# Standard Deviation Calculator

Calculate standard deviation and variance for any data set. Supports both population and sample standard deviation. Free online statistics calculator.

## What this calculates

Calculate the standard deviation and variance of any data set with this free online tool. Enter your numbers separated by commas, choose between population and sample calculation, and get instant results including the coefficient of variation.

## Inputs

- **Data Set** — Enter numbers separated by commas. Example: 10, 12, 23, 23, 16, 23, 21, 16
- **Calculation Type** — options: Population (divide by N), Sample (divide by N-1) — Use Population if data represents the entire group. Use Sample if data is a subset of a larger group.

## Outputs

- **Standard Deviation** — The square root of the variance, measuring data spread.
- **Variance** — The average of squared deviations from the mean.
- **Mean** — The arithmetic average of the data set.
- **Count (N)** — The number of data points.
- **Sum of Squared Deviations** — The total of (each value - mean)².
- **Coefficient of Variation** — Standard deviation as a percentage of the mean.

## Details

Standard deviation is the most widely used measure of statistical dispersion, quantifying how spread out data points are from the mean.

Population vs. Sample Standard Deviation

- Population standard deviation (sigma): Used when your data set includes the entire population. Variance = Sum of squared deviations / N.

- Sample standard deviation (s): Used when your data is a sample from a larger population. Variance = Sum of squared deviations / (N - 1). The N-1 denominator (Bessel's correction) provides an unbiased estimate of the population variance.

Step-by-Step Calculation

- Calculate the mean (average) of the data set.

- Subtract the mean from each value to get the deviation.

- Square each deviation.

- Sum all the squared deviations.

- Divide by N (population) or N-1 (sample) to get the variance.

- Take the square root of the variance to get the standard deviation.

Coefficient of Variation (CV)

The CV is the standard deviation expressed as a percentage of the mean: CV = (SD / |Mean|) x 100%. It allows comparison of variability between data sets with different units or scales.

Interpreting Standard Deviation

For a normal distribution, approximately 68% of data falls within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs (the 68-95-99.7 rule, or empirical rule).

## Frequently Asked Questions

**Q: What is standard deviation?**

A: Standard deviation measures how spread out numbers are from their average (mean). A low standard deviation means the data points are clustered close to the mean, while a high standard deviation means they are spread out over a wider range. It is the square root of the variance.

**Q: What is the difference between population and sample standard deviation?**

A: Population standard deviation divides the sum of squared deviations by N (the total count), while sample standard deviation divides by N-1. Use population when your data includes every member of the group you are studying. Use sample when your data is a subset. Dividing by N-1 (Bessel's correction) compensates for the bias of estimating a population parameter from a sample.

**Q: What does the coefficient of variation tell me?**

A: The coefficient of variation (CV) expresses standard deviation as a percentage of the mean. It allows you to compare variability across data sets with different scales or units. For example, a CV of 10% means the standard deviation is 10% of the mean. A higher CV indicates more relative variability.

**Q: What is the 68-95-99.7 rule?**

A: For data that follows a normal (bell-curve) distribution, approximately 68% of values fall within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This is also called the empirical rule and is useful for identifying outliers.

**Q: Why is variance useful if standard deviation exists?**

A: Variance (the square of standard deviation) has mathematical properties that make it easier to work with in statistical formulas. Variances of independent variables add directly, which is essential for ANOVA, regression analysis, and probability theory. Standard deviation, however, is more interpretable because it is in the same units as the original data.

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Source: https://vastcalc.com/calculators/statistics/standard-deviation
Category: Statistics
Last updated: 2026-04-21