Calculate the confidence interval for a population's standard deviation given the sample standard deviation, sample size, and confidence level.

## Understanding Confidence Intervals for Standard Deviation

When calculating a confidence interval for a population's standard deviation, we use the sample standard deviation and the chi-square distribution, which depends on the sample size and the desired confidence level. The chi-square distribution is asymmetric, so we need to use different critical values for the upper and lower bounds.

### Why We Use Chi-Square Values for Upper and Lower Bounds

In a two-tailed confidence interval, the significance level \( \alpha \) is divided equally between the two tails of the chi-square distribution:

- For example, with a 95% confidence level, we want 95% of the distribution’s area to lie between the bounds, leaving a total of 5% (or \( 0.05 \)) outside the interval.
- This remaining 5% is split equally between the upper and lower tails: 2.5% in the lower tail and 2.5% in the upper tail.

Therefore:

- To find the
**upper bound**, we use the chi-square critical value that leaves 2.5% in the right tail of the distribution, corresponding to \( 1 - 0.025 = 0.975 \) of the cumulative area to the left. - To find the
**lower bound**, we use the chi-square critical value that leaves 2.5% in the left tail of the distribution, corresponding to \( 0.025 \) of the cumulative area to the left.

### Formula for Confidence Interval of Standard Deviation

The confidence interval for the population standard deviation \( \sigma \) is calculated as:

Where:

- \( n \): Sample size
- \( s \): Sample standard deviation
- \( \chi^2_{\alpha/2} \) and \( \chi^2_{1 - \alpha/2} \): Chi-square critical values for the chosen confidence level

### Real-Life Example: Measuring Machine Consistency

Suppose a manufacturer wants to verify the consistency of a machine that produces bolts by estimating the standard deviation of bolt weights. They take a sample of 30 bolts and find a sample standard deviation of 0.02 grams.

- With a 95% confidence level, we split the 5% significance level into 2.5% for each tail.
- The upper chi-square critical value corresponds to \( \chi^2_{0.975} \), and the lower to \( \chi^2_{0.025} \).
- Using these critical values, they calculate the confidence interval bounds for the standard deviation.

This interval gives the manufacturer a range within which they can be 95% confident that the true standard deviation of bolt weights lies, helping them assess the machine's consistency.

Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.