Confidence Interval for Standard Deviation

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Confidence Interval for Standard Deviation: Definition, Formula, and Interpretation

What is a Confidence Interval for Standard Deviation?

A confidence interval for the standard deviation provides a range of plausible values for the population standard deviation based on a sample. It gives both an estimate of the standard deviation and a measure of the uncertainty associated with that estimate.

Formulas

Exact Confidence Interval

The confidence interval for the population standard deviation σ is given by:

\left[\sqrt{\frac{(n-1)s^2}{\chi^2_{\alpha/2, n-1}}}, \sqrt{\frac{(n-1)s^2}{\chi^2_{1-\alpha/2, n-1}}}\right]

Standard Error Approximation

For large samples, the standard error of the sample standard deviation is:

SE(s) = \frac{s}{\sqrt{2(n-1)}}

This can be used to construct approximate confidence intervals: $s \pm z_{\alpha/2} \cdot SE(s)$

Where:

$n$ is the sample size
$s$ is the sample standard deviation
$\chi^2_{(p, n - 1)}$ is the p-th percentile of the chi-square distribution with n-1 degrees of freedom
$\alpha$ is the significance level (e.g., 0.05 for a 95% confidence interval)
$z_{\alpha/2}$ is the critical value from the standard normal distribution

Note: The standard error approximation is simpler but less accurate for small samples. Use the exact confidence interval when possible, especially for n < 30.

Interpretation

A 95% confidence interval for the standard deviation means that if we were to repeat the sampling process many times and calculate the confidence interval each time, about 95% of these intervals would contain the true population standard deviation.

It's important to note that the confidence interval provides a range of plausible values for the population standard deviation, not a single point estimate. The width of the interval gives us an idea of how precise our estimate is – narrower intervals indicate more precise estimates.

Assumptions

To correctly interpret and apply the confidence interval for the standard deviation, the following assumptions should hold:

The sample is randomly selected from the population
The population is normally distributed
The observations are independent of each other

If these assumptions are violated, the confidence interval may not be reliable or interpretable. In such cases, alternative methods or transformations of the data might be necessary.

Applications

Confidence intervals for the standard deviation are useful in various fields:

Quality control: Estimating variability in manufacturing processes
Finance: Assessing the volatility of financial instruments
Biology: Measuring the variability in biological traits
Psychology: Evaluating the consistency of psychological measurements
Environmental science: Estimating the variability of pollutant concentrations