Standard deviation distribution

1. Sablon:Matematika Sample standard deviation and its properties in the context of a normal distribution. Here’s a breakdown of the key points and equations you’ve provided:

Sample Standard Deviation

The sample standard deviation $s$ is calculated using the formula:

$$s=\sqrt{\frac{1}{N}\sum _{i=1}^N(x_i-\overline{x}{)}^2}$$

where: - $N$ is the number of samples, - $x_i$ are the individual sample values, - $\overline{x}$ is the sample mean.

Distribution of $s$

The distribution of the sample standard deviation $s$ for a normal population is given by:

$$f_N(s)=2{\left(\frac{N}{2{\sigma }^2}\right)}^{(N-1)/2}\frac{1}{\mathrm{\Gamma }(\frac{1}{2}(N-1))}{e}^{-\frac{N{s}^2}{2{\sigma }^2}}{s}^{N-2}$$

where: - ${\sigma }^2$ is the population variance, - $\mathrm{\Gamma }(z)$ is the gamma function.

Mean of $s$

The mean of the sample standard deviation is:

$$⟨s⟩=b(N)\sigma$$

where $b(N)$ is given by:

$$b(N)=\sqrt{\frac{2}{N}}\frac{\mathrm{\Gamma }(N/2)}{\mathrm{\Gamma }((N-1)/2)}$$

Approximation for $b(N)$

Romanovsky provided an asymptotic expansion for $b(N)$:

$$b(N)=1-\frac{3}{4N}-\frac{7}{32N^2}-\frac{9}{128N^3}+\dots$$

Raw Moments

The $r$-th raw moments of $s$ are given by:

$${{\mu }_r}^{\prime }={\left(\frac{2}{N}\right)}^{r/2}\frac{\mathrm{\Gamma }\left(\frac{N-1+r}{2}\right)}{\mathrm{\Gamma }\left(\frac{N-1}{2}\right)}{\sigma }^r$$

Variance of $s$

The variance of $s$ can be expressed as:

$$\text{var}(s)={{\mu }_2}^{\prime }-{\mu }^2=\frac{1}{N}[N-1-\frac{2{\mathrm{\Gamma }}^2(N/2)}{{\mathrm{\Gamma }}^2((N-1)/2)}]{\sigma }^2$$

Unbiased Estimator

Finally, the term $\frac{s}{b(N)}$ serves as an unbiased estimator of the population standard deviation $\sigma$.

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