# Sampe Standard deviation, what distribution does it follow?

Suppose we have $$k$$ groups of $$n$$ i.i.d observations which follow the $$N(0,1)$$ parent distribution. The sample standard deviation $$S$$ is a random variable (with $$k$$ available observations), do we know what distribution does it follow? I also have the same question when the parent distribution is $$N^2(0,1)$$, which basically means that the parent distribution is a $$\chi^2_1$$.

• I just upvoted the answer by @BruceET, but this might be of interest if you actually want the somewhat ugly distribution: stats.stackexchange.com/a/410311/247352.
– Ed V
Apr 16, 2020 at 20:29

The relevant distribution here is called the chi distribution:

$$S \sim \frac{\sigma}{\sqrt{n-1}} \cdot \text{Chi}(\text{df} = n-1).$$

Using the rules for transformations of random variables, the density function for the standard deviation is:

\begin{aligned} f_S(s) &= \text{Chi} \Bigg( \frac{\sqrt{n-1} \cdot s}{\sigma} \Bigg| \text{df} = n-1 \Bigg) \cdot \frac{\sqrt{n-1}}{\sigma} \\[6pt] &= \frac{(n-1)^{n/2}}{2^{(n-3)/2} \cdot \sigma \cdot \Gamma(\tfrac{n-1}{2})} \cdot \Big( \frac{s}{\sigma} \Big)^{n-2} \cdot \exp \Big( - \frac{n-1}{2} \cdot \frac{s^2}{\sigma^2} \Big). \\[6pt] \end{aligned}

The resulting mean and variance are:

\begin{aligned} \mathbb{E}(S) &= \sigma \cdot \sqrt{\frac{2}{n-1}} \cdot \frac{\Gamma(\tfrac{n}{2})}{\Gamma(\tfrac{n-1}{2})}, \\[6pt] \mathbb{V}(S) &= \sigma^2 \Bigg[ 1 - \frac{2}{n-1} \cdot \frac{\Gamma(\tfrac{n}{2})^2}{\Gamma(\tfrac{n-1}{2})^2} \Bigg]. \\[6pt] \end{aligned}

• Thanks for the answer. Is it possible to derive the mean and the variance of S from this p.d.f? (calculus problem) Also, what happens when the parent distribution is from a $X^2_1$ distribution? Apr 17, 2020 at 12:40
• From quality control books, I believe that the mean of S which is described by the p.d.f you mentioned above is $c_4*\sigma$ and the variance is $\sigma^2 * (1-c_4)$ but I do not know why. Does the p.d.f. above have a name? I am interested in deriving the c.d.f Apr 17, 2020 at 12:49

It is easier to discuss the distribution of the sample variance $$S^2 = \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar X)^2,$$ where $$X_i$$ are a random sample from $$\mathsf{Norm}(\mu, \sigma)$$ and $$\bar X$$ is the sample mean. In that case, $$\frac{(n-1)S^2}{\sigma^2} \sim \mathsf{Chisq}(\nu = n-1).$$ Thus, the distribution of $$S^2$$ is a multiple of $$\mathsf{Chisq}(\nu = n-1).$$ The second displayed relationship can be proved using a multivariate transformaion or by using moment generating functions. [After an appropriate linear transformation of an $$n$$-dimensional multivariate normal distribution, $$\bar X$$ has a one dimensional marginal distribution and, independently, $$S^2$$ is a function of $$n-1$$ dimensions.]

You don't say why you seek the distribution of $$S.$$ The relationship above, using $$\mathsf{Chisq}(\nu = n-1),$$ can be used to find a 95% confidence interval for $$\sigma$$ as follows.

First, a 95% CI for $$\sigma^2$$ is of the form $$\left(\frac{(n-1)S^2}{U},\frac{(n-1)S^2}{L}\right),$$ where $$L$$ and $$U$$ cut probabiltiy 0.025 from the left and right tails, respectively, of $$\mathsf{Chisq}(\nu = n-1).$$ Then, to find a 95% CI for $$\sigma,$$ take square roots of the endpoints of the above CI for $$\sigma^2.$$

Of course, the distribution of $$S$$ can be found by taking the square root of the relevant chi-squared distribution. But historically, printed tables of percentage points of chi-squared distributions have been available, so the distribution of $$S^2$$ is more commonly used.

Notes: (1) For normal data $$E(S^2_n) = \sigma^2,$$ but because of the linear nature of expectation, this equality does not survive taking square roots: $$E(S_n) = \sigma\sqrt{\frac{2}{n-1}}\Gamma(\frac{n}{2})/\Gamma(\frac{n-1}{2}) < \sigma,$$ where $$S_n$$ is the standard deviation of a sample of size $$n.$$ The bias of $$S_n$$ as an estimator of $$\sigma$$ is small; except for very small samples. For samples of moderate or large size, the bias is often ignored in practice. For example: $$E(S_5) = 0.9400\sigma$$ and $$E(S_{50}) = 0.9949\sigma.$$

n = c(5,25,50);  round(sqrt((2/(n-1)))*gamma(n/2)/gamma((n-1)/2),4)
 0.9400 0.9896 0.9949


(2) For a random sample from the standard normal distribution, the distribution of $$S_{25}$$ is simulated below:

set.seed(2020)
s = replicate( 10^6, sd(rnorm(25)) )
summary(s)
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
0.4191  0.8905  0.9862  0.9897  1.0850  1.7311
hist(s, prob=T, br=30, col="skyblue2", main="Dist'n of Sample SD") 