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Standard Deviation Calculator with steps

Choose sample or population, paste your numbers and follow the calculation from the mean to the square root of the variance.

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Answer

Enter your numbers or pick an example — the answer and the steps appear here instantly.

Sample vs population standard deviation

  • Population: divide by N

    When your list is the whole group, the variance is the average squared distance from the mean: σ² = Σ(x − μ)² / N. The standard deviation σ is its square root.

  • Sample: divide by n − 1

    When the list is a sample from a bigger group, use s² = Σ(x − x̄)² / (n − 1). Dividing by one less than the count is Bessel's correction, which offsets a sample's tendency to underestimate the variance.

  • Which one to pick

    Ask whether the group is larger than your list. Survey answers from 40 of 900 students are a sample; the heights of every player on the team you are describing are a population.

  • How much it matters

    The gap shrinks as the list grows. With 8 values the sample variance is 8/7 of the population variance; with 500 values the factor is only 500/499.

Worked example: 2, 4, 4, 4, 5, 5, 7, 9

The mean is 40/8 = 5. Subtracting it from each value gives deviations −3, −1, −1, −1, 0, 0, 2 and 4, and squaring them gives 9, 1, 1, 1, 0, 0, 4 and 16, which add up to 32.

For the population version, divide by N = 8: the variance is 4 and the standard deviation is exactly 2, since 4 is a perfect square. For the sample version, divide by n − 1 = 7: the variance is 32/7 and the standard deviation is √(32/7) ≈ 2.138.

The calculator's outline follows the same path: mean, squared deviations, their sum, the division and finally the square root. When the variance is a perfect square the answer is exact; otherwise it is rounded and shown with ≈.

How to use the calculator

  1. Choose sample or population

    Sample divides by n − 1, population by N. The choice changes the variance and everything after it.

  2. Paste the data

    Use commas, spaces or line breaks between values; up to 500 numbers, including decimals and fractions.

  3. Calculate

    Read the mean, each squared deviation, the variance and the standard deviation in the result card.

  4. Ask what it means

    “Explain step by step” opens a written explanation in the AI Math Solver, useful when you want to know what the number says about your data.

Frequently asked questions

It is a typical distance between the values and their mean, in the same units as the data. A small value means the numbers cluster tightly; a large one means they are spread out.

Dividing by n − 1 instead of n makes the variance a little bigger. A sample's own mean sits closer to its values than the true population mean usually does, and the correction compensates for that.

The square root of the variance is usually irrational, like √(32/7). It is then rounded and flagged with ≈ and a Rounded tag; when the variance is a perfect square, as with 4, the result is exact.

Variance is the standard deviation squared. It is shown in the result too, but it is measured in squared units, which is why the standard deviation is usually easier to interpret.

No account or payment is needed for the calculation; it runs in your browser and sends nothing to a server. A written AI explanation is included in the Pro plans* (*fair use: up to 300 runs a day across the writing, study and code tools) or takes 2 credits without a plan, and signed-out visitors can read its opening for free.

Use the steps to understand and check your own work, and follow your instructor's policy on AI tools. Keep in mind the AI explanation can contain errors, while the calculator's numbers are computed directly from your data.

Related statistics help

Measure the spread of your data

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Calculate standard deviation
  • Sample and population
  • Squared deviations listed
  • Exact when possible