Mean, Standard Deviation and Variance
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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 \(\sigma^2 / $N\).
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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().
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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,
\[{\hat{\sigma}}^2 = {1 \over (N-1)} \sum (x_i - mean)^2\]
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gsl_stats_sd(data)
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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.
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gsl_stats_tss(data)
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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().
\[{\rm TSS} = \sum(x_i - mean) ^ 2\]
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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 \(\mu\),
\[{\hat{\sigma}} ^ 2 = { 1 \over N } \sum(x_i - \mu) ^ 2\]
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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.