Test whether the mean of a population is zero:

The full test table:

Test whether the means of two populations differ by 2:

The mean difference :

At the 0.05 level, is significantly different from 2:

Compare the locations of multivariate populations:

The mean difference vector :

At the 0.05 level, is not significantly different from {1,2}:

Test whether the means of some populations are equal:

The means of the first two populations are similar:

The mean of the third population is different from the first:

The "third series" of measurements of the passage time of light was recorded by Newcomb in 1882. The given values divided by 1000 plus 24 give the time in millionths of a second for light to traverse a known distance. The true value is now considered to be 33.02:

Use Chauvenet's criterion to identify outlying observations:

A -test on the bulk of the data suggests that Newcomb's measure of the speed of light was significantly lower than reality:

The vitamin C content and head weight were recorded for 30 samples from each of two experimental cabbage cultivars:

Plots of the head weight and vitamin C content by cultivar:

The vitamin C content is significantly higher for the c52 cultivar:

The weight data is not normally distributed for c52, so MannWhitneyTest is used to show that a significantly lighter cabbage produced significantly more vitamin C:

Fifty samples from each of three species of iris flowers were collected. The samples consist of measures of the length and width of the irises' sepals and petals. It is difficult to distinguish the species virginica and versicolor from one another:

A Hotelling test suggests a difference in the measures for the two similar species:

A visualization of the data suggests this difference is most prominent in the petal dimensions:

For univariate data, the test statistic follows StudentTDistribution under :

For multivariate data, the test statistic follows HotellingTSquareDistribution under :

The degrees of freedom are data-dependent for univariate data:

One sample:

Two samples with equal variances:

Two samples with unequal variances (Satterthwaite approximation):

The type of degrees of freedom used can be controlled using VerifyTestAssumptions:

Explicitly assume equal variances and test for normality:

Explicitly assume unequal variances to use the Satterthwaite approximation:

For multivariate data, the squared Mahalanobis distance is used to compute Hotelling's statistic:

Under , the test statistic follows HotellingTSquareDistribution[p,n-1]:

If the population variance is known, the more powerful ZTest can be used:

ZTest correctly rejects more frequently than TTest:

TTest is robust to mild deviations from normality:

The -value can still be interpreted in the usual way:

Large deviations from normality require the use of median-based tests:

The -value can be interpreted in the usual way for SignedRankTest but not TTest:

For two-sample testing of non-normal data, use MannWhitneyTest:

For non-normal data, MannWhitneyTest can be more powerful than TTest:

TTest works with the values only when the input is a TimeSeries:

TTest works with all the values together when the input is a TemporalData:

Test all the values only:

Test whether the means of the two paths are equal:

TTest assumes that the data is normally distributed:

Use a median-based test that does not assume normality:

The covariance matrix of multivariate data may not be invertible:

Compute the statistic when the null hypothesis is true:

The test statistic given a particular alternative:

Compare the distributions of the test statistics: