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Standard Deviation Calculator | Compute mean, variance & std dev

Calculate the standard deviation and variance of any data set, choosing sample or population mode, with the squared deviations shown. The complete spread analysis — mean, variance, standard deviation — makes it easy to understand how consistent your data really is.

Last updated: July 2026 · Free · No sign-up required

Results

Enter values above and click Calculate to see your result instantly.

Quick reference

MetricPopulationSample
Standard deviationσs
Mean symbolμx̄
DenominatorNn − 1
Use caseAll valuesSubset estimating a larger group

functions Shows the working, not just the answer

For students and teachers, the final number is only half the value. Every math tool here exposes the formula it applied, the intermediate steps, and the rounding rule, so you can follow along, check your homework, or use the answer in a proof or report with confidence.

calculate Accurate to the spec

Calculations use 64-bit floating point with sensible rounding for the domain (currency to 2 decimals, percentages to 4 decimals, algebra to 6 significant figures). Where exact rational arithmetic matters — fractions, factorials, simplification — we use a dedicated BigNumber path so 1/3 + 1/6 returns ½, not 0.49999.

school Free for classroom use

Educators are welcome to link to any math calculator on CalcaTools from a class site, Google Classroom, or worksheet. The pages are mobile-friendly, free, ad-supported (so we can keep them free) and have no sign-up wall — students just click and use them in class or at home.

tips_and_updates Pair with the spoke articles

Below the calculator we link a small set of plain-English explainer pages — "What is a percentage?", "Why does PEMDAS matter?", and so on. They cover the underlying concept in 4–6 short paragraphs. Read those before the calculator if the topic is new, or after if you want the extra context.

Interpretation guide

Rule of thumbNormal distribution meaning
±1 SDAbout 68% of values
±2 SDAbout 95% of values
±3 SDAbout 99.7% of values

lightbulb A real-world example — analyzing test scores

Test scores for 8 students: 72, 85, 91, 67, 78, 88, 95, 82.

Result: Mean: 82.25. Sample Standard Deviation (s): 9.45. Population Standard Deviation (σ): 8.84. Variance (sample): 89.36.

What this means: A standard deviation of ~9.5 around a mean of 82 means most students scored between 73 and 92 (one σ below to above). Sample vs population SD: use sample (n-1 denominator) when your data is a subset of a larger group (these 8 students out of all the class's possible scores); use population (n denominator) when the data is the entire group of interest.

Formula & methodology

Formula: Population σ = √(Σ(x−μ)^2/N); sample s = √(Σ(x−x̄)^2/(n−1))

The Standard Deviation Calculator is built on a well-established calculation method. It uses the formula Population σ = √(Σ(x−μ)^2/N); sample s = √(Σ(x−x̄)^2/(n−1)) to turn your inputs into a reliable result. Calculates the standard deviation and variance of a data set. Enter your numbers and choose whether they represent a sample or the whole population, and the tool returns the mean, variance, standard deviation, and The steps are shown on the page so you can follow the reasoning from input to output.

Standard deviation is the square root of variance. Use population SD when your dataset is complete; use sample SD with n−1 (Bessel's correction) when estimating a larger population from a sample.

What is the Standard Deviation Calculator?

The Standard Deviation Calculator is a free, browser-based math calculator tool that helps you Calculates the standard deviation and variance of a data set. Enter your numbers and choose whether they represent a sample or the whole population, and the too. Instead of working through the math by hand or in a spreadsheet, you enter your values and the calculator returns an accurate result instantly — while still showing the formula and the steps so you can verify the reasoning. It is designed for quick everyday use: no sign-up, no installation, and everything runs locally in your browser for complete privacy.

How to use the Standard Deviation Calculator

  1. Enter the required values into the input fields.
  2. Press the calculate button — the result appears immediately, updated live as you change any value.
  3. Read the step-by-step breakdown below the result to see exactly how the calculation was performed.
  4. Use the interpretation guide to understand what the result means for your situation, and try different inputs to see how they change the outcome.

How to use the Standard Deviation Calculator

Paste or type your numbers, then choose whether they represent a sample or the entire population. The tool computes the mean, the squared deviation of each value from it, and the standard deviation: for a sample, s = √(Σ(x−x̄)² ÷ (n−1)); for a population, the divisor is N. The sample version's n−1 adjustment corrects for the fact that a sample underestimates the true spread. For eight test scores like 72, 85, 91, 67, 78, 88, 95, 82, the sample standard deviation is about 9.4 points.

