p-Value Calculator

A p-value (probability value) is a number between 0 and 1 used in hypothesis testing. It represents the probability of observing your test result — or one more extreme — assuming the null hypothesis is true. A small p-value (typically below 0.05) suggests your result is unlikely to have occurred by chance alone, providing evidence against the null hypothesis.

If your p-value is less than or equal to your significance threshold (α, usually 0.05), you reject the null hypothesis and call the result statistically significant. If it is greater than α, you fail to reject the null hypothesis. A p-value does NOT measure the size of an effect or the practical importance of a finding — it only reflects the compatibility of your data with the null hypothesis.

A Z score measures how many standard deviations an observation is from the mean of a standard normal distribution. You use a Z test (and Z score) when your sample size is large (typically n > 30) and the population standard deviation is known. The resulting p-value tells you how likely that Z score is under the null hypothesis.

A T score comes from Student's t-distribution and is used when your sample size is small or the population standard deviation is unknown. The t-distribution is wider than the normal distribution, reflecting greater uncertainty with small samples. You must also supply degrees of freedom (usually n − 1 for a one-sample test) to compute the correct p-value.

An F statistic is the ratio of two variance estimates and is used in ANOVA tests and regression analysis to compare group means or model fit. It follows an F-distribution defined by two degrees of freedom values: the numerator df (related to the number of groups or predictors) and the denominator df (related to the total sample size minus the number of groups).

Pearson's r measures the linear relationship between two variables, ranging from −1 (perfect negative correlation) to +1 (perfect positive correlation). Its p-value tests the null hypothesis that the true population correlation is zero. To compute it you need r and degrees of freedom (n − 2, where n is the number of paired observations).

A chi-square (χ²) statistic tests whether observed frequencies in categorical data differ from expected frequencies. It is commonly used in goodness-of-fit tests and tests of independence in contingency tables. The chi-square distribution is defined by its degrees of freedom, which depend on the number of categories or cells in your table.

P-values have several well-known limitations: they do not measure effect size or practical significance, they are heavily influenced by sample size (very large samples can produce tiny p-values for trivial effects), and they are frequently misinterpreted. A p-value below 0.05 does not mean there is a 95% probability that your hypothesis is true. For this reason, researchers are encouraged to report confidence intervals and effect sizes alongside p-values.