Mean, Standard Deviation and Variance¶

gsl_stats_mean(data)¶

This function returns the arithmetic mean of data, a dataset of length n with stride stride. The arithmetic mean, or sample mean, is denoted by $\hat{\mu}$ and defined as,

\[\hat{\mu}= {1 \over N} \sum x_i\]

where $x_i$ are the elements of the dataset data. For samples drawn from a gaussian distribution the variance of $\hat{\mu}$ is $\sigma^2 / $N$.

gsl_stats_variance(data)¶

This function returns the estimated, or sample, variance of data a dataset of length n. The estimated variance is denoted by $\hat{\sigma^2}$ and is defined by,

\[{\hat{\sigma}}^2 = {1 \over (N-1)} \sum (x_i - {\hat{\mu}})^2\]

where $x_i$ are the elements of the dataset data. Note that the normalization factor of $1/(N-1)$ results from the derivation of $\hat{\sigma}^2$ as an unbiased estimator of the population variance $\sigma^2$. For samples drawn from a Gaussian distribution the variance of $\hat{\sigma}^2$ itself is $2 \sigma^4 / N$.

This function computes the mean via a call to gsl_stats_mean(). If you have already computed the mean then you can pass it directly to gsl_stats_variance_m().

gsl_stats_variance_m(data, mean)¶

This function returns the sample variance of data relative to the given value of mean. The function is computed with $\hat{\mu}$ replaced by the value of mean that you supply,

\[{\hat{\sigma}}^2 = {1 \over (N-1)} \sum (x_i - mean)^2\]

gsl_stats_sd(data)¶

gsl_stats_sd_m(data, mean)¶

The standard deviation is defined as the square root of the variance. These functions return the square root of the corresponding variance functions above.

gsl_stats_tss(data)¶

gsl_stats_tss_m(data, mean)¶

These functions return the total sum of squares(TSS) of data about the mean.For gsl_stats_tss_m() the user - supplied value of mean is used, and for gsl_stats_tss() it is computed using gsl_stats_mean().

\[{\rm TSS} = \sum(x_i - mean) ^ 2\]

gsl_stats_variance_with_fixed_mean(data, mean)¶

This function computes an unbiased estimate of the variance of data when the population mean mean of the underlying distribution is known a priori .In this case the estimator for the variance uses the factor $1/N$ and the sample mean $\hat{\mu}$ is replaced by the known population mean $\mu$,

\[{\hat{\sigma}} ^ 2 = { 1 \over N } \sum(x_i - \mu) ^ 2\]

gsl_stats_sd_with_fixed_mean(data, mean)¶

This function calculates the standard deviation of data for a fixed population mean mean. The result is the square root of the corresponding variance function.

Statistics

Absolute deviation