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Common test statistics

See legend defining symbols at bottom of table. The statistics for some other tests have their own articles, including the Wald test and the likelihood ratio test.

Name	Formula	Assumptions or notes
One-sample z-test	$z={\frac {{\overline {x}}-\mu _{0}}{(\sigma /{\sqrt {n}})}}$	(Normal population or n > 30) and σ known. (z is the distance from the mean in relation to the standard deviation of the mean). For non-normal distributions it is possible to calculate a minimum proportion of a population that falls within k standard deviations for any k (see: Chebyshev's inequality).
Two-sample z-test	$z={\frac {({\overline {x}}_{1}-{\overline {x}}_{2})-(\mu _{1}-\mu _{2})_{0}}{\sqrt {{\frac {\sigma _{1}^{2}}{n_{1}}}+{\frac {\sigma _{2}^{2}}{n_{2}}}}}}$	Normal population and independent observations and both (σ1 and σ2 known)
One-sample t-test	$t={\frac {{\overline {x}}-\mu _{0}}{(s/{\sqrt {n}})}},$ $df=n-1\$	(Normal population or n > 30) and σ unknown
Paired t-test	$t={\frac {{\overline {d}}-d_{0}}{(s_{d}/{\sqrt {n}})}},$ $df=n-1\$	(Normal population of differences or n > 30) and σ unknown
Two-sample pooled t-test, equal variances*	$t={\frac {({\overline {x}}_{1}-{\overline {x}}_{2})-(\mu _{1}-\mu _{2})_{0}}{s_{p}{\sqrt {{\frac {1}{n_{1}}}+{\frac {1}{n_{2}}}}}}},$ $s_{p}^{2}={\frac {(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}},$ $df=n_{1}+n_{2}-2\$	(Normal populations or n1 + n2 > 40) and independent observations and σ1 = σ2 and (σ1 and σ2 unknown)
Two-sample unpooled t-test, unequal variances*	$t={\frac {({\overline {x}}_{1}-{\overline {x}}_{2})-(\mu _{1}-\mu _{2})_{0}}{\sqrt {{\frac {s_{1}^{2}}{n_{1}}}+{\frac {s_{2}^{2}}{n_{2}}}}}},$ $df={\frac {\left({\frac {s_{1}^{2}}{n_{1}}}+{\frac {s_{2}^{2}}{n_{2}}}\right)^{2}}{{\frac {\left({\frac {s_{1}^{2}}{n_{1}}}\right)^{2}}{n_{1}-1}}+{\frac {\left({\frac {s_{2}^{2}}{n_{2}}}\right)^{2}}{n_{2}-1}}}}$	(Normal populations or n1 + n2 > 40) and independent observations and σ1 ≠ σ2 and (σ1 and σ2 unknown)
One-proportion z-test	$z={\frac {{\hat {p}}-p_{0}}{\sqrt {\frac {p_{0}(1-p_{0})}{n}}}}$	n .p0 > 10 and n (1 − p0) > 10 and it is a SRS (Simple Random Sample).
Two-proportion z-test, equal variances*	$z={\frac {({\hat {p}}_{1}-{\hat {p}}_{2})-(p_{1}-p_{2})_{0}}{\sqrt {{\hat {p}}(1-{\hat {p}})({\frac {1}{n_{1}}}+{\frac {1}{n_{2}}})}}}$ ${\hat {p}}={\frac {x_{1}+x_{2}}{n_{1}+n_{2}}}$	n1 p1 > 5 and n1(1 − p1) > 5 and n2 p2 > 5 and n2(1 − p2) > 5 and independent observations
Two-proportion z-test, unequal variances*	$z={\frac {({\hat {p}}_{1}-{\hat {p}}_{2})-(p_{1}-p_{2})_{0}}{\sqrt {{\frac {{\hat {p}}_{1}(1-{\hat {p}}_{1})}{n_{1}}}+{\frac {{\hat {p}}_{2}(1-{\hat {p}}_{2})}{n_{2}}}}}}$	n1 p1 > 5 and n1(1 − p1) > 5 and n2 p2 > 5 and n2(1 − p2) > 5 and independent observations
One-sample chi-square test	$\chi ^{2}={\frac {(n-1)s^{2}}{\sigma _{0}^{2}}}$	One of the following • All expected counts are at least 5 • All expected counts are > 1 and no more that 20% of expected counts are less than 5
*Two-sample F test for equality of variances	$F={\frac {s_{1}^{2}}{s_{2}^{2}}}$	Arrange so $s_{1}^{2}$ > $s_{2}^{2}$ and reject H0 for $F>F(\alpha /2,n_{1}-1,n_{2}-1)$
Definition of symbols	$\alpha$, the probability of Type I error (rejecting a null hypothesis when it is in fact true) $n$ = sample size $n_{1}$ = sample 1 size $n_{2}$ = sample 2 size ${\overline {x}}$ = sample mean $\mu _{0}$ = hypothesized population mean $\mu _{1}$ = population 1 mean $\mu _{2}$ = population 2 mean $\sigma$ = population standard deviation $\sigma ^{2}$ = population variance $s$ = sample standard deviation $s^{2}$ = sample variance $s_{1}$ = sample 1 standard deviation $s_{2}$ = sample 2 standard deviation $t$ = t statistic $df$ = degrees of freedom ${\overline {d}}$ = sample mean of differences $d_{0}$ = hypothesized population mean difference $s_{d}$ = standard deviation of differences ${\hat {p}}$ = x/n = sample proportion, unless specified otherwise $p_{0}$ = hypothesized population proportion $p_{1}$ = proportion 1 $p_{2}$ = proportion 2 $\min\{n_{1},n_{2}\}$ = minimum of n1 and n2 $x_{1}=n_{1}p_{1}$ $x_{2}=n_{2}p_{2}$ $\chi ^{2}$ = Chi-squared statistic $F$ = F statistic	In general, the subscript 0 indicates a value taken from the null hypothesis, H0, which should be used as much as possible in constructing its test statistic.