Standard Deviation Calculator

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Individual Deviations

Understanding Standard Deviation

What is Standard Deviation?

Standard deviation (σ or s) measures the spread or dispersion of data points around the mean. A low standard deviation means data points are clustered close to the mean, while a high standard deviation indicates data is more spread out.

Population vs Sample Standard Deviation

Sample standard deviation (s) divides by (n-1), which corrects for bias when estimating population variance from a sample. Use this when your data is a subset of a larger population—which is almost always the case in real-world scenarios.

Population standard deviation (σ) divides by n. Use this only when you have data for the entire population.

Variance

Variance is simply the square of standard deviation. It represents the average squared deviation from the mean.

Coefficient of Variation (CV)

The CV expresses standard deviation as a percentage of the mean, allowing comparison of variability between datasets with different units or scales.

Standard Error (SE)

Standard error estimates how much sample means would vary if you repeated the sampling. It decreases as sample size increases.

Interpreting Standard Deviation

• Empirical Rule (68-95-99.7): For normally distributed data, ~68% falls within 1σ, ~95% within 2σ, and ~99.7% within 3σ of the mean
• Low CV (<15%): Data points are relatively consistent
• High CV (>30%): High variability in the data

For a deeper walkthrough with worked examples, see our guide: Standard Deviation Explained — Step by Step.

Frequently Asked Questions

What is standard deviation?

Standard deviation is a measure of how spread out values are from their average (mean). A small standard deviation means data points cluster tightly around the mean. A large one means values are widely scattered.

What is the difference between population and sample standard deviation?

Population standard deviation (σ) divides by n and is used when you have data for every member of a group. Sample standard deviation (s) divides by (n − 1) and is used when your data is a subset of a larger population. Most real-world data is a sample, so (n − 1) is the safer default.

Can standard deviation be zero or negative?

Standard deviation is zero when all values in the dataset are identical. It can never be negative because the formula squares every deviation and then takes a square root.

How do outliers affect standard deviation?

Outliers have a large impact because deviations are squared. A single extreme value can inflate the standard deviation significantly. If your data has outliers, consider also reporting the interquartile range (IQR).

What is the relationship between variance and standard deviation?

Variance is the square of standard deviation: Variance = σ². Standard deviation is in the same units as the original data, making it easier to interpret. Variance is useful in further statistical calculations.

What does the coefficient of variation (CV) tell me?

CV expresses standard deviation as a percentage of the mean (CV = σ/x̄ × 100%). It lets you compare variability between datasets with different units or scales. A CV below 15% generally indicates low variability.

How do I calculate standard deviation in Excel or Google Sheets?

Use =STDEV.S(range) for sample standard deviation or =STDEV.P(range) for population standard deviation. Replace "range" with your cell range, such as A1:A20.

Why divide by (n − 1) instead of n for samples?

This is called Bessel's correction. The sample mean is estimated from the same data, which "uses up" one degree of freedom. Dividing by (n − 1) produces an unbiased estimate of the true population variance.

What is standard error and how is it different?

Standard deviation measures the spread of individual data points. Standard error (SE = σ/√n) measures how much the sample mean would vary across repeated samples. Standard error decreases as sample size grows; standard deviation does not.

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