How to Calculate Confidence Interval for Standard Deviation in Excel

Calculating a confidence interval for standard deviation in Excel helps you understand the range within which the true population standard deviation is likely to fall. This guide explains how to perform this calculation using Excel's built-in functions and provides a step-by-step calculator.

What is a Confidence Interval for Standard Deviation?

A confidence interval for standard deviation provides a range of values that is likely to contain the true population standard deviation. It's calculated based on a sample standard deviation and the sample size, with a specified confidence level (typically 90%, 95%, or 99%).

The confidence interval gives you a range of values that you can be confident contains the population standard deviation. For example, if you calculate a 95% confidence interval for standard deviation, you can be 95% confident that the true population standard deviation falls within that range.

Note: The confidence interval for standard deviation is not symmetric around the sample standard deviation like the confidence interval for the mean. The lower bound can be zero, but the upper bound will always be higher than the sample standard deviation.

Confidence Interval Formula

The confidence interval for standard deviation is calculated using the chi-square distribution. The formula for the lower and upper bounds of the confidence interval is:

Lower bound = s × √(n-1 / χ²_{α/2, n-1})

Upper bound = s × √(n-1 / χ²_{1-α/2, n-1})

Where:

s = sample standard deviation
n = sample size
χ²_{α/2, n-1} = chi-square critical value for α/2 with n-1 degrees of freedom
χ²_{1-α/2, n-1} = chi-square critical value for 1-α/2 with n-1 degrees of freedom
α = significance level (1 - confidence level)

Excel provides the CHISQ.INV.RT function to calculate the chi-square critical values.

Calculating in Excel

Step-by-Step Instructions

Enter your sample data in a column of cells.
Calculate the sample standard deviation using the STDEV.P function:

=STDEV.P(A2:A100)
Determine your confidence level (e.g., 95%).
Calculate the significance level (α) as 1 - confidence level (e.g., 0.05 for 95%).
Calculate the degrees of freedom as n-1 (where n is your sample size).
Calculate the chi-square critical values using CHISQ.INV.RT:

=CHISQ.INV.RT(α/2, degrees of freedom)

=CHISQ.INV.RT(1-α/2, degrees of freedom)
Calculate the lower and upper bounds of the confidence interval:

Lower bound = s × SQRT((n-1)/χ²_{α/2, n-1})

Upper bound = s × SQRT((n-1)/χ²_{1-α/2, n-1})

Example Excel Formulas

Step	Formula
Sample standard deviation	=STDEV.P(A2:A100)
Degrees of freedom	=COUNTA(A2:A100)-1
Chi-square critical value (lower)	=CHISQ.INV.RT(0.025, B2)
Chi-square critical value (upper)	=CHISQ.INV.RT(0.975, B2)
Lower bound	=A2*SQRT((B2-1)/C2)
Upper bound	=A2*SQRT((B2-1)/D2)

Worked Example

Let's calculate a 95% confidence interval for standard deviation for the following sample data: 10, 12, 15, 18, 20, 22, 25, 28, 30, 35.

Sample standard deviation (s) = 7.21
Sample size (n) = 10
Degrees of freedom = 9
α = 0.05 (for 95% confidence)
χ²_0.025,9 ≈ 2.70
χ²_0.975,9 ≈ 19.02
Lower bound = 7.21 × √(9/2.70) ≈ 11.36
Upper bound = 7.21 × √(9/19.02) ≈ 5.16

The 95% confidence interval for standard deviation is approximately 5.16 to 11.36.

FAQ

What is the difference between confidence interval for mean and standard deviation?

The confidence interval for the mean provides a range of values that is likely to contain the true population mean, while the confidence interval for standard deviation provides a range of values that is likely to contain the true population standard deviation. The formulas and calculations are different for each.

How do I interpret the confidence interval for standard deviation?

The confidence interval for standard deviation gives you a range of values that you can be confident contains the true population standard deviation. For example, a 95% confidence interval means you can be 95% confident that the true population standard deviation falls within that range.

What if my sample size is small?

With a small sample size, the confidence interval for standard deviation will be wider because there is more uncertainty in estimating the population standard deviation. You may need to collect more data to get a more precise estimate.

Can I calculate a confidence interval for standard deviation in Excel without using the CHISQ.INV.RT function?

No, the CHISQ.INV.RT function is required to calculate the chi-square critical values needed for the confidence interval for standard deviation. This function is available in Excel 2010 and later versions.