Written and maintained by CASRAI Editorial Board
Published · Last updated
On this page
A test statistic is a single number computed from your sample that a hypothesis test uses to decide whether the data are consistent with the null hypothesis. The four you meet most often are the t, z, F and chi-square (χ²) statistics. Each is a different formula with a different reference distribution, and that distribution, not the software’s label, determines the critical value and the p-value. This reference gives the formula, the null distribution and the rejection rule for each, so you can read the statistic in a results table and report it correctly.
If you are still deciding which test fits your design, start with how to choose a statistical test; this page assumes you already have a statistic and want to know what it is.
What a test statistic is, and what it is not
A test statistic is a function of the sample data whose distribution is known when the null hypothesis is true. You compute it, compare it with the critical value set by your significance level α, and reject the null hypothesis if it is more extreme than that value in the direction of the alternative (Penn State STAT Online, hypothesis testing, critical value approach).
- The test statistic is the number (for example T = 2.31).
- The test is the whole procedure: hypotheses, statistic, reference distribution and decision rule. See the dictionary entry for statistical test.
- The p-value is derived from the statistic and its distribution, not the other way around. See p-value.
The four common test statistics at a glance
The formulas below are for the standard one- and two-sample cases published in the sources listed at the end. Notation: Ȳ or x̄ is a sample mean, s a sample standard deviation, N or n a sample size, ν degrees of freedom.
| Statistic | Formula (case shown) | Null hypothesis | Reference distribution under H₀ | Source |
|---|---|---|---|---|
| t (one sample) | T = (Ȳ − μ₀) / (s/√N) | μ = μ₀ | t with N − 1 degrees of freedom | NIST e-Handbook 1.3.5.2 |
| t (two sample, equal variances) | T = (Ȳ₁ − Ȳ₂) / (sp·√(1/N₁ + 1/N₂)), with sp² = [(N₁−1)s₁² + (N₂−1)s₂²] / (N₁+N₂−2) | μ₁ = μ₂ | t with N₁ + N₂ − 2 degrees of freedom | NIST e-Handbook 1.3.5.3 |
| t (two sample, unequal variances) | T = (Ȳ₁ − Ȳ₂) / √(s₁²/N₁ + s₂²/N₂) | μ₁ = μ₂ | t with ν from the Welch–Satterthwaite approximation | NIST e-Handbook 1.3.5.3 |
| z (mean, known σ) | z = (x̄ − μ) / (σ/√n) | population mean equals μ | standard normal | OpenLearn, Open University |
| F (two variances) | F = s₁² / s₂² | σ₁² = σ₂² | F with N₁ − 1 and N₂ − 1 degrees of freedom | NIST e-Handbook 1.3.5.9 |
| χ² (goodness of fit) | χ² = Σ (Oi − Ei)² / Ei over k bins | data follow the specified distribution | approximately χ² with k − c degrees of freedom (c = number of estimated parameters + 1) | NIST e-Handbook 1.3.5.15 |
The t statistic
The t statistic compares a sample mean with a hypothesised value, or two sample means with each other, measured in units of an estimated standard error. Because the standard deviation is estimated from the sample, the statistic follows a t distribution rather than the normal. For a one-sample test, T = (Ȳ − μ₀)/(s/√N) with N − 1 degrees of freedom; the two-sided rule is to reject H₀ if |T| exceeds the critical value t(1 − α/2, ν) (NIST e-Handbook).
The two-sample version has two forms. With equal population variances the denominator uses the pooled standard deviation sp and the degrees of freedom are N₁ + N₂ − 2. If you do not assume equal variances, the denominator is √(s₁²/N₁ + s₂²/N₂) and the degrees of freedom come from the Welch–Satterthwaite approximation (NIST e-Handbook). Reports should say which form was used. For the shape of the distribution, see t distribution vs normal distribution.
