Sample Standard Deviation Calculator with Confidence Interval

This calculator helps you determine the sample standard deviation and its confidence interval. Sample standard deviation measures the dispersion of data points in a sample from the mean, while the confidence interval provides a range within which the true population standard deviation is likely to fall.

What is Sample Standard Deviation?

Sample standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. Unlike population standard deviation, which uses the entire population, sample standard deviation uses a subset of the population (a sample) to estimate the population's standard deviation.

The formula for sample standard deviation (s) is:

s = √[Σ(xi - x̄)² / (n - 1)] where: - xi = each individual data point - x̄ = sample mean - n = number of data points in the sample

This formula divides by (n - 1) instead of n to provide an unbiased estimate of the population standard deviation.

How to Calculate Sample Standard Deviation

Step-by-Step Calculation

Collect your sample data points.
Calculate the sample mean (x̄) by summing all data points and dividing by the number of data points (n).
For each data point, subtract the sample mean and square the result (this is the squared deviation).
Sum all the squared deviations.
Divide the sum of squared deviations by (n - 1).
Take the square root of the result to get the sample standard deviation.

Note: When calculating standard deviation for a population, you would divide by n instead of (n - 1). This calculator uses the sample formula with (n - 1) in the denominator.

Understanding Confidence Interval

A confidence interval provides a range of values that is likely to contain the true population standard deviation. The confidence level (usually 90%, 95%, or 99%) indicates the probability that the interval contains the true value.

The formula for the confidence interval for standard deviation is:

Lower bound = s × √[1 - (χ²α/2, n-1)/(n-1)] Upper bound = s × √[1 - (χ²1-α/2, n-1)/(n-1)] where: - s = sample standard deviation - χ²α/2, n-1 = critical value from chi-square distribution - α = significance level (1 - confidence level)

For example, with a 95% confidence level, α = 0.05, and you would use the chi-square critical values that leave 2.5% in each tail of the distribution.

Example Calculation

Let's calculate the sample standard deviation and confidence interval for the following sample data: 5, 7, 9, 11, 13.

Sample mean (x̄) = (5 + 7 + 9 + 11 + 13) / 5 = 45 / 5 = 9
Squared deviations:
- (5-9)² = 16
- (7-9)² = 4
- (9-9)² = 0
- (11-9)² = 4
- (13-9)² = 16
Sum of squared deviations = 16 + 4 + 0 + 4 + 16 = 40
Sample standard deviation (s) = √(40 / 4) = √10 ≈ 3.162
For a 95% confidence interval:
- Critical values: χ²0.025,4 = 0.484, χ²0.975,4 = 11.143
- Lower bound = 3.162 × √[1 - (0.484/4)] ≈ 3.162 × 0.88 ≈ 2.78
- Upper bound = 3.162 × √[1 - (11.143/4)] ≈ 3.162 × 1.33 ≈ 4.24

The 95% confidence interval for the population standard deviation is approximately 2.78 to 4.24.

FAQ

What is the difference between sample and population standard deviation?

Sample standard deviation estimates the population standard deviation using a subset of data. The key difference is in the denominator of the formula: sample uses (n-1) while population uses n. This adjustment makes the sample estimate unbiased.

How does sample size affect the confidence interval?

Larger sample sizes generally result in narrower confidence intervals because they provide more precise estimates of the population parameters. With more data, the standard error decreases, making the interval more reliable.

What does a 95% confidence interval mean?

If you were to take many samples and calculate 95% confidence intervals for each, approximately 95% of those intervals would contain the true population standard deviation. It doesn't mean there's a 95% probability that any single interval contains the true value.