Standard Deviation Calculator

Calculate standard deviation, variance, mean, and other statistical measures for both population and sample data sets. Perfect for statistics homework and data analysis.

Enter at least 2 values. Separate with commas, spaces, or new lines.

What is Standard Deviation?

Standard deviation (σ or s) is a measure of how spread out numbers are from the mean. A low standard deviation indicates that values tend to be close to the mean, while a high standard deviation indicates that values are spread out over a wider range.

Population vs Sample

There are two types of standard deviation depending on your data:

  • Population (σ): Use when you have data for the entire population. Divide by N (total count).
  • Sample (s):Use when you have a sample from a larger population. Divide by (N-1) to correct for bias (Bessel's correction).

Formulas

Population Standard Deviation (σ)

σ = √[Σ(xᵢ - μ)² / N]

Sample Standard Deviation (s)

s = √[Σ(xᵢ - x̄)² / (n-1)]

Variance

Variance is the square of standard deviation. It represents the average of squared deviations from the mean. While mathematically useful, standard deviation is easier to interpret because it's in the same units as the original data.

Variance = (Standard Deviation)²

Coefficient of Variation (CV)

The coefficient of variation is the ratio of standard deviation to mean, expressed as a percentage. It allows comparison of variability between datasets with different units or scales.

CV = (σ / μ) × 100%

Interpreting Standard Deviation

For normally distributed data (bell curve):

  • ~68% of values fall within 1 standard deviation of the mean
  • ~95% of values fall within 2 standard deviations
  • ~99.7% of values fall within 3 standard deviations

Frequently Asked Questions

What does standard deviation tell you?
Standard deviation measures how spread out your data is from the mean. A low value means data points cluster near the average. A high value means data is widely dispersed. For normally distributed data, about 68% of values fall within 1 standard deviation of the mean.
When should I use population vs sample standard deviation?
Use population (σ) when your data includes every member of the group you are studying — for example, all students in a class. Use sample (s) when you have a subset — for example, 100 voters polled from a city of 1 million. Sample uses n-1 (Bessel's correction) to avoid underestimating variability.
What is a good coefficient of variation?
The coefficient of variation (CV) expresses standard deviation as a percentage of the mean. A CV below 15% indicates low variability, 15-30% is moderate, and above 30% is high. Lower CV values generally indicate more consistent and predictable data.

Related Tools