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P-Value Calculator

Q: One-tailed or two-tailed?

Two-tailed unless the direction of the effect was specified before looking at the data. A two-tailed p is twice the one-tailed p in the observed direction.

Q: Why is the chi-square p-value always upper-tailed?

The chi-square statistic measures total discrepancy, so only large values contradict H0; goodness-of-fit and independence tests use the upper tail.

Q: What does p < 0.05 mean?

If H0 were true, a statistic at least this extreme would occur in fewer than 5 % of repetitions. It is not the probability that H0 is true.

Converts a test statistic into a p-value for the standard normal (z), Student t or chi-square distribution, with the chosen tails, the cumulative probability and the critical value at a given significance level.

When to use

You already have a z, t or chi-square statistic (and its degrees of freedom) from a test or a paper and need the p-value or the critical value.

Do not use when: You have the raw data and still need the statistic (use t-test, chi-square-test or proportion-z-test), or the statistic follows another distribution such as F.

Formula

two-tailed (z, t): p = 2 × (1 − F(|statistic|)); right: p = 1 − F(statistic); left: p = F(statistic); chi_square: p = 1 − F(statistic) where F is the CDF of the reference distribution

The normal CDF uses the Abramowitz & Stegun erf approximation (error < 1.5·10⁻⁷); the t CDF uses the regularized incomplete beta function and the chi-square CDF the regularized incomplete gamma function (Numerical Recipes continued fractions). Critical values are found by bisection.

Inputs

Parameter	Type	Required	Description
`statistic`	number	yes	Observed value of the statistic (z, t or χ²).
`distribution`	enum: z \| t \| chi_square	default z	Reference distribution of the statistic under the null hypothesis.
`degrees_of_freedom`	number	no	Required for t and chi_square; ignored for z. May be fractional (Welch t-test). Range: > 0, ≤ 10000000
`tails`	enum: two \| right \| left	default two	Which tail(s) count as at least as extreme. For chi_square the p-value is always the upper tail.
`alpha`	number	default 0.05	Significance level as a probability (0.05 = 5 %), not a percentage. Confidence intervals and critical values use level 1 − alpha. Range: > 0

Outputs

Output	Type	Description
`p_value`	number	Probability, under H0, of a statistic at least as extreme as the observed one in the chosen tail(s).
`cumulative_probability`	number	P(X ≤ statistic) under the reference distribution.
`critical_value`	number	Statistic value at which p would equal alpha (two-tailed: the positive bound ±value; left-tailed: negative).
`distribution_used`	string	Reference distribution and degrees of freedom.
`significant`	boolean	true when p_value < alpha.
`decision`	string	Plain-language conclusion.

Example

t = 2.0 with 10 df, two-tailed: {"statistic":2,"distribution":"t","degrees_of_freedom":10,"tails":"two"} → {"p_value":0.073388,"cumulative_probability":0.963306,"critical_value":2.2281,"significant":false}

Chi-square = 3.84 with 1 df: {"statistic":3.84,"distribution":"chi_square","degrees_of_freedom":1} → {"p_value":0.050044,"cumulative_probability":0.949956,"critical_value":3.8415,"significant":false}

GET https://tttkmbb.com/api/v1/calculate/p-value?statistic=2&distribution=t&degrees_of_freedom=10&tails=two

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/p-value (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/p-value · Markdown: https://tttkmbb.com/statistics/p-value.md · JSON definition: https://tttkmbb.com/statistics/p-value.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="p-value"
OpenAPI operationId: calculate_p_value
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – p-value (reference)
NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.4 t Distribution (government)
NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.6 Chi-Square Distribution (government)

FAQ

One-tailed or two-tailed?

Two-tailed unless the direction of the effect was specified before looking at the data. A two-tailed p is twice the one-tailed p in the observed direction.

Why is the chi-square p-value always upper-tailed?

The chi-square statistic measures total discrepancy, so only large values contradict H0; goodness-of-fit and independence tests use the upper tail.

What does p < 0.05 mean?

If H0 were true, a statistic at least this extreme would occur in fewer than 5 % of repetitions. It is not the probability that H0 is true.

Related calculators

T-Test Calculator — Compute t and its p-value from data.
Chi-Square Test Calculator — Chi-square statistic from observed counts.
Z-Score Calculator — Standardise a value to obtain a z statistic.