Almost every quantitative thesis reports p-values, yet the p-value is one of the most misunderstood numbers in research. Many students, and many published papers, describe it as ‘the probability that the results are due to chance’ or ‘the probability that the hypothesis is true’. Neither is correct. This guide explains in plain language what a p-value is, how to interpret it, the most common misconceptions, and how to report it properly in APA style.
A plain-language definition
A p-value is the probability of obtaining a result at least as extreme as the one you observed, assuming that the null hypothesis is true (and that all the other assumptions of the statistical model hold). The null hypothesis usually states that there is no effect: no difference between groups, or no relationship between variables.
In other words, the p-value answers this question: ‘If there were really no effect, how surprising would data like mine be?’ A small p-value means your data would be unusual if the null hypothesis were true, which counts as evidence against the null hypothesis. A large p-value means your data are quite compatible with the null hypothesis. It does not prove the null hypothesis true.
An example
Suppose you compare exam scores of students who used a revision app with those who did not, and an independent samples t test gives t(60) = 2.50, p = .015. This means that if the app truly made no difference to average scores, you would expect a difference as large as the one you observed, in either direction, in only about 1.5% of studies like yours. That is fairly unusual, so you conclude that the data provide evidence of a difference.
The .05 threshold and statistical significance
Before the analysis, researchers choose a significance level, α (alpha), usually .05. If p is below α, the result is called statistically significant and the null hypothesis is rejected; otherwise it is not rejected. The .05 level is a convention, not a law of nature. Some researchers have proposed a stricter default of .005 for claims of new discoveries (Benjamin et al., 2018), and fields such as particle physics use far stricter thresholds.
Using α = .05 means accepting that, when the null hypothesis is actually true, you will wrongly reject it about 5% of the time. This is a Type I error (a false positive). Failing to detect an effect that really exists is a Type II error (a false negative). Its probability, β, depends on the sample size and the effect size. Power (1 − β) is the probability of detecting a real effect; see sample size for a thesis.
What the p-value does not tell you
In 2016 the American Statistical Association published a statement on p-values to address widespread misuse. It set out six principles (Wasserstein & Lazar, 2016):
- P-values can indicate how incompatible the data are with a specified statistical model.
- P-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone.
- Scientific conclusions and business or policy decisions should not be based only on whether a p-value passes a specific threshold.
- Proper inference requires full reporting and transparency.
- A p-value, or statistical significance, does not measure the size of an effect or the importance of a result.
- By itself, a p-value does not provide a good measure of evidence regarding a model or hypothesis.
Common misconceptions
Greenland et al. (2016) catalogued many misinterpretations. The ones that appear most often in theses are:
- ‘p = .03 means there is a 3% chance the null hypothesis is true.’ No. The p-value is calculated assuming the null hypothesis is true, so it cannot be the probability that the null is true.
- ‘p = .03 means there is a 3% chance the result happened by chance.’ This is a different wording of the same error.
- ‘p > .05 proves there is no effect.’ No. A non-significant result may reflect a small sample or a small effect. Absence of evidence is not evidence of absence. Look at the effect size and its confidence interval.
- ‘A smaller p-value means a bigger or more important effect.’ No. With a large sample, a trivial effect can have a tiny p-value. Report and interpret effect sizes; see effect size interpretation.
- ‘p = .049 and p = .051 are fundamentally different.’ They represent almost identical evidence. Avoid treating .05 as a cliff edge, and avoid labels such as ‘marginally significant’ or ‘approaching significance’.
- ‘If p < .05, the result will replicate.’ Not necessarily. Replication depends on power, effect size and the quality of the study.
Why confidence intervals and effect sizes help
A 95% confidence interval shows the range of effect sizes that are reasonably compatible with your data. If the interval for a mean difference excludes zero, the two-tailed p-value is below .05, so the interval contains the same information as the significance test and more. A narrow interval shows a precise estimate; a wide one shows substantial uncertainty. APA 7 recommends reporting confidence intervals and effect sizes alongside p-values (APA, 2020). All ScholarCalc calculators report them, for example the t test calculator and the correlation calculator.
One-tailed and two-tailed p-values
A two-tailed test looks for an effect in either direction; a one-tailed test looks only in a direction stated in advance. A one-tailed p-value is half the two-tailed value when the effect is in the predicted direction. Use one-tailed tests only when you made a clear directional prediction before seeing the data and would treat an effect in the opposite direction as no different from zero. Most theses report two-tailed tests, and switching to one-tailed after seeing the results is not acceptable.
