Standard Deviation Calculator

What Is Standard Deviation?

Standard deviation is a measure of how spread out the values in a dataset are around their mean, or average.

A smaller standard deviation means that the values tend to be closer to the mean. A larger standard deviation means that the values are more spread out.

Standard deviation is commonly used in statistics, research, finance, quality control, and data analysis to describe the variability of a dataset.

Population vs. Sample Standard Deviation

The correct standard deviation formula depends on whether your dataset represents an entire population or a sample taken from a larger population.

Standard Deviation Formulas

Population Standard Deviation

Here, σ represents population standard deviation, μ is the population mean, x represents each value, and N is the number of values.

Sample Standard Deviation

In the sample formula, s represents sample standard deviation, x̄ is the sample mean, and n is the number of observations.

Example: Dataset 2, 4, 6, 8, 10

First, calculate the mean:

Mean = (2 + 4 + 6 + 8 + 10) ÷ 5 = 6

The population standard deviation for this dataset is approximately 2.83.

The sample standard deviation is approximately 3.16.

The two values differ because sample standard deviation uses n − 1 in the denominator rather than n.

Standard Deviation and Variance

Variance and standard deviation are closely related measures of spread.

Variance is calculated from the average squared distance of the values from the mean. Standard deviation is the square root of the variance.

Because standard deviation is expressed in the same units as the original data, it is often easier to interpret than variance.

How to Use the Standard Deviation Calculator

1. Enter your numerical dataset separated by commas, spaces, or line breaks.
2. Choose population or sample standard deviation.
3. Click Calculate Standard Deviation.
4. Review the mean, variance, and standard deviation.
5. Click Reset to calculate another dataset.

How to Interpret Standard Deviation

Standard deviation should generally be interpreted relative to the scale and context of the dataset.

If two datasets have the same mean, the dataset with the smaller standard deviation has values that are more tightly grouped around the mean.

A standard deviation of zero occurs when every value in the dataset is identical.

Standard deviation by itself does not indicate whether a dataset is good or bad. Its meaning depends on the data and the question being studied.

Common Standard Deviation Mistakes

• Using population standard deviation when the data is a sample from a larger population.
• Forgetting that sample standard deviation uses n − 1.
• Calculating the deviations from the wrong mean.
• Confusing variance with standard deviation.
• Rounding values too early during a manual calculation.

Frequently Asked Questions

What does standard deviation measure?

Standard deviation measures how much the values in a dataset vary or spread around the mean.

What is the difference between population and sample standard deviation?

Population standard deviation divides the sum of squared deviations by the number of values, N. Sample standard deviation divides by n − 1 because the data represents a sample from a larger population.

Can standard deviation be zero?

Yes. Standard deviation is zero when every value in the dataset is exactly the same.

What is variance?

Variance is the average squared deviation from the mean. Standard deviation is the square root of variance.

How many numbers do I need?

Enter at least two numerical values. Sample standard deviation requires at least two observations.