p-Value

The mechanics run in one direction: sample data produce a test statistic, and the statistic is mapped onto the null distribution to produce a tail area. For a mean test with unknown variance, the statistic is the difference between the sample mean and the hypothesised mean, divided by the standard error, which is the sample standard deviation over the square root of the sample size. That value is then located on a t distribution with the appropriate degrees of freedom, and the p-value is the probability mass lying beyond it.

Direction matters. For a one-sided alternative, the p-value is the area in the single relevant tail. For a two-sided alternative, it is the area beyond the absolute value of the statistic in both tails, which for a symmetric distribution means doubling the one-tailed area. A large-sample two-tailed test with a statistic near 1.96 sits close to a p-value of 0.05; near 2.58, close to 0.01.

Statistical software rarely reports critical values; it reports coefficients, standard errors, t-statistics and p-values side by side. Each coefficient's p-value tests the null that the true slope equals zero against a two-sided alternative, so the analyst can scan a column and identify which explanatory variables carry evidence of a non-zero relationship. The F-statistic reported for the whole regression carries its own p-value, testing whether all slope coefficients are jointly zero.

The same convention appears throughout applied finance: unit-root tests on price series, tests of serial correlation in residuals, tests of equality between two portfolio means, and chi-square tests on variances all report a p-value rather than a critical value. Because the p-value already embeds the degrees of freedom and the tail structure, it makes results comparable across tests with different distributions.

Treating the p-value as the smallest alpha at which rejection occurs is more useful than treating it as a single accept-or-reject verdict, because it lets a reader apply their own tolerance for Type I error. A result with a p-value of 0.03 is rejected at the 5 percent level but not at the 1 percent level, and reporting the number rather than a binary label preserves that information.

The link to confidence intervals is direct. For a two-sided test, a p-value below 0.05 corresponds to a 95 percent confidence interval that excludes the hypothesised value, and a p-value above 0.05 corresponds to an interval that contains it. The interval adds what the p-value omits: the magnitude and precision of the estimated effect, which is why serious analysis reports both.

Three tasks recur. First, given a p-value and a stated significance level, state the decision in correct language: reject the null, or fail to reject it. The null is never accepted or proven. Second, given a test statistic and a distribution, reason about whether the corresponding p-value is above or below conventional thresholds, and adjust correctly between one-tailed and two-tailed framing. Third, interpret a table of regression output without recomputing anything.

A further discipline is separating the statistical conclusion from the investment conclusion. Significance testing answers whether an observed pattern is plausibly attributable to sampling variation. It does not establish that the estimated alpha, spread or beta is large enough to survive transaction costs, capacity limits or out-of-sample decay.