Standard Deviation Distribution — Definition, Formula & Examples

The standard deviation distribution (or sampling distribution of the standard deviation) describes how the sample standard deviation varies when you repeatedly draw random samples of the same size from a population. It shows the spread of all possible sample standard deviation values and helps you understand how reliable any single sample's standard deviation is as an estimate of the population parameter.

Given a population with standard deviation $\sigma$ , the sampling distribution of the sample standard deviation $s$ is the probability distribution of $s$ computed over all possible samples of size $n$ drawn from that population. For normally distributed populations, the related quantity $(n-1)s^2 / \sigma^2$ follows a chi-squared distribution with $n - 1$ degrees of freedom, from which the distribution of $s$ can be derived.

Key Formula

\frac{(n-1)s^2}{\sigma^2} \sim \chi^2_{n-1}

Where:

$n$ = Sample size
$s^2$ = Sample variance
$\sigma^2$ = Population variance
$\chi^2_{n-1}$ = Chi-squared distribution with n − 1 degrees of freedom

How It Works

When you take a single sample and compute its standard deviation, that value is just one estimate of the true population standard deviation. If you took many samples of the same size, each would yield a slightly different

s

. The collection of all these values forms the sampling distribution of the standard deviation. For large sample sizes, this distribution becomes approximately normal and centers near

\sigma

. For small samples drawn from a normal population, the distribution is right-skewed. The spread of this distribution shrinks as

n

increases, meaning larger samples give more precise estimates.

Worked Example

Problem: A population is normally distributed with σ = 10. You draw samples of size n = 25. Find the value of the chi-squared statistic if a particular sample has s = 12.

Step 1: Compute the sample variance.

s^2 = 12^2 = 144

Step 2: Plug into the chi-squared formula with n − 1 = 24 degrees of freedom.

\chi^2 = \frac{(25-1)(144)}{100} = \frac{24 \times 144}{100} = 34.56

Step 3: Interpret: compare 34.56 to the chi-squared distribution with 24 degrees of freedom to assess how unusual this sample standard deviation is. The critical value at the 0.05 upper tail is approximately 36.42, so s = 12 is within the typical range.

Answer: The chi-squared statistic is 34.56 with 24 degrees of freedom, indicating that a sample standard deviation of 12 is not unusually far from the population value of 10.

Why It Matters

Understanding how sample standard deviations vary is essential for constructing confidence intervals for population variance and for hypothesis testing in courses like inferential statistics. Quality control engineers use this concept to determine whether manufacturing variability has changed, by comparing observed sample standard deviations against expected distributions.

Common Mistakes

Mistake: Assuming the sampling distribution of the standard deviation is always normal.

Correction: It is right-skewed for small samples. The chi-squared relationship holds exactly only when sampling from a normal population, and approximate normality of the sampling distribution requires large n.