Multiple testing
Every test you run carries a chance of a false positive. If you run 20 independent tests at α = .05 when no real effects exist, you would expect about one significant result by chance alone. This is why post hoc tests after ANOVA use adjusted p-values (Tukey, Bonferroni, Dunn), and why you should be cautious about results from many unplanned comparisons. Report all the tests you ran, not just the significant ones (Wasserstein & Lazar, 2016).
How to report p-values in APA 7 style
- Report exact p-values to two or three decimal places, for example p = .031 or p = .12.
- For values below .001, write p < .001. Never write p = .000, which is how SPSS displays very small values.
- Do not use a leading zero, because p cannot exceed 1: write .031, not 0.031.
- Italicise p.
- Report the test statistic, its degrees of freedom and an effect size together with p, for example t(60) = 2.50, p = .015, d = 0.64.
- State your α level once in the method section, for example ‘An alpha level of .05 was used for all tests’.
Students who used the revision app scored higher than those who did not, t(60) = 2.50, p = .015, d = 0.64, 95% CI of the difference [1.06, 9.54].
There was no statistically significant relationship between age and satisfaction, r(98) = .09, p = .373, 95% CI [−.11, .28].
Writing about p-values in your thesis
Good practice is to describe what the result shows, not just whether p crossed a threshold. For example, instead of ‘the result was significant (p < .05), proving that the app works’, write ‘students who used the app scored on average 5.3 marks higher, a medium-sized effect, and this difference was statistically significant, p = .015′. For non-significant results, avoid ‘no effect’ and say ‘no statistically significant difference was found’, then discuss the effect size, the confidence interval and possible reasons such as limited power.
Roman Urdu mein khulasa
p-value ka matlab hai: agar null hypothesis (yani koi farq ya rishta nahi) sach ho, to itna ya is se zyada extreme nateeja aane ka kitna imkan hai. p .05 se kam ho to nateeja statistically significant kehlata hai. Lekin p-value ye nahi batati ke hypothesis sach hone ka kitna imkan hai, aur na hi ye effect ka size batati hai. p > .05 ka matlab ye nahi ke effect bilkul nahi hai. APA mein exact p likhein (p = .031), .000 ke bajaye p < .001 likhein, aur saath mein effect size aur confidence interval bhi dein.
Frequently asked questions
What does p < .05 mean?
If the null hypothesis were true, a result at least as extreme as yours would occur less than 5% of the time. By convention, such results are called statistically significant.
Is the p-value the probability that the null hypothesis is true?
No. The p-value is calculated assuming the null hypothesis is true, so it cannot tell you the probability that it is true (Wasserstein & Lazar, 2016).
Does a non-significant result mean there is no effect?
No. It means the data did not provide enough evidence to reject the null hypothesis. Check the effect size and confidence interval; the study may have lacked power.
How do I report a p-value of .000?
Write p < .001. SPSS shows .000 when the value is below .0005.
Should I say ‘marginally significant’ for p = .06?
It is better to report the exact p-value and effect size without such labels and interpret the result cautiously.
Related calculators
- Independent Samples t Test Calculator
- Pearson Correlation Calculator
- Power Analysis Calculator
- Test chooser: which statistical test should I use?
References
- American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). https://doi.org/10.1037/0000165-000
- Benjamin, D. J., Berger, J. O., Johannesson, M., Nosek, B. A., Wagenmakers, E.-J., Berk, R., … Johnson, V. E. (2018). Redefine statistical significance. Nature Human Behaviour, 2(1), 6–10. https://doi.org/10.1038/s41562-017-0189-z
- Greenland, S., Senn, S. J., Rothman, K. J., Carlin, J. B., Poole, C., Goodman, S. N., & Altman, D. G. (2016). Statistical tests, P values, confidence intervals, and power: A guide to misinterpretations. European Journal of Epidemiology, 31(4), 337–350. https://doi.org/10.1007/s10654-016-0149-3
- Wasserstein, R. L., & Lazar, N. A. (2016). The ASA statement on p-values: Context, process, and purpose. The American Statistician, 70(2), 129–133. https://doi.org/10.1080/00031305.2016.1154108
Written and maintained by the ScholarCalc team. Last reviewed: November 2026.