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,
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,
\[{\hat{\sigma}}^2 = {1 \over (N1)} \sum (x_i  {\hat{\mu}})^2\]
where \(x_i\) are the elements of the dataset data. Note that the normalization factor of \(1/(N1)\) 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,

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()
.

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 \(\mu\),

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.