What is standard deviation?

Standard deviation measures how spread out numbers are around the mean. Population SD is denoted by σ and sample SD by s.

When should I use N or N−1?

Use N for full populations (σ). Use N−1 for samples (s) to correct bias (Bessel’s correction).

Does standard deviation assume normality?

No. It is defined for any numeric data, though many rules of thumb (68-95-99.7) assume roughly normal distributions.

How do I compute variance and SD by hand?

Compute the mean, subtract it from each value to get deviations, square them, average (divide by N or N−1), then take the square root.

Standard Deviation Calculator: Mean, Variance, σ & s with Steps

Standard Deviation Calculator - Mean, Variance & Data Spread Analysis

See how tightly your data clusters around the average. This Standard Deviation Calculator quickly measures spread so you can gauge consistency or volatility-useful for exam scores, lab results, quality control, and even investment return swings-without hand-cranking formulas.

Paste or type your numbers, choose whether you’re treating the data as a sample or a full population, and get an instant dispersion read you can compare across datasets. The goal is to turn raw values into clear insight-helping you spot outliers, benchmark stability, and make decisions with confidence.

Enter your dataset to instantly compute standard deviation, variance, and mean - with full solution steps.

Standard Deviation Results interpretation

Small SD → values cluster tightly around the mean (low spread).

Large SD → values are more dispersed (high spread or outliers).

Who it’s for: students, researchers, analysts comparing variability across groups, or anyone summarizing spread.

The familiar 68-95-99.7 rule of thumb applies only when data are roughly normal.

Prefer sample SD (s) unless you truly have the full population.

How this calculator works

We parse your numbers, compute the mean, then square each deviation from the mean. For a population we divide by N; for a sample we divide by N−1 (Bessel’s correction). The square root of that variance gives standard deviation.

Show formulas

Mean: μ = (Σxᵢ)/N

Population variance: σ² = (Σ(xᵢ−μ)²)/N

Sample variance: s² = (Σ(xᵢ−x̄)²)/(N−1)

Standard deviation: σ or s = √variance

Limitations: SD is sensitive to outliers and assumes numeric, comparable units. Skewed or multimodal data may be better summarized with robust dispersion measures (e.g., IQR).

FAQ

Is 0 a valid standard deviation?

Yes-when all values are identical.

Can I mix commas, spaces, and new lines?

Yes. The input accepts any whitespace or commas as separators.

Which should I report, σ or s?

Use s (sample SD) unless your numbers represent an entire population.

Is SD the same as standard error?

No. SD summarizes data spread; standard error summarizes uncertainty of an estimate.

How many data points do I need?

Two or more. With very small samples, SD is unstable-collect more data when possible.

What this Standard Deviation Calculator does

Our standard deviation calculator is a quick way to turn a list of numbers into clear measures of spread: mean, variance, and the standard deviation (population σ or sample s). Paste values, choose the data type, and the result updates instantly with step-by-step math. This article explains what SD means, when to use population vs sample calculations, common assumptions and caveats, and how to apply the results.

Why standard deviation matters

Two datasets can have the same mean but very different spreads. SD captures that spread in original units, which makes it easy to compare consistency across products, students, machines, or experiments. A small SD indicates values cluster near the mean; a large SD signals variability, potential outliers, or a heterogeneous process.

Population vs sample calculations

If your list contains the entire population, divide the sum of squared deviations by N to get σ². If it’s a sample, divide by N−1 to correct bias-that’s the sample variance s². The square root gives σ or s. Our calculator lets you switch types to see the impact.

Interpreting SD with the 68-95-99.7 rule

For roughly normal data, about 68% of values lie within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD. Use this only as a heuristic; real data can be skewed or heavy-tailed. Always visualize your data and check for outliers.

Limitations & robust alternatives

SD is sensitive to outliers. For skewed distributions, consider the interquartile range (IQR) or median absolute deviation (MAD). When reporting uncertainty of a mean, use the standard error or a confidence interval rather than raw SD.

How to use this calculator effectively

Clean data (consistent units), paste values, select population/sample, and review the steps. If the SD looks large, check the frequency table or a quick plot (not shown here) for outliers. Consider transformations (e.g., log) for multiplicative data before computing SD.

Standard Deviation Calculator - Mean, Variance & Data Spread Analysis

Your data (select sample or full population)

Result