Interpreting your result

The standard deviation is the average distance of values from the mean, in the original units — a test average of 82 with a standard deviation of 9.4 means most scores fall within about ±9.4 points of 82. For bell-shaped data, roughly 68% of values lie within one standard deviation of the mean and 95% within two. It is the standard measure of spread because it weights all data and is stable across samples, unlike the range, which only uses the extremes. Comparing standard deviations across groups tells you which is more consistent: the same mean with a smaller standard deviation means more predictable results.

Common mistakes to avoid

The most common error is using the population formula on a sample — dividing by n instead of n−1 understates the true variability. Second, confusing standard deviation with the mean absolute deviation: SD squares the deviations, which penalizes outliers more heavily. Third, comparing standard deviations across data sets with different means — use the coefficient of variation (SD ÷ mean) for that. Finally, forgetting that the standard deviation is zero only when all values are identical, and it cannot be negative.

Tips for best results

Use the sample option whenever your data is a subset of a larger group, which covers most real situations. Report the standard deviation alongside the mean, never alone — together they summarize the data set. For skewed data or outliers, consider the median and IQR instead, since the standard deviation is sensitive to extreme values. When pooling data from multiple sources, calculate the standard deviation on the combined set, not by averaging the individual standard deviations. Check your units: the standard deviation is in the same units as the data.

Authoritative source: Wolfram MathWorld

Frequently asked questions

How do I use the standard deviation calculator?
Enter your numbers, then choose whether they are a sample or the entire population. For a sample the tool uses s = √(Σ(x−x̄)²/(n−1)); for a population it divides by N instead. It returns the mean, variance, and standard deviation with the squared deviations shown. For example, eight test scores such as 72, 85, 91, 67, 78, 88, 95, 82 are handled in one paste.
What formula does the standard deviation calculator use?
The Standard Deviation Calculator uses the standard Population σ = √(Σ(x−μ)^2/N); sample s = √(Σ(x−x̄)^2/(n−1)). You enter your values, and the calculator applies the formula step by step, showing the working so you can verify the math and understand how the result is derived rather than trusting a black box.
Is the standard deviation calculator accurate?
The Standard Deviation Calculator applies the standard formula: Population σ = √(Σ(x−μ)^2/N); sample s = √(Σ(x−x̄)^2/(n−1)). It is tested against published worked examples before launch and uses native double-precision arithmetic, so results are accurate to typical precision for the values you enter — with no intermediate rounding that could distort the answer.
Why do my results differ from another calculator?
The Standard Deviation Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Calculates the standard deviation and variance of a data set. Enter your numbers and choose whether they represent a sample or the whole population, and the tool returns the mean, variance, standard deviation, and range, using the correct formula for each There is no limit on usage, and it works on any device.
How do I calculate a pooled standard deviation from two groups?
Pool the variances weighted by degrees of freedom: s²ₚ = ((n₁−1)s₁² + (n₂−1)s₂²) ÷ (n₁+n₂−2), then take the root. Groups of 12 (SD 4.1) and 15 (SD 3.6) pool to √((11×16.81 + 14×12.96) ÷ 25) = 3.83. It's the SD the two-sample t-test uses, and it's only appropriate when the groups' spreads are genuinely similar.
What does the Standard Deviation Calculator do?
Calculates the standard deviation and variance of a data set. Enter your numbers and choose whether they represent a sample or the whole population, and the tool returns the mean, variance, standard deviation, and range, using the correct formula for each. The calculator takes your inputs, applies the standard calculation, and returns a clear result so you can make an informed decision without doing the math by hand.
Is the Standard Deviation Calculator free and private?
Yes — the Standard Deviation Calculator. There is no sign-up, no paywall, no trial, and no limit on how many calculations you can run. CalcaTools covers its costs with non-intrusive display advertising, so the calculator itself never asks for payment or restricts any feature.
How accurate is the Standard Deviation Calculator?
The standard deviation is computed exactly from your inputs using the definitional formulas — population σ = √(Σ(x−μ)²/N) and sample s = √(Σ(x−x̄)²/(n−1)) — with full precision arithmetic. The choice of sample versus population divisor is the only assumption you control, and the tool applies it consistently. Results are correct to the displayed precision.

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