Calculating Standard Deviation

Algebra 1 · Axiom Academy

EXAMPLE Calculating Standard Deviation A complete walkthrough: from a set of data to how spread out it is around the mean. Five students each take a quiz scored out of 10. Find the (population) standard deviation of their scores — a single number describing how far the scores typically fall from the average. What an outlier does to the spread Keep the same scores but swap the 10 for a 2 — an extreme low value. Recomputing from scratch (new mean 6.4): Key insight: one extreme value pushed the standard deviation up by about 70%. Because each deviation is squared , values far from the mean count far more — so standard deviation is sensitive to outliers. Nice work — you computed a standard deviation end to end. The pieces to keep: It measures spread: standard deviation says how far the scores typically sit from the mean. The five steps: mean → deviations → square them → average the squares (variance) → square root. Why we square: the raw deviations always sum to zero, so squaring makes them positive and weights large gaps more heavily. Why the square root: it undoes the squaring so the answer is back in the original units (here, quiz points). Population vs. sample: here we divided the sum of squares by N = 5 (population). For a sample you'd divide by N - 1 = 4 instead, giving a slightly larger value. Result: a standard deviation of points — the scores cluster fairly tightly around the average of 8.

This is the written version of the interactive lesson above. See the full Algebra 1 course.