Standard Deviation Calculator

Calculate population and sample standard deviation for any dataset

Method and source check

Last checked: Jul 31, 2026

Source and implementation check; this is not professional review or advice.

Method

The calculator finds the mean and squared deviations, then reports both population standard deviation (divide by N) and sample standard deviation (divide by n − 1).

σ = √(Σ(xᵢ − μ)² / N)

s = √(Σ(xᵢ − x̄)² / (n − 1))

Variables and units

xᵢ: each observed value
μ / x̄: population mean / sample mean
N / n: population size / sample size

Worked example

For 2, 4, 4, 4, 5, 5, 7, 9: the mean is 5, population standard deviation is 2, and sample standard deviation is about 2.1381.

Limitations

Use the population result only when the values are the complete population. Use the sample result when the values are a sample; neither result proves that data are normally distributed.

Sources

NIST/SEMATECH e-Handbook: Standard Deviation — National Institute of Standards and Technology(opens in a new tab)

Table of Contents

How to Use

Enter your dataset as numbers separated by commas, spaces, or semicolons
Click calculate to compute the standard deviation
View both population (σ) and sample (s) standard deviation results
Review the deviations and squared deviations for each data point

What is Standard Deviation?

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a dataset. It tells you how spread out the numbers are from the mean (average). A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.

Standard deviation is one of the most commonly used measures of variability in statistics and is essential for understanding data distribution, making predictions, and conducting statistical tests.

Population vs Sample Standard Deviation

There are two types of standard deviation, depending on whether you're working with the entire population or a sample:

**Population Standard Deviation (σ)**: Used when you have data for the entire population. The variance is calculated by dividing the sum of squared deviations by N (the number of data points).
**Sample Standard Deviation (s)**: Used when you have data from a sample of the population. The variance is calculated by dividing the sum of squared deviations by N-1 (Bessel's correction), which provides an unbiased estimate of the population variance.

How to Calculate Standard Deviation

The standard deviation is calculated using the following steps:

Calculate the mean (average) of all data points
Subtract the mean from each data point to get the deviation
Square each deviation
Calculate the variance by averaging the squared deviations (divide by N for population, N-1 for sample)
Take the square root of the variance to get the standard deviation

**Formula for Population Standard Deviation:** σ = √(Σ(x - μ)² / N)

**Formula for Sample Standard Deviation:** s = √(Σ(x - x̄)² / (N - 1))

Where: σ (sigma) is population standard deviation, s is sample standard deviation, x is each data point, μ (mu) is population mean, x̄ (x-bar) is sample mean, N is the number of data points, and Σ (sigma) means sum.

Interpreting Standard Deviation

Understanding what standard deviation tells you about your data:

**Small Standard Deviation**: Data points are clustered close to the mean, indicating consistency and low variability
**Large Standard Deviation**: Data points are spread out over a wide range, indicating high variability or diversity
**Zero Standard Deviation**: All data points are identical (no variation)
**Empirical Rule (68-95-99.7)**: In a normal distribution, approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations

Common Applications

Standard deviation is widely used across many fields:

**Finance**: Measuring investment risk and portfolio volatility
**Quality Control**: Monitoring manufacturing processes and product consistency
**Research**: Analyzing experimental data and testing hypotheses
**Education**: Evaluating test score distributions and student performance
**Weather**: Assessing temperature variability and climate patterns
**Healthcare**: Analyzing patient data and treatment outcomes
**Sports**: Evaluating player performance consistency

Relationship with Variance

Variance and standard deviation are closely related measures of dispersion. Variance is the average of the squared deviations from the mean, while standard deviation is the square root of variance.

Standard deviation is often preferred over variance because it's expressed in the same units as the original data, making it more interpretable. For example, if you're measuring heights in centimeters, the standard deviation will also be in centimeters, while variance would be in square centimeters.

Frequently Asked Questions

What's the difference between population and sample standard deviation?: Population standard deviation (σ) is used when you have data for the entire population and divides by N. Sample standard deviation (s) is used for a subset of the population and divides by N-1 (Bessel's correction) to provide an unbiased estimate. Use sample standard deviation when working with a sample to make inferences about the larger population.
Why is standard deviation more useful than variance?: While both measure dispersion, standard deviation is in the same units as the original data, making it more intuitive to interpret. For example, if measuring test scores (0-100), a standard deviation of 15 points is easier to understand than a variance of 225 points².
What does a high standard deviation indicate?: A high standard deviation indicates that data points are spread out over a wide range of values, showing high variability or diversity in the dataset. This could mean inconsistent data, diverse samples, or a wide distribution of values.
Can standard deviation be negative?: No, standard deviation is always zero or positive. It's calculated as the square root of variance, which is itself the average of squared values. A standard deviation of zero means all values in the dataset are identical.
What is the empirical rule for standard deviation?: The empirical rule (68-95-99.7 rule) applies to normal distributions: approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This helps you understand how data is distributed around the mean.

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