The z statistic
The z statistic has the same structure as t but uses a known population standard deviation, so its reference distribution is the standard normal. For a test of a mean with known σ, z = (x̄ − μ)/(σ/√n), and the statistic follows a standard normal distribution (OpenLearn, Open University, “Z-test: known population standard deviation”). The practical difference from t is whether σ is known or estimated; z-test vs t-test works through that choice.
The F statistic
The F statistic is a ratio of two variances. In the NIST F test for equality of variances, F = s₁²/s₂², with N₁ − 1 and N₂ − 1 degrees of freedom under H₀: σ₁² = σ₂². The rejection region depends on the alternative: for an upper one-tailed test reject if F exceeds the upper critical value; for a two-tailed test reject if F is below the lower or above the upper critical value (NIST e-Handbook).
One convention trap: NIST states that its Fα denotes the upper critical value, the reverse of some textbooks and software. When you copy a critical value from a table or package, check which tail it refers to. F ratios also appear in analysis of variance; see ANOVA and one-way vs two-way ANOVA.
The chi-square statistic
The chi-square statistic sums squared differences between observed and expected counts, each scaled by the expected count. In the NIST goodness-of-fit form, χ² = Σ(Oi − Ei)²/Ei over k bins; it is approximately chi-square distributed with k − c degrees of freedom, where c is the number of estimated parameters plus one, and H₀ is rejected when χ² exceeds the critical value χ²(1 − α, k − c) (NIST e-Handbook). Because it is built from squares, it cannot be negative, and large values indicate a poor fit. See chi-square test.
How to read and report a test statistic
- Name the statistic and its degrees of freedom exactly as the test defines them (for example t(194), F(9, 11), χ²(4)).
- Check the reference distribution matches the version of the test you ran: pooled vs Welch t, known vs estimated σ.
- Confirm the tail. A two-sided t test rejects on |T|; a one-sided test uses one critical value. See rejection region.
- Give the exact p-value and the decision at your stated α, not only the statistic.
- Follow the formatting rules of your style guide. APA statistics reporting covers how to typeset these in a manuscript.
Common mistakes
- Treating the statistic as the effect size. A large T or χ² reflects the data relative to a null model and the sample size; it is not a measure of how large an effect is.
- Using z when σ was estimated. If the standard deviation comes from your sample, the statistic is t, not z.
- Reporting the wrong degrees of freedom for equal-variance vs Welch t tests.
- Reading an F critical value from the wrong tail because a package and a textbook define Fα differently.
- Comparing a statistic with the wrong table. Each statistic is only meaningful against its own reference distribution.
Checklist
- Statistic named, with degrees of freedom.
- Null hypothesis and alternative stated, including whether one- or two-sided.
- Version of the formula identified (pooled or Welch, known or estimated σ).
- α, critical value or exact p-value, and decision reported.
- Assumptions of the test checked. See parametric vs non-parametric tests if they fail.
Frequently asked questions
What are the main types of test statistics?
The most common are t, z, F and chi-square. They differ in what they measure (mean differences, standardised means with known σ, variance ratios, and observed-versus-expected count discrepancies) and in the distribution they follow when the null hypothesis is true.
What is the difference between a test statistic and a p-value?
The test statistic is computed from the data. The p-value is the probability, under the null hypothesis, of a statistic at least as extreme as the one observed, found from the statistic’s reference distribution.
How do I know which test statistic to use?
That follows from the research question, the data type and the design, not from the statistic. Use how to choose a statistical test and then compute the statistic that test defines.
Can a test statistic be negative?
t and z can be negative or positive, which is why two-sided tests use the absolute value. F and χ² are built from squared quantities and cannot be negative.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods: 1.3.5.2 One-sample t-test; 1.3.5.3 Two-sample t-test; 1.3.5.9 F-test for equality of two variances; 1.3.5.15 Chi-square goodness-of-fit test. Read 2026-11-08.
- Open University, OpenLearn: Data analysis: hypothesis testing, “Z-test: known population standard deviation”. Read 2026-11-08.
- Penn State STAT Online: Hypothesis testing, critical value approach. Read 2026-11-08.
