How to Calculate Standard Deviation

1.What standard deviation is

Standard deviation is a typical distance from the mean. If every value were identical, the deviation would be zero. If values fan out, the number grows. It is not the range (max − min), not the mean itself, and not a claim that the data are “good” or “bad.” Variance is the square of this measure; we take the square root so the result sits in the same units as the original numbers.

2.The formula

Find the mean, subtract it from each value, square those gaps, then average and take a square root. Sample form divides by n−1 (Bessel’s correction) when the data are a sample from a larger population. Population form divides by n when the list is the whole group you care about. The n−1 divisor is slightly larger in spread for the same numbers; that is the whole difference.

\[ \begin{aligned}s &= \sqrt{\frac{1}{n-1}\sum (x_i-\bar{x})^2} \\\sigma &= \sqrt{\frac{1}{n}\sum (x_i-\mu)^2}\end{aligned} \]

3.Worked example

Take 1, 2, 2, 3, 4, 5 — the same defaults as the standard deviation calculator. The sum is 17 and n = 6, so the mean is 17/6 ≈ 2.833. Squared deviations from that mean add to 65/6 ≈ 10.833. Sample (n−1 = 5): √(10.833 / 5) ≈ 1.472. Population (n = 6): √(10.833 / 6) ≈ 1.344. Same points, two divisors, two spreads.

\[ s=\sqrt{10.833/5}\approx 1.472 \]

4.See the spread

Edit the six values and toggle sample (n−1) versus population (n). The plot places each point on a number line, marks the mean, and shades ±1 standard deviation. Watch the band widen or shrink when the divisor changes. This is a teaching widget — no comma-separated paste box, no full calculator chrome. Open the standard deviation calculator when you want the standalone tool.

5.Open the full calculator

The standard deviation calculator uses the same sample and population formulas and the same 1, 2, 2, 3, 4, 5 example, with How it works and related statistics tools on one page. Browse other statistics calculators if you need a z-score, an average, or mean–median–mode next.