Softonic

Standard Deviation Calculator

Step 1 / 4 Standard deviation

Paste a list of numbers and the calculator returns the mean, variance, standard deviation (both sample s with n−1 denominator and population σ with n denominator), the coefficient of variation, and z-scores for each value. Handy when you want to know how dispersed your data is around its average — a key diagnostic before running any parametric test.

How standard deviation is computed

  1. 1

    Paste your numbers

    Separated by commas, spaces or newlines. Non-numeric entries are skipped.

  2. 2

    Mean x-bar is computed

    Sum divided by count.

  3. 3

    Squared deviations are summed

    sum((x − x-bar)²).

  4. 4

    Divide and take the root

    Sample: divide by (n−1), take √. Population: divide by n, take √.

Sample vs population — when to use which

Use population (n divisor) Use sample (n−1 divisor)
You have the entire population You have a sample drawn from a larger population
Full census of employees 20 customers sampled from thousands
All 10 rolls of a die in a specific session Measurements from a production line

The n−1 divisor (Bessel’s correction) produces an unbiased estimator of the population variance from sample data. With n as divisor you systematically underestimate the true population variance. For large n the difference shrinks but it matters at small sample sizes.

Standard deviation intuition

If a set has mean 100 and SD 15, then (assuming roughly normal distribution):

That is the 68-95-99.7 rule, also called the empirical rule. IQ scores, human heights, and many natural measurements follow it closely.

Coefficient of variation

CV = SD / mean. A unitless measure of dispersion — useful when comparing variability across datasets with different means. A CV of 0.1 (10%) means the SD is 10% of the mean, roughly. Not meaningful for data that can cross zero.

Z-scores

For each value x: z = (x − mean) / SD. Tells you how many SDs above or below the mean that value sits. |z| > 2 is often flagged as outlier-ish; |z| > 3 is quite rare in normal data.

Common mistakes

Frequently Asked Questions

Excel has two functions: STDEV (sample, n−1 denominator) and STDEVP (population, n denominator). Make sure you are using the one that matches the sample-or-population assumption you want.

Yes — SD has the same units as your measurements (cm, dollars, seconds). Variance is in squared units, which is why SD is more readable.

Sample SD is defined for n >= 2. Below roughly n=30, consider reporting confidence intervals around the SD or using a robust alternative.

SD is still defined. For a proportion p, SD = sqrt(p × (1−p)). A sample with 60% ones has SD = sqrt(0.6 × 0.4) ~= 0.49 regardless of how many observations.

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