A p-value is the probability of observing a test statistic at least as extreme as yours, assuming the null hypothesis is true. A small p-value (typically below 0.05) means your data are unlikely under H0, so you reject H0. It is NOT the probability that H0 is true or false — that is a common misreading.

How is the p-value for a Z-test calculated?

The standard normal CDF Phi(z) gives the probability that a standard normal variable is at most z. For a two-tailed test p = 2 * min(Phi(z), 1 - Phi(z)). For a right-tailed test p = 1 - Phi(z). For a left-tailed test p = Phi(z). This tool implements Phi via the error function: Phi(z) = 0.5 * (1 + erf(z / sqrt(2))).

How is the p-value for a t-test calculated?

The t-distribution CDF is derived from the regularised incomplete beta function: P(T <= t | df) = I_{df/(df+t^2)}(df/2, 1/2). The tail probability is then handled the same way as the Z-test. For large df the t-distribution converges to the standard normal, so the p-values converge too.

How is the chi-squared p-value calculated?

The chi-squared CDF with k degrees of freedom equals the regularised lower incomplete gamma function P(k/2, chi2/2). Because chi-squared statistics are always non-negative and the test is directional (large values indicate deviation), p = 1 - P(k/2, chi2/2). This is equivalent to the right-tail area of the chi-squared distribution.

What degrees of freedom should I use?

For a one-sample t-test use df = n - 1. For an independent two-sample t-test (equal variances) use df = n1 + n2 - 2. For a chi-squared goodness-of-fit test use df = k - 1 where k is the number of categories. For a chi-squared test of independence use df = (rows - 1)(columns - 1).

Is p < 0.05 always statistically significant?

The 0.05 threshold is a convention, not a law. Many fields use stricter thresholds (particle physics requires p < 0.000001; clinical trials often use 0.01). Some use looser thresholds for exploratory work. You can set any alpha in this calculator. Remember also that statistical significance is not the same as practical significance — a tiny effect can be highly significant with a large enough sample.

P-Value Calculator — Gera Tools

Q: How is the chi-squared p-value calculated?

The chi-squared CDF with k degrees of freedom equals the regularised lower incomplete gamma function P(k/2, chi2/2). Because chi-squared statistics are always non-negative and the test is directional (large values indicate deviation), p = 1 - P(k/2, chi2/2). This is equivalent to the right-tail area of the chi-squared distribution.

Q: What degrees of freedom should I use?

For a one-sample t-test use df = n - 1. For an independent two-sample t-test (equal variances) use df = n1 + n2 - 2. For a chi-squared goodness-of-fit test use df = k - 1 where k is the number of categories. For a chi-squared test of independence use df = (rows - 1)(columns - 1).

Q: Is p < 0.05 always statistically significant?

The 0.05 threshold is a convention, not a law. Many fields use stricter thresholds (particle physics requires p < 0.000001; clinical trials often use 0.01). Some use looser thresholds for exploratory work. You can set any alpha in this calculator. Remember also that statistical significance is not the same as practical significance — a tiny effect can be highly significant with a large enough sample.

A p-value calculator that covers the three workhorses of inferential statistics: the Z-test (standard normal distribution), Student’s t-test (t-distribution), and the chi-squared test (χ²-distribution). Paste in your test statistic, set the degrees of freedom and tail type, and get the exact p-value with the full formula working shown — no look-up tables, no Excel, no Python needed. Everything runs in your browser; no numbers are sent anywhere.

How the maths works

Z-test (standard normal)

The Z-test applies when you know the population standard deviation or have a large enough sample that the central limit theorem makes the normal approximation valid. The standard normal CDF Φ(z) is:

Φ(z) = ½ · (1 + erf(z / √2))

where erf is the error function. The tool evaluates erf via the Horner-form polynomial approximation from Abramowitz & Stegun (equation 7.1.26), accurate to within 1.5 × 10⁻⁷.

Two-tailed: p = 2 · min(Φ(z), 1 − Φ(z))
Right-tailed: p = 1 − Φ(z)
Left-tailed: p = Φ(z)

t-test (Student’s t-distribution)

When the population variance is unknown and estimated from the sample, the t-distribution with df degrees of freedom is appropriate. Its CDF is expressed via the regularised incomplete beta function:

P(T ≤ t | df) = 1 − ½ · I_{df/(df+t²)}(df/2, 1/2)

The incomplete beta function is evaluated via Lentz’s continued-fraction algorithm (Numerical Recipes §6.4) with the symmetry relation applied when x > (a+1)/(a+b+2) for better convergence. As df → ∞ the t-distribution converges to the standard normal, so the p-values also converge.

Chi-squared test (χ²-distribution)

The chi-squared statistic with k degrees of freedom follows a distribution whose CDF is the regularised lower incomplete gamma function:

P(χ² ≤ x | k) = γ(k/2, x/2) / Γ(k/2) = P(k/2, x/2)

The p-value is the right-tail area: p = 1 − P(k/2, x/2). The tool evaluates γ via a series expansion when x < a+1 and via a continued fraction otherwise, using the log-Gamma function (Lanczos approximation, g=7) for numerical stability.

Worked example — t-test

A nutritionist measures the daily calorie intake for a sample of 20 adults and reports a sample mean of 2,150 kcal. The assumed population mean is 2,000 kcal and the sample standard deviation is 320 kcal. The one-sample t-statistic is:

t = (x̄ − μ₀) / (s / √n) = (2150 − 2000) / (320 / √20) = 150 / 71.55 ≈ 2.096

With df = 19, the two-tailed p-value is approximately 0.050 — right on the conventional threshold. Entering t = 2.096, df = 19, two-tailed into the calculator reproduces this.

Scenario	t	df	Tail	p-value	Significant at 5%?
Nutrition study (above)	2.096	19	Two	0.0499	Yes (barely)
Drug efficacy	3.21	30	Two	0.0031	Yes
Exam score increase	1.45	14	Right	0.0841	No
Height difference	−2.58	25	Left	0.0080	Yes

Formula note — why these algorithms?

Direct numerical evaluation of these CDFs requires special functions (erf, incomplete beta, incomplete gamma) that are not built in to JavaScript. The implementations here follow standard numerical analysis references:

erf: Abramowitz & Stegun 7.1.26 — five-coefficient Horner polynomial, max error < 1.5 × 10⁻⁷.
Incomplete beta: Lentz continued fraction (Numerical Recipes §6.4) — the standard approach in scientific computing libraries including SciPy.
Log-Gamma: Lanczos approximation with g = 7, accurate to ~15 significant figures.

All three algorithms are validated against the worked examples in this page and against R’s pt(), pnorm(), and pchisq() functions.