Teststatistic is a quantity derived from thesample forstatistical hypothesis testing.^[1] A hypothesis test is typically specified in terms of a test statistic, considered as a numerical summary of a data-set that reduces the data to one value that can be used to perform the hypothesis test. In general, a test statistic is selected or defined in such a way as to quantify, within observed data, behaviours that would distinguish thenull from thealternative hypothesis, where such an alternative is prescribed, or that would characterize the null hypothesis if there is no explicitly stated alternative hypothesis.

An important property of a test statistic is that itssampling distribution under the null hypothesis must be calculable, either exactly or approximately, which allowsp-values to be calculated. Atest statistic shares some of the same qualities of adescriptive statistic, and many statistics can be used as both test statistics and descriptive statistics. However, a test statistic is specifically intended for use in statistical testing, whereas the main quality of a descriptive statistic is that it is easily interpretable. Some informative descriptive statistics, such as thesample range, do not make good test statistics since it is difficult to determine their sampling distribution.

Two widely used test statistics are thet-statistic and theF-statistic.

Suppose the task is to test whether a coin isfair (i.e. has equal probabilities of producing a head or a tail). If the coin is flipped 100 times and the results are recorded, the raw data can be represented as a sequence of 100 heads and tails. If there is interest in themarginal probability of obtaining a tail, only the numberT out of the 100 flips that produced a tail needs to be recorded. ButT can also be used as a test statistic in one of two ways:

• the exactsampling distribution ofT under the null hypothesis is thebinomial distribution with parameters 0.5 and 100.
• the value ofT can be compared with its expected value under the null hypothesis of 50, and since the sample size is large, anormal distribution can be used as an approximation to the sampling distribution either forT or for the revised test statisticT−50.

Using one of these sampling distributions, it is possible to compute either aone-tailed or two-tailed p-value for the null hypothesis that the coin is fair. The test statistic in this case reduces a set of 100 numbers to a single numerical summary that can be used for testing.

One-sample tests are appropriate when a sample is being compared to thepopulation from a hypothesis. The population characteristics are known from theory or are calculated from the population.

Two-sample tests are appropriate for comparing two samples, typically experimental and control samples from ascientifically controlled experiment.

Paired tests are appropriate for comparing two samples where it is impossible to control important variables. Rather than comparing two sets, members are paired between samples so the difference between the members becomes the sample. Typically the mean of the differences is then compared to zero. The common example scenario for when apaired difference test is appropriate is when a single set of test subjects has something applied to them and the test is intended to check for an effect.

Z-tests are appropriate for comparing means under stringent conditions regarding normality and a known standard deviation.

At-test is appropriate for comparing means under relaxed conditions (less is assumed).

Tests of proportions are analogous to tests of means (the 50% proportion).

Chi-squared tests use the same calculations and the same probability distribution for different applications:

• Chi-squared tests for variance are used to determine whether a normal population has a specified variance. The null hypothesis is that it does.
• Chi-squared tests of independence are used for deciding whether two variables are associated or are independent. The variables arecategorical rather than numeric. It can be used to decide whetherleft-handedness is correlated with height (or not). The null hypothesis is that the variables are independent. The numbers used in the calculation are the observed and expected frequencies of occurrence (fromcontingency tables).
• Chi-squaredgoodness of fit tests are used to determine the adequacy of curves fit to data. The null hypothesis is that the curve fit is adequate. It is common to determine curve shapes to minimize themean square error, so it is appropriate that the goodness-of-fit calculation sums the squared errors.

F-tests (analysis of variance, ANOVA) are commonly used when deciding whether groupings of data by category are meaningful. If the variance of test scores of the left-handed in a class is much smaller than the variance of the whole class, then it may be useful to study lefties as a group. The null hypothesis is that two variances are the same – so the proposed grouping is not meaningful.

In the table below, the symbols used are defined at the bottom of the table. Many other tests can be found inother articles. Proofs exist that the test statistics are appropriate.^[2]

Name	Formula	Assumptions or notes
One-sample${\displaystyle z}$ -test	${\displaystyle z=\frac{\overline{x}-{\mu }_0}{(\sigma /\sqrt{n})}}$	(Normal populationorn large)and σ known. (z is the distance from the mean in relation to thestandard deviation of the mean). For non-normal distributions it is possible to calculate a minimum proportion of a population that falls withink standard deviations for anyk (see:Chebyshev's inequality).
Two-sample z-test	${\displaystyle z=\frac{({\overline{x}}_1-{\overline{x}}_2)-d_0}{\sqrt{\frac{{\sigma }_1^2}{n_1}+\frac{{\sigma }_2^2}{n_2}}}}$	Normal populationand independent observationsand σ1 and σ2 are known where${\displaystyle d_0}$ is the value of${\displaystyle {\mu }_1-{\mu }_2}$ under the null hypothesis
One-samplet-test	${\displaystyle t=\frac{\overline{x}-{\mu }_0}{(s/\sqrt{n})},}$ ${\displaystyle df=n-1}$	(Normal populationorn large)and${\displaystyle \sigma }$ unknown
Pairedt-test	${\displaystyle t=\frac{\overline{d}-d_0}{(s_d/\sqrt{n})},}$ ${\displaystyle df=n-1}$	(Normal population of differencesorn large)and${\displaystyle \sigma }$ unknown
Two-sample pooledt-test, equal variances	${\displaystyle t=\frac{({\overline{x}}_1-{\overline{x}}_2)-d_0}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}},}$ ${\displaystyle s_p^2=\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2},}$ ${\displaystyle df=n_1+n_2-2}$[3]	(Normal populationsorn1 + n2 > 40)and independent observationsand σ1 = σ2 unknown
Two-sample unpooledt-test, unequal variances (Welch'st-test)	${\displaystyle t=\frac{({\overline{x}}_1-{\overline{x}}_2)-d_0}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}},}$ ${\displaystyle df=\frac{{\left({\displaystyle \frac{s_1^2}{n_1}}+{\displaystyle \frac{s_2^2}{n_2}}\right)}^2}{{\displaystyle \frac{{\left({\displaystyle \frac{s_1^2}{n_1}}\right)}^2}{n_1-1}}+{\displaystyle \frac{{\left({\displaystyle \frac{s_2^2}{n_2}}\right)}^2}{n_2-1}}}}$[3]	(Normal populationsorn1 + n2 > 40)and independent observationsand σ1 ≠ σ2 both unknown
One-proportion z-test	${\displaystyle z=\frac{\hat{p}-p_0}{\sqrt{p_0(1-p_0)}}\sqrt{n}}$	n · p0 > 10andn (1 − p0) > 10and it is a SRS (Simple Random Sample), seenotes.
Two-proportion z-test, pooled for${\displaystyle H_0:p_1=p_2}$	${\displaystyle z=\frac{({\hat{p}}_1-{\hat{p}}_2)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}}$ ${\displaystyle \hat{p}=\frac{x_1+x_2}{n_1+n_2}}$	n1p1 > 5andn1(1 − p1) > 5andn2p2 > 5andn2(1 − p2) > 5and independent observations, seenotes.
Two-proportion z-test, unpooled for${\displaystyle |d_0|>0}$	${\displaystyle z=\frac{({\hat{p}}_1-{\hat{p}}_2)-d_0}{\sqrt{\frac{{\hat{p}}_1(1-{\hat{p}}_1)}{n_1}+\frac{{\hat{p}}_2(1-{\hat{p}}_2)}{n_2}}}}$	n1p1 > 5andn1(1 − p1) > 5andn2p2 > 5andn2(1 − p2) > 5and independent observations, seenotes.
Chi-squared test for variance	${\displaystyle {\chi }^2=(n-1)\frac{s^2}{{\sigma }_0^2}}$	df = n-1 • Normal population
Chi-squared test forgoodness of fit	${\displaystyle {\chi }^2=\sum _k\frac{(\text{observed}-\text{expected}{)}^2}{\text{expected}}}$	df = k − 1 − # parameters estimated, and one of these must hold. • All expected counts are at least 5.[4]: 350  • All expected counts are > 1 and no more than 20% of expected counts are less than 5[5]
Two-sample F test for equality of variances	${\displaystyle F=\frac{s_1^2}{s_2^2}}$	Normal populations Arrange so${\displaystyle s_1^2\ge s_2^2}$ and reject H0 for${\displaystyle F>F(\alpha /2,n_1-1,n_2-1)}$[6]
Regressiont-test of${\displaystyle H_0:R^2=0.}$	${\displaystyle t=\sqrt{\frac{R^2(n-k-1^{\ast })}{1-R^2}}}$	RejectH0 for${\displaystyle t>t(\alpha /2,n-k-1^{\ast })}$[4]: 288  *Subtract 1 for intercept;k terms contain independent variables.
In general, the subscript 0 indicates a value taken from thenull hypothesis, H0, which should be used as much as possible in constructing its test statistic.... Definitions of other symbols: ${\displaystyle \alpha }$, theprobability ofType I error (rejecting anull hypothesis when it is in fact true) ${\displaystyle n}$ =sample size ${\displaystyle n_1}$ = sample 1 size ${\displaystyle n_2}$ = sample 2 size ${\displaystyle \overline{x}}$ =sample mean ${\displaystyle {\mu }_0}$ = hypothesizedpopulation mean ${\displaystyle {\mu }_1}$ = population 1 mean ${\displaystyle {\mu }_2}$ = population 2 mean ${\displaystyle \sigma }$ =population standard deviation ${\displaystyle {\sigma }^2}$ =population variance ${\displaystyle s}$ =sample standard deviation ${\displaystyle \sum ^k}$ = sum (of$k$ numbers) ${\displaystyle s^2}$ =sample variance ${\displaystyle s_1}$ = sample 1 standard deviation ${\displaystyle s_2}$ = sample 2 standard deviation ${\displaystyle t}$ =t statistic ${\displaystyle df}$ =degrees of freedom ${\displaystyle \overline{d}}$ = sample mean of differences ${\displaystyle d_0}$ = hypothesized population mean difference ${\displaystyle s_d}$ = standard deviation of differences ${\displaystyle {\chi }^2}$ =Chi-squared statistic ${\displaystyle \hat{p}=\frac{x}{n}}$ = sampleproportion, unless specified otherwise ${\displaystyle p_0}$ = hypothesized population proportion ${\displaystyle p_1}$ = proportion 1 ${\displaystyle p_2}$ = proportion 2 ${\displaystyle d_p}$ = hypothesized difference in proportion ${\displaystyle min\{n_1,n_2\}}$ = minimum of$n_1$ and$n_2$ ${\displaystyle x_1=n_1{p}_1}$ ${\displaystyle x_2=n_2{p}_2}$ ${\displaystyle F}$ =F statistic

• Null distribution
• Likelihood-ratio test
• Neyman–Pearson lemma
• ${\displaystyle R^2}$ =coefficient of determination
• Sufficiency (statistics)

• Statistical hypothesis testing
• Sample statistics

• Articles with short description
• Short description matches Wikidata