Standard Deviation Calculator

How to use this calculator

• Enter numbers separated by commas, spaces, or line breaks. Results update automatically after a short delay while you type; you can also press Calculate for an immediate run.
• Use Sort ascending to rewrite your list in order, Reset to clear, and Copy results to copy every statistic (same order as the on-page cards, plus sorted values).
• Adjust decimal places and optional thousands separators. On large screens the result cards use a four-column grid; on smaller screens they stack for easier reading.

What the results cards show

Each metric below matches a card in the tool (order may differ slightly from the theoretical derivation—SS is listed after CV for layout, but SS is the same value used in variance).

• Count (n), Sum, Mean — basic size and average of your list.
• Range, Minimum, Maximum — range = max − min; min and max come from the sorted data.
• MAD — mean absolute deviation: (1/n) Σ|xᵢ − x̄|.
• Degrees of freedom (n − 1) — the divisor used for sample variance (integer; N/A if n < 2).
• Variance & standard deviation — population (÷ n) and sample (÷ (n − 1)); SD is the square root of the matching variance.
• CV — population / sample — σ / |x̄| and s / |x̄|; N/A if the mean is 0.
• Sum of squares (SS) — Σ(xᵢ − x̄)²; ties directly to both variance formulas.
• Standard error of the mean — s / √n (N/A if n < 2).
• Sorted data — full list in ascending order for checking inputs.

How to calculate standard deviation step by step

1. Find the mean. Add all values and divide by the count n:

   x̄ = (x₁ + x₂ + … + xₙ) / n

2. Compute deviations. Subtract the mean from each observation: (xᵢ − x̄).
3. Square each deviation. Use (xᵢ − x̄)² so positive and negative differences both contribute.
4. Sum the squared deviations.

   SS = Σ (xᵢ − x̄)²

5. Divide to get variance.
   • Population variance (entire group): divide by n — σ² = SS / n.
   • Sample variance (data from a larger population): divide by n − 1 — s² = SS / (n − 1) (requires n ≥ 2).

   The denominator is the clearest difference between population (n) and sample (n − 1) variance; the standard deviation formula is the square root of whichever variance you chose.

6. Take the square root.

   σ = √(SS / n)   |   s = √(SS / (n − 1))

After σ and s are known, you can read the same SS from the calculator and verify σ² = SS/n and s² = SS/(n−1). CV and SEM build on σ, s, and x̄ as described in the results list above.

Population vs Sample Standard Deviation

Use population formulas when your list is the complete set you want to describe (every team member’s score, every item in this batch). Use sample formulas when the list is only part of a larger group and you want to generalize; dividing by (n − 1) corrects bias in the variance estimate. Academic assignments and software (e.g. VAR.P vs VAR.S) label these differently—compare both outputs above with your rubric.

Common questions (FAQ)

• What is Population vs Sample Standard Deviation?

  Population standard deviation uses every member of the group you care about. Its variance divides squared deviations by n (the count). Sample standard deviation is for data drawn from a larger population; its variance divides by (n − 1) (Bessel’s correction) so the estimate is unbiased. This tool shows population and sample variance, standard deviation, and CV side by side so you can match your textbook, Excel VAR.P / VAR.S, or course notes.

• What is the standard deviation formula?

  Start with the mean μ (or x̄). For each value xᵢ, compute the deviation (xᵢ − mean), square it, and sum those squares (SS). For population variance σ², divide SS by n; for sample variance s², divide SS by (n − 1). Standard deviation is the square root of variance: σ or s. The standard deviation formula is therefore √(variance), with the divisor n or (n − 1) chosen to match population vs sample.

• How to calculate standard deviation step by step?

  Use the numbered guide below: (1) compute the mean, (2) subtract the mean from each value, (3) square each deviation, (4) sum the squares (SS), (5) divide SS by n (population) or (n − 1) (sample) to get variance, (6) take the square root for standard deviation. The on-page Step-by-Step section walks through the same logic with symbols.

• What is Sum of Squares (SS), and why does it appear after CV on the screen?

  SS = Σ(xᵢ − x̄)² is the sum of squared deviations from the mean. It is exactly what you divide by n or (n − 1) to obtain variance. The calculator computes SS first internally; the SS card is placed after the CV cards in the layout for readability, but the numeric SS is the same quantity used in the variance formulas above.

• What are Range, MAD, CV, and degrees of freedom (n − 1) on this page?

  Range is max − min (spread of the extremes). MAD (mean absolute deviation) is (1/n) Σ|xᵢ − x̄|, a scale in the same units as your data and often less sensitive to outliers than squaring. Coefficient of variation (CV) is standard deviation divided by |mean|: σ/|x̄| for population and s/|x̄| for sample—useful for comparing relative variability when means differ. If the mean is 0, CV is undefined (N/A). Degrees of freedom for the usual sample variance is n − 1 (shown as N/A when n < 2).

• What does “Standard error of the mean (s / √n)” mean?

  The standard error of the mean measures how much the sample mean would vary if you repeated sampling. We report SEM = s / √n, where s is the sample standard deviation (SS divided by n − 1, then square root). If n < 2, s is undefined, so SEM is not shown. When your list is the full population, some authors use σ / √n with population σ; our primary SEM line follows the usual sample-based definition.

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