methodology is more publishable than an overclaimed significant result from a flawed design. Reviewers also assess whether the student demonstrates understanding of the limitations of their analysis, including sample size constraints common in high school research.

What kinds of research does PJPCR publish, and does it accept quantitative student work?

PJPCR publishes original research across the sciences, social sciences, humanities, and interdisciplinary fields, including quantitative studies using statistical analysis. Submission and peer review are free. A publication fee applies for accepted papers. Papers are evaluated on originality, methodological rigor, and accuracy of interpretation. Students can review the full scope of accepted work and examine the criteria for evaluating a high school research journal before deciding whether PJPCR is the right venue for their work.

Conclusion

A p-value measures the probability of observing your data if the null hypothesis is true. It does not prove a hypothesis, measure practical importance, or confirm that an effect is real. Reporting a p-value correctly requires stating the full test result, including the test statistic and degrees of freedom, reporting effect size alongside the p-value, and interpreting the finding accurately in the discussion section. Avoiding p-hacking, selecting the correct statistical test in advance, and acknowledging non-significant results honestly are the markers of statistically sound student research.

These are not minor technical details. They are the criteria by which peer reviewers assess quantitative work. Students who internalize these principles produce research that is credible, reproducible, and appropriate for academic publication. If your research is complete and statistically sound, submit it to PJPCR at princeton-jpcr.org.