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.