Title: What Is a P‑Value — Practical Guide, Interpretation, Calculation, and Reporting
Source: Investopedia — “P‑Value” (https://www.investopedia.com/terms/p/p-value.asp). Additional guidance from the American Statistical Association: “Statement on Statistical Significance and P‑Values” (2016).
Key takeaways
– A p‑value quantifies how compatible the observed data are with a specified null hypothesis: it is the probability of obtaining results at least as extreme as those observed if the null hypothesis were true.
– Smaller p‑values indicate stronger evidence against the null hypothesis, but a p‑value alone does not measure effect size or the probability that the null hypothesis is true.
– Use p‑values together with pre‑specified significance levels, effect sizes, confidence intervals, and transparent reporting to draw reliable conclusions.
1. What a p‑value is (plain language)
– Definition: The p‑value is the probability of observing the current data—or data showing a more extreme departure from the null hypothesis—assuming the null hypothesis is true.
– Role: It provides a continuous measure of evidence against the null hypothesis and offers the smallest significance level (α) at which the null would be rejected.
2. Formal description and the three common test types
– Let T be the test statistic and t_obs its observed value. Under H0 the distribution of T is known (or approximated). The p‑value is:
– For an upper‑tailed test: p = P(T ≥ t_obs | H0).
– For a lower‑tailed test: p = P(T ≤ t_obs | H0).
– For a two‑tailed test: p = 2 × P(T ≥ |t_obs| | H0) (or the equivalent area in both tails).
– Practical note: The exact calculations depend on the test statistic’s distribution (e.g., normal, t, chi‑square) and on degrees of freedom.
3. How a p‑value is calculated (practical steps)
1. State hypotheses:
– Null hypothesis (H0): the baseline claim (e.g., mean difference = 0).
– Alternative hypothesis (H1): what you want to test (e.g., mean difference ≠ 0).
2. Choose an appropriate test and test statistic (z, t, chi‑square, etc.) based on data type and assumptions.
3. Check assumptions (independence, distributional shape, variance equality, sample size).
4. Compute the test statistic from your sample (e.g., t = (x̄ − μ0) / (s / √n)).
5. Using the null distribution of the statistic, calculate the tail probability corresponding to the observed statistic (this is the p‑value). In practice use statistical software or lookup tables.
6. Interpret: compare to a pre‑specified α (if using a decision rule) or report the exact p‑value and interpret with context.
4. Example (investor vs. S&P 500 — paraphrased)
– Scenario: An investor tests whether their portfolio’s returns equal the S&P 500’s returns.
– H0: Portfolio mean return = S&P mean return. H1 (two‑tailed): they are not equal.
– After computing the appropriate test statistic and p‑value:
– If p = 0.001 → very unlikely to observe such a difference if H0 were true; strong evidence against H0.
– If p = 0.08 → moderate evidence against H0 that may be considered significant at α = 0.10 but not at α = 0.05.
– Practical lesson: different pre‑set α levels (e.g., 0.05 vs 0.10) can change a reject/retain decision, so reporting the exact p‑value lets readers judge.
5. Common interpretations and shorthand thresholds
– p < 0.05: commonly called “statistically significant” (but this is a convention, not law).
– p < 0.01 or p 0.05: insufficient evidence to reject H0 at the 5% level (but not proof that H0 is true).
– Note: These thresholds are arbitrary; context and consequences of Type I/II errors should guide choice of α.
6. What p = 0.001 means
– If the null hypothesis were true, there is a 0.1% chance of observing data at least as extreme as what you observed.
– This indicates strong evidence against H0, but it is not the probability that H0 is false, nor does it say anything about the practical size of the effect.
7. Is p = 0.05 “significant”?
– Historically and conventionally, p < 0.05 is used as a cutoff for significance. However:
– The 0.05 threshold is arbitrary.
– A p slightly below 0.05 (e.g., 0.049) and a p slightly above 0.05 (e.g., 0.051) provide very similar evidence but can lead to different decisions if one relies strictly on the cutoff.
– Always report the exact p‑value and the effect size; avoid binary "significant/not significant" thinking.
8. Comparing two p‑values: what’s valid and what’s misleading
– Rule of thumb: a smaller p‑value generally indicates stronger evidence against H0.
– Caveats:
– You cannot infer that a lower p‑value implies a larger or more important effect without examining effect sizes and sample size.
– p‑values depend on sample size: with very large samples, tiny effects can produce very small p‑values.
– When comparing two studies, compare effect estimates and confidence intervals rather than p‑values alone.
– Example: pA = 0.04 vs pB = 0.06 → A shows slightly stronger evidence, but the difference is small; check effect sizes and CIs.
9. Practical checklist for using p‑values (for researchers & analysts)
1. Predefine the hypothesis and the significance level (α) when applicable.
2. Choose an appropriate test and verify its assumptions. If assumptions fail, use robust or nonparametric methods.
3. Compute the test statistic and the exact p‑value (use software: R, Python, SAS, SPSS, Stata, Excel).
4. Report fully:
– Exact p‑value (not just “p < 0.05”).
– Test statistic and degrees of freedom.
– Sample size and sample summary (means, SDs).
– Effect size and confidence interval.
– Any data exclusions, multiple comparisons, or pre‑registration status.
5. Interpret in context: consider prior evidence, plausibility, and practical significance.
6. Consider corrections for multiple testing if many hypotheses are tested (Bonferroni, FDR).
7. Avoid p‑hacking and selective reporting; prefer pre‑registration and transparent analysis.
10. How to compute p‑values in common tools (brief)
– R example (two‑sample t‑test): t.test(x, y) → gives t, df, and p‑value.
– Python (SciPy): from scipy import stats; stats.ttest_ind(x, y) → returns t and p.
– Excel: use T.DIST.2T (two‑tailed) or built‑in Data Analysis tool.
– Statistical packages will compute exact p‑values for standard tests; for custom tests you may need simulation (permutation/bootstrap) to obtain p‑values.
11. Limitations and common misunderstandings
– A p‑value is not:
– The probability H0 is true.
– The probability the observed effect is due to chance (that wording is imprecise).
– Sensitive to sample size: larger samples can produce tiny p‑values for trivial effects.
– Not a measure of effect size or practical importance.
– Multiple comparisons inflate false positive risk if uncorrected.
– Selective reporting and post‑hoc hypothesis testing (“p‑hacking”) can produce misleading p‑values.
– For nuanced decision making consider complementary approaches: confidence intervals, effect sizes, Bayesian methods (Bayes factors), replication.
12. Good reporting practice (recommended)
– Always report: exact p‑value, test statistic, degrees of freedom, n, effect size (e.g., mean difference, odds ratio), and a confidence interval.
– State assumptions and any violations.
– If multiple tests were performed, state correction methods.
– If possible, share raw data and code to support reproducibility.
13. Alternatives and complements to p‑values
– Confidence intervals — show range of plausible effect sizes.
– Effect size measures — quantify magnitude (Cohen’s d, R², odds ratios).
– Bayesian posterior probabilities and Bayes factors — provide a different framing for evidence about hypotheses.
– Pre‑registered analysis and replication — strengthen credibility.
14. Bottom line
P‑values are a useful tool for quantifying evidence against a null hypothesis, but they are one input — not a final verdict. Use p‑values together with effect sizes, confidence intervals, transparent reporting, and domain knowledge. Always be cautious about over‑interpreting small p‑values and avoid treating conventional cutoffs as automatic proof.
References
– Investopedia: “P‑Value” — https://www.investopedia.com/terms/p/p-value.asp
– American Statistical Association: “ASA Statement on Statistical Significance and P‑Values” (2016) — https://www.amstat.org/asa/files/pdfs/P-ValueStatement.pdf
Editor’s note: The following topics are reserved for upcoming updates and will be expanded with detailed examples and datasets.