Standard Deviation Calculator

What Is Standard Deviation?

Standard deviation is a measure of how spread out a set of numbers is around its mean. A smaller standard deviation means the values tend to be closer to the average, while a larger standard deviation indicates more variation.

The result is expressed in the same units as the original data. For example, if the values represent dollars, the standard deviation is also expressed in dollars.

How to Use the Standard Deviation Calculator

1. Enter your numbers separated by commas, spaces, or new lines.
2. Choose population or sample standard deviation.
3. Review the standard deviation and supporting statistics.

The calculator also shows the arithmetic mean, variance, number of values, and sum of the data set.

Population vs. Sample Standard Deviation

Population standard deviation is used when the numbers represent the complete population you want to describe. Sample standard deviation is used when the numbers are a sample taken from a larger population.

The main difference is the divisor used when calculating variance. Population variance divides the sum of squared differences by the number of values, while sample variance divides it by one less than the number of values.

Standard Deviation Formula

For a population, the standard deviation is the square root of the average squared distance from the mean.

Population SD = √(Σ(x − μ)² / N)

For a sample, the denominator is one less than the number of values.

Sample SD = √(Σ(x − x̄)² / (n − 1))

How the Calculation Works

1. Calculate the mean of the data set.
2. Subtract the mean from each value.
3. Square each difference.
4. Find the average of those squared differences to get variance.
5. Take the square root of the variance.

Worked Example

Consider the values 10, 12, 14, 16, and 18. Their mean is 14. The squared differences from the mean are 16, 4, 0, 4, and 16, which add up to 40.

For the population calculation, 40 is divided by 5, giving a variance of 8. The population standard deviation is therefore approximately 2.83.

If the same five values are treated as a sample, 40 is divided by 4. The sample variance is 10 and the sample standard deviation is approximately 3.16.

What Does a Standard Deviation Tell You?

Standard deviation describes the amount of variation in the data. It does not by itself tell you whether a data set is good, bad, normal, or unusual. Its meaning depends on the context, the units, and the distribution of the values.

Standard Deviation and the Mean

The mean is the center used by the standard deviation calculation. Standard deviation measures the typical size of the values' deviations from that center using squared differences.

Standard Deviation vs. Variance

Variance is measured in squared units, while standard deviation is the square root of variance and therefore returns to the original units. This makes standard deviation easier to interpret alongside the original values.

When Should You Use Sample Standard Deviation?

Use sample standard deviation when the data represents a sample and you want to estimate variability in the larger population. The calculation uses n − 1 in the denominator, commonly called the degrees-of-freedom adjustment.

When Should You Use Population Standard Deviation?

Use population standard deviation when your data contains every member of the population being analyzed and you are describing that complete population rather than estimating a larger one.

Frequently Asked Questions