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The Student's t distribution plays a role in a number of widely used statistical analyses, including Student's t test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis.
When only a sample of data from a population is available, the term standard deviation of the sample or sample standard deviation can refer to either the above-mentioned quantity as applied to those data, or to a modified quantity that is an unbiased estimate of the population standard deviation (the standard deviation of the entire population).
This algorithm can easily be adapted to compute the variance of a finite population: simply divide by n instead of n − 1 on the last line.. Because SumSq and (Sum×Sum)/n can be very similar numbers, cancellation can lead to the precision of the result to be much less than the inherent precision of the floating-point arithmetic used to perform the computation.
A one-sample Student's t-test is a location test of whether the mean of a population has a value specified in a null hypothesis. In testing the null hypothesis that the population mean is equal to a specified value μ 0 , one uses the statistic
If () is a general scalar-valued function of a normal vector, its probability density function, cumulative distribution function, and inverse cumulative distribution function can be computed with the numerical method of ray-tracing (Matlab code). [17]
MATLAB includes an implementation of the Jarque–Bera test, the function "jbtest". Python statsmodels includes an implementation of the Jarque–Bera test, "statsmodels.stats.stattools.py". R includes implementations of the Jarque–Bera test: jarque.bera.test in the package tseries, [3] for example, and jarque.test in the package moments. [4]
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In the examples below, we will take the values given as randomly chosen from a larger population of values.. The data set [100, 100, 100] has constant values. Its standard deviation is 0 and average is 100, giving the coefficient of variation as 0 / 100 = 0