scipy.stats.ttest_1samp#

scipy.stats.ttest_1samp(a, popmean, axis=0, nan_policy='propagate', alternative='two-sided', *, keepdims=False)[source]#

Calculate the T-test for the mean of ONE group of scores.

This is a test for the null hypothesis that the expected value (mean) of a sample of independent observations a is equal to the given population mean, popmean.

Parameters:

aarray_like

Sample observations.

popmeanfloat or array_like

Expected value in null hypothesis. If array_like, then its length along axis must equal 1, and it must otherwise be broadcastable with a.

axisint or None, default: 0

If an int, the axis of the input along which to compute the statistic. The statistic of each axis-slice (e.g. row) of the input will appear in a corresponding element of the output. If None, the input will be raveled before computing the statistic.

nan_policy{‘propagate’, ‘omit’, ‘raise’}

Defines how to handle input NaNs.

propagate: if a NaN is present in the axis slice (e.g. row) along which the statistic is computed, the corresponding entry of the output will be NaN.
omit: NaNs will be omitted when performing the calculation. If insufficient data remains in the axis slice along which the statistic is computed, the corresponding entry of the output will be NaN.
raise: if a NaN is present, a ValueError will be raised.

alternative{‘two-sided’, ‘less’, ‘greater’}, optional

Defines the alternative hypothesis. The following options are available (default is ‘two-sided’):

‘two-sided’: the mean of the underlying distribution of the sample is different than the given population mean (popmean)
‘less’: the mean of the underlying distribution of the sample is less than the given population mean (popmean)
‘greater’: the mean of the underlying distribution of the sample is greater than the given population mean (popmean)

keepdimsbool, default: False

If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array.

Returns:

resultTtestResult

An object with the following attributes:

statisticfloat or array: The t-statistic.
pvaluefloat or array: The p-value associated with the given alternative.
dffloat or array: The number of degrees of freedom used in calculation of the t-statistic; this is one less than the size of the sample (a.shape[axis]).

Added in version 1.10.0.

The object also has the following method:

confidence_interval(confidence_level=0.95): Computes a confidence interval around the population mean for the given confidence level. The confidence interval is returned in a namedtuple with fields low and high.

Added in version 1.10.0.

Notes

The statistic is calculated as (np.mean(a) - popmean)/se, where se is the standard error. Therefore, the statistic will be positive when the sample mean is greater than the population mean and negative when the sample mean is less than the population mean.

Beginning in SciPy 1.9, np.matrix inputs (not recommended for new code) are converted to np.ndarray before the calculation is performed. In this case, the output will be a scalar or np.ndarray of appropriate shape rather than a 2D np.matrix. Similarly, while masked elements of masked arrays are ignored, the output will be a scalar or np.ndarray rather than a masked array with mask=False.

Examples

Suppose we wish to test the null hypothesis that the mean of a population is equal to 0.5. We choose a confidence level of 99%; that is, we will reject the null hypothesis in favor of the alternative if the p-value is less than 0.01.

When testing random variates from the standard uniform distribution, which has a mean of 0.5, we expect the data to be consistent with the null hypothesis most of the time.

>>> import numpy as np
>>> from scipy import stats
>>> rng = np.random.default_rng()
>>> rvs = stats.uniform.rvs(size=50, random_state=rng)
>>> stats.ttest_1samp(rvs, popmean=0.5)
TtestResult(statistic=2.456308468440, pvalue=0.017628209047638, df=49)

As expected, the p-value of 0.017 is not below our threshold of 0.01, so we cannot reject the null hypothesis.

When testing data from the standard normal distribution, which has a mean of 0, we would expect the null hypothesis to be rejected.

>>> rvs = stats.norm.rvs(size=50, random_state=rng)
>>> stats.ttest_1samp(rvs, popmean=0.5)
TtestResult(statistic=-7.433605518875, pvalue=1.416760157221e-09, df=49)

Indeed, the p-value is lower than our threshold of 0.01, so we reject the null hypothesis in favor of the default “two-sided” alternative: the mean of the population is not equal to 0.5.

However, suppose we were to test the null hypothesis against the one-sided alternative that the mean of the population is greater than 0.5. Since the mean of the standard normal is less than 0.5, we would not expect the null hypothesis to be rejected.

>>> stats.ttest_1samp(rvs, popmean=0.5, alternative='greater')
TtestResult(statistic=-7.433605518875, pvalue=0.99999999929, df=49)

Unsurprisingly, with a p-value greater than our threshold, we would not reject the null hypothesis.

Note that when working with a confidence level of 99%, a true null hypothesis will be rejected approximately 1% of the time.

>>> rvs = stats.uniform.rvs(size=(100, 50), random_state=rng)
>>> res = stats.ttest_1samp(rvs, popmean=0.5, axis=1)
>>> np.sum(res.pvalue < 0.01)
1

Indeed, even though all 100 samples above were drawn from the standard uniform distribution, which does have a population mean of 0.5, we would mistakenly reject the null hypothesis for one of them.

ttest_1samp can also compute a confidence interval around the population mean.

>>> rvs = stats.norm.rvs(size=50, random_state=rng)
>>> res = stats.ttest_1samp(rvs, popmean=0)
>>> ci = res.confidence_interval(confidence_level=0.95)
>>> ci
ConfidenceInterval(low=-0.3193887540880017, high=0.2898583388980972)

The bounds of the 95% confidence interval are the minimum and maximum values of the parameter popmean for which the p-value of the test would be 0.05.

>>> res = stats.ttest_1samp(rvs, popmean=ci.low)
>>> np.testing.assert_allclose(res.pvalue, 0.05)
>>> res = stats.ttest_1samp(rvs, popmean=ci.high)
>>> np.testing.assert_allclose(res.pvalue, 0.05)

Under certain assumptions about the population from which a sample is drawn, the confidence interval with confidence level 95% is expected to contain the true population mean in 95% of sample replications.

>>> rvs = stats.norm.rvs(size=(50, 1000), loc=1, random_state=rng)
>>> res = stats.ttest_1samp(rvs, popmean=0)
>>> ci = res.confidence_interval()
>>> contains_pop_mean = (ci.low < 1) & (ci.high > 1)
>>> contains_pop_mean.sum()
953