Understanding P-Value

What a p-value is

A p-value is a probability that quantifies how compatible your observed data are with a specified null hypothesis. It answers the question: if the null hypothesis were true, how probable would it be to see results as extreme as—or more extreme than—what was actually observed?

Think of it as a measure of surprise. A small p-value means the data would be unexpected under the null hypothesis; a large p-value means the data are plausible under that assumption.

Why p-values matter

P-values play a central role in deciding whether an observed pattern likely reflects a real effect or is plausibly due to random chance. This is useful across fields that rely on data-driven decisions, including finance, medicine, economics, and public policy.

For investors and analysts, p-values help evaluate whether differences in returns, risk metrics, or strategy performances are likely meaningful—or whether they could be artifacts of sampling variation.

How p-values are used in practice

Common uses of p-values include:

• Testing whether one investment strategy outperforms another.
• Determining if a change in policy or market condition correlates with different outcomes.
• Assessing whether a measured relationship between variables (for example, volatility and returns) is likely real.

Regulatory and professional organizations often require reporting p-values to support claims. They also set conventions on which p-values are acceptable for publication or official reporting.

Conceptual overview: how p-values are calculated

Calculating a p-value involves three elements: a null hypothesis, a test statistic that summarizes the data, and the probability distribution of that test statistic under the null hypothesis.

In plain terms, you compute a test statistic from your sample, then determine the probability—given the null model—of obtaining a statistic at least as extreme as the observed value. That probability is the p-value.

Tails and directions of tests

Tests are classified by the direction of “extremeness” they consider:

• Lower-tailed test: counts outcomes below the observed value as extreme.
• Upper-tailed test: counts outcomes above the observed value.
• Two-tailed test: counts outcomes on both sides that are as extreme or more extreme.

Choosing one- or two-tailed tests should be guided by the research question before seeing the data. Changing the direction after examining results inflates false positive risk.

The role of sampling distribution and degrees of freedom

The test statistic has a sampling distribution determined by the null hypothesis and sample design. For example, t-distributions apply when testing means with unknown variance and small samples; normal distributions appear with large samples under many conditions.

Degrees of freedom affect the shape of these distributions. More degrees of freedom generally make the distribution narrower, which changes how surprising a particular observation is.

Interpreting p-values

A p-value is not the probability that the null hypothesis is true. It is the probability of observing results at least as extreme as the observed ones assuming the null hypothesis is correct.

Common interpretations and thresholds:

• p < 0.05: conventionally considered statistically significant in many disciplines.
• p < 0.01 or p < 0.001: stronger evidence against the null hypothesis.
• p > 0.05: insufficient evidence to reject the null hypothesis at that threshold.

These thresholds are conventions, not laws. What matters for decisions is context: the cost of incorrect conclusions, prior information, and the plausibility of effects.

Practical context: what a p-value tells you

Small p-values suggest the data are unlikely under the null model. That can justify further investigation or action, but it does not prove a causal mechanism or guarantee replicability.

Large p-values indicate the data are consistent with the null hypothesis, but they do not prove the null is true. Lack of evidence is not evidence of absence.

Worked example: comparing a portfolio to the S&P 500

Suppose an investor claims their portfolio performs the same as the S&P 500. The null hypothesis states “portfolio returns equal S&P returns.” The alternative states “portfolio returns differ from S&P returns.”

After collecting return data over a time window, the investor calculates a test statistic and derives a p-value. If the p-value is very small—say, 0.001—then under the null hypothesis, observing such a difference would be rare (about 1 in 1,000). That provides strong evidence against the null.

If the p-value equals 0.08, and the researcher uses a conventional 0.05 cutoff, they would not reject the null. Yet a p-value of 0.08 still suggests some evidence against the null; different stakeholders might treat it differently depending on risk tolerance.

Comparing two p-values

If two analyses produce different p-values, the smaller p-value indicates stronger evidence against the respective null hypothesis—provided the tests and sample sizes are comparable.

For example, if Portfolio A vs. benchmark gives p = 0.10 and Portfolio B vs. benchmark gives p = 0.01, Portfolio B’s performance differs from the benchmark with stronger statistical evidence. Still, this does not indicate the size or practical importance of the difference.

Common misconceptions and pitfalls

A few frequent mistakes can lead to wrong conclusions from p-values:

• Equating p-value with effect size. A tiny p-value can result from a trivial effect if the sample is large.
• Treating p < 0.05 as definitive proof. Statistical significance does not imply practical importance.
• Interpreting a non-significant result as proof of no effect. Insufficient power can hide real effects.
• P-hacking: trying many analyses and reporting only those with small p-values inflates false positives.
• Ignoring multiple comparisons. Running many tests increases the chance of false positives unless adjustments are made.

Why sample size matters

Sample size influences p-values because larger samples reduce sampling noise. With more observations, smaller differences can become statistically detectable, producing smaller p-values for modest effects.

This means you should interpret p-values together with sample size and the estimated magnitude of the effect, not in isolation.

Limitations of p-values

P-values are a single number summarizing compatibility with a null model; they do not capture uncertainty about effect magnitude or the practical consequences of a decision.

They are sensitive to model assumptions, measurement error, and analysis choices. Different reasonable analytical choices can yield different p-values from the same raw data.

Multiple testing and false discoveries

When many hypotheses are tested, some will produce small p-values purely by chance. Corrections such as the Bonferroni adjustment, false discovery rate procedures, or pre-specifying hypotheses help control this problem.

Complementary tools: effect sizes and confidence intervals

To make better-informed decisions, combine p-values with effect sizes and confidence intervals. Effect sizes quantify how large an observed difference or relationship is; confidence intervals show a range of plausible values for that estimate.

Reporting all three elements—p-value, effect size, and confidence interval—gives a fuller picture of statistical and practical significance.

Best practices for using p-values

Follow these practices to reduce common errors and improve the reliability of conclusions:

• Pre-register hypotheses and analysis plans where possible to avoid selective reporting.
• Specify one- or two-tailed tests in advance and justify the choice.
• Report effect sizes and confidence intervals alongside p-values.
• Adjust for multiple comparisons when testing many hypotheses.
• Consider statistical power before collecting data—ensure the sample size can detect meaningful effects.
• Be transparent about data-cleaning steps and alternative analyses that were tried.

Practical advice for investors and analysts

When you see p-values in research or performance reports, ask these questions:

• What is the effect size and is it economically meaningful?
• How large was the sample and how were data collected?
• Were multiple tests performed and were corrections applied?
• Are results robust across reasonable alternative analyses?
• Has the analysis been replicated on independent data?

Small p-values can guide attention, but investment decisions should also weigh costs, risks, and the practical importance of the estimated effect.

How regulators and institutions commonly treat p-values

Many agencies and journals expect p-values to be reported and often endorse conventional thresholds for publication or policy statements. However, growing consensus encourages presenting p-values as part of a broader set of evidence rather than as a binary decision rule.

In high-stakes contexts, combining statistical evidence with domain expertise and replication strengthens confidence in conclusions.

Summary: what to take away

P-values quantify how surprising observed data are under a specified null hypothesis. They are useful for signaling evidence against a null model but do not measure effect size, probability the null is true, or practical importance by themselves.

Use p-values together with effect sizes, confidence intervals, and transparent methods. Be mindful of sample size, multiple testing, and selective reporting. When interpreted carefully, p-values are a valuable tool for separating plausible signals from noise.

Disclaimer: This article is compiled from publicly available
information and is for educational purposes only. MEXC does not guarantee the
accuracy of third-party content. Readers should conduct their own research.