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Standard Deviation Calculator – Sample and Population

Paste a set of numbers to get the standard deviation, variance, mean and standard error. Choose sample or population, and see a step-by-step table of each value’s deviation from the mean and its square, so you can follow — or check — the working.

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How to use the standard deviation calculator

  1. Paste or type your data, separated by commas, spaces or new lines.
  2. Choose whether your data is a sample or the whole population.

You’ll see the standard deviation, the variance, the mean, the count, the other kind of standard deviation for comparison, the standard error and the sum of squares. For up to 100 values, a table shows each step of the calculation.

The formulas

Population: σ = √( Σ(x − μ)² ÷ N )
Sample: s = √( Σ(x − x̄)² ÷ (n − 1) )
Standard error = s ÷ √n

Worked example

Data: 2, 4, 4, 4, 5, 5, 7, 9

  1. Mean = 40 ÷ 8 = 5
  2. Deviations from the mean: −3, −1, −1, −1, 0, 0, 2, 4
  3. Squared: 9, 1, 1, 1, 0, 0, 4, 16 — sum = 32
  4. Population variance = 32 ÷ 8 = 4 → σ = 2
  5. Sample variance = 32 ÷ 7 ≈ 4.571 → s ≈ 2.138
  6. Standard error = 2.138 ÷ √8 ≈ 0.756

Interpreting standard deviation

Standard deviation is in the same units as your data. If a class’s test scores have a mean of 70 and a standard deviation of 10, a typical score is within about 10 points of 70. If another class has the same mean but a standard deviation of 3, its scores are much more tightly bunched.

For roughly bell-shaped data:

  • about 68% of values fall within 1 standard deviation of the mean,
  • about 95% within 2,
  • about 99.7% within 3.

So in the first class, about 95% of students scored between 50 and 90.

Where standard deviation is used

  • Science and engineering — measuring the precision of experiments and manufacturing.
  • Finance — volatility of investment returns is their standard deviation.
  • Education — comparing spreads of test scores and calculating z-scores.
  • Quality control — control charts flag values far from the mean.
  • Weather and sport — how consistent temperatures or player performances are.

Tips

  • Check for typing errors. A single wrong value, such as 500 instead of 50, can make the standard deviation much larger.
  • Use enough data. With very few values, the standard deviation is a rough estimate.
  • Report it with the mean, for example “70 ± 10”.

For the mean, median and mode on their own, use the average calculator. The calculation runs in your browser; your data isn’t uploaded.

Frequently asked questions

What is standard deviation?

A measure of how spread out numbers are around their mean. A small standard deviation means values sit close to the average; a large one means they are widely scattered.

Should I use sample or population standard deviation?

Use population (σ) when your data includes every member of the group you care about — for example every student in one class. Use sample (s) when your data is a sample used to estimate a bigger group, such as a survey. Sample standard deviation divides by n − 1 instead of n, which corrects for the sample tending to underestimate the spread.

What is the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9?

The mean is 5 and the squared deviations add up to 32. Population standard deviation = √(32 ÷ 8) = 2. Sample standard deviation = √(32 ÷ 7) ≈ 2.138.

What is variance?

The average of the squared deviations from the mean — standard deviation squared. Variance is in squared units, which is why standard deviation (back in the original units) is easier to interpret.

What is the standard error?

The standard error of the mean is s ÷ √n. It estimates how much the sample mean would vary between different samples, and is used for confidence intervals.

What is the 68-95-99.7 rule?

For data that follows a normal (bell-shaped) distribution, about 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three.

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