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Learn Index · Data Management and Statistics Grade 11–12

Variance and standard deviation

Measures of how spread out data are around the mean.

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Measuring spread around the mean

Two classes both average 70% on a test. In one, everyone scored 65–75. In the other, scores ranged from 30 to 100. The mean is the same; the spread isn’t. Variance and standard deviation measure it.

Step by step

Data: 4, 8, 6, 5, 7 (mean = 6)

Value x Deviation (x − mean) Squared
4 −2 4
8 2 4
6 0 0
5 −1 1
7 1 1
Sum 10
  • Variance (population): σ² = 10 ÷ 5 = 2
  • Standard deviation: σ = √2 ≈ 1.41

Why square the deviations?

Deviations above and below the mean cancel out (they always sum to 0). Squaring makes them all positive. The square root at the end returns to the original units.

Population vs sample

Formula Divide by
Population standard deviation σ √[Σ(x − μ)²/N] N
Sample standard deviation s √[Σ(x − x̄)²/(n − 1)] n − 1

Dividing by n − 1 for a sample corrects for the tendency of samples to underestimate spread. For the data above as a sample: s² = 10/4 = 2.5, s ≈ 1.58.

Interpreting

  • Small standard deviation: data clustered near the mean.
  • Large standard deviation: data spread out.
  • The units are the same as the data (cm, $, points).

Example: Two machines fill 500 mL bottles. Machine A: mean 500, σ = 2 mL. Machine B: mean 500, σ = 9 mL. Machine A is more consistent.

The normal distribution connection

For bell-shaped (normal) data:

  • About 68% of values lie within 1 σ of the mean.
  • About 95% within 2 σ.
  • About 99.7% within 3 σ.

Using technology

Calculators (σx and sx) and spreadsheets (=STDEV.P and =STDEV.S) compute these instantly. Know which one you need!

Worked examples

Example 1: Data 2, 4, 4, 4, 5, 5, 7, 9 (mean 5). Population σ? Squared deviations sum to 32 → σ² = 4 → σ = 2.

Example 2: All values are equal (e.g., 7, 7, 7). Standard deviation? 0.

Example 3: Test scores: mean 72, σ = 8. A score of 88 is how many standard deviations above the mean? (88 − 72)/8 = 2.

Common mistakes

  • Forgetting to square the deviations, or forgetting the square root at the end.
  • Mixing up σ (divide by N) and s (divide by n − 1).
  • Thinking standard deviation can be negative.

Try it yourself

  1. Data 1, 3, 5 (mean 3). Population variance and σ?
  2. Which has more spread: σ = 3 or σ = 12?
  3. Add 10 to every value in a data set. What happens to σ?
  4. Mean 50, σ = 5. About what percent of normal data lies between 40 and 60?

Answers: 1) variance 8/3 ≈ 2.67; σ ≈ 1.63 2) σ = 12 3) It stays the same. 4) about 95%

Quick recap

  • Variance = average squared deviation; standard deviation = its square root.
  • Use N for a population, n − 1 for a sample.
  • Small σ: tight data; large σ: spread out.