## Confidence Interval for Variance

Let $X_1, X_2, \cdots , X_n$ be a random sample of size $n$ from $N(\mu, \sigma^2)$.

## Formula

$100(1-\alpha)$% confidence interval estimate of population variance $\sigma^2$ is

### $\bigg(\dfrac{(n-1)s^2}{\chi^2_{(\alpha/2,n-1)}}, \dfrac{(n-1)s^2}{\chi^2_{(1-\alpha/2,n-1)}}\bigg)$

where,

• $\overline{X}=\frac{1}{n} \sum X_i$ is the sample mean,
• $s^2= \frac{1}{n-1}\sum (X_i-\overline{X})^2$ is the sample variance,
• $1-\alpha$ is the confidence coefficient,
• $\chi^2_{(1-\alpha/2,n-1)}$ and $\chi^2_{(1-\alpha/2,n-1)}$ are the table values of $\chi^2$.