Confidence Interval for a Standard Deviation

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A confidence interval for a standard deviation is a range of values that is likely to contain a population standard deviation with a certain level of confidence.

This tutorial explains the following:

• The motivation for creating this confidence interval.
• The formula to create this confidence interval.
• An example of how to calculate this confidence interval.
• How to interpret this confidence interval.

Confidence Interval for a Standard Deviation: Motivation

The reason to create a confidence interval for a standard deviation is because we want to capture our uncertainty when estimating a population standard deviation.

For example, suppose we want to estimate the standard deviation of weight of a certain species of turtle in Florida. Since there are thousands of turtles in Florida, it would be extremely time-consuming and costly to go around and weigh each individual turtle.

Instead, we might take a simple random sample of 50 turtles and use the standard deviation of weight of the turtles in this sample to estimate the true population standard deviation:

The problem is that the standard deviation in the sample is not guaranteed to exactly match the standard deviation in the whole population. So, to capture this uncertainty we can create a confidence interval that contains a range of values that are likely to contain the true standard deviation in the population.

Confidence Interval for a Standard Deviation: Formula

We use the following formula to calculate a confidence interval for a mean:

Confidence Interval = [√(n-1)s²/X²_α/2, √(n-1)s²/X²_1-α/2]

where:

• n: sample size
• s: sample standard deviation
• X²: Chi-square critical value with n-1 degrees of freedom.

Confidence Interval for a Standard Deviation: Example

Suppose we collect a random sample of turtles with the following information:

• Sample size n = 27
• Sample standard deviation s = 6.43

Here is how to find various confidence intervals for the true population standard deviation:

90% Confidence Interval: [√(27-1)*6.43²/38.885,   √(27-1)*6.43²/15.379) = [5.258, 8.361]

95% Confidence Interval: [√(27-1)*6.43²/41.923,   √(27-1)*6.43²/13.844) = [5.064, 8.812]

99% Confidence Interval: [√(27-1)*6.43²/48.289,   √(27-1)*6.43²/11.160) = [4.718, 9.814]

Note: You can also find these confidence intervals by using the Confidence Interval for a Standard Deviation Calculator.

Confidence Interval for a Standard Deviation: Interpretation

The way we would interpret a confidence interval is as follows:

There is a 95% chance that the confidence interval of [5.064, 8.812] contains the true population standard deviation.

Another way of saying the same thing is that there is only a 5% chance that the true population standard deviation lies outside of the 95% confidence interval. That is, there’s only a 5% chance that the true population standard deviation is greater than 8.812 or less than 5.064.