🎲Standard Deviation

Two students average 70 over five quizzes. Are they equally good? Student A scored 68, 69, 70, 71, 72; student B scored 50, 60, 70, 80, 90. Identical means, wildly different tempers: A is steady, B rides a roller coaster. Describing that difference needs more than mean, median and mode — those report the center, not the spread. Meet the second eye: standard deviation.

Same average, different temper

Subtract the mean from every value; each difference is a deviation. A's deviations are −2, −1, 0, +1, +2, while B's are −20, −10, 0, +10, +20 — same center, ten times the swing. Unfortunately deviations always sum to zero: positives cancel negatives, and averaging them wastes the whole calculation (for A: $-2 - 1 + 0 + 1 + 2 = 0$). To let deviation survive, square it first — the sign disappears, the size stays.

Square, then average, then take the root

The recipe has three steps:

1. Square every deviation;
2. Average them to get the variance;
3. Take the square root to return to the standard deviation.

Apply it to both students:

• A: squares 4, 1, 0, 1, 4, summing to 10; variance $= 10 \div 5 = 2$; standard deviation $= \sqrt{2} \approx 1.4$;
• B: squares 400, 100, 0, 100, 400, summing to 1000; variance $= 1000 \div 5 = 200$; standard deviation $= \sqrt{200} \approx 14.1$.

The final square root restores the units: scores are measured in points, and squaring squared the units too, so the root pulls them back to points. One-sentence intuition: the standard deviation is the typical distance from the mean. A's scores stray about a point or so; B's stray by a dozen.

One full calculation

+🗑 Clear4865389

Count

7

Mean

6.14

Median

6

Mode

8

Add a few values, remove a few, and watch how wide the dot plot opens: dots huddled together mean a small standard deviation, dots flung wide mean a large one. The component computes mean, median and mode — judging "how spread out" with your own eyes is exactly what standard deviation formalizes.

Population, sample, and the bell

Two quick notes. First, calculators offer both $\sigma$ and $s$: use $\sigma$ (dividing by $n$) when your data is the entire population, and $s$ (dividing by $n-1$) when it is only a sample drawn from one — samples sit artificially close together, and the adjusted divisor compensates. Details later; pick the right key for now. Second, data like heights or test scores often piles up in a bell shape: most values crowd near the mean, thinning out toward the edges. The larger the standard deviation, the shorter and wider the bell; the smaller, the taller and narrower. For bell-shaped data, about seventy percent falls within one standard deviation of the mean, and almost everything within two. So "average height 170, standard deviation 6" already lets you picture the whole crowd. You can see distribution shapes with your own eyes in Histograms; the Range measures spread using only the two extremes; and Misusing Averages shows how one extreme value drags a mean astray.

Check yourself

✅

Quick quiz

↺ Try again

1. 1. A small standard deviation means the data?

   clusters near the meanhas a large meanhas few values

2. 2. What is the standard deviation of [2, 6]?

   248

3. 3. A dataset has variance 200. Its standard deviation is about?

   14.120020