Mean Deviation and Standard Deviation · Hook

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Ever wondered which sprinter is the more consistent runner?

Kemi and Zainab both average 13 seconds for the 100 m at inter-house sports. But Kemi's times are always close to 13 s, while Zainab's jump between 11 s and 15 s. The average alone can't show the difference — a measure of spread can.

Revision notes

📒 Mean Deviation and Standard Deviation: lesson notes

Measure how spread out data is with the range, mean deviation, variance and standard deviation.

Kemi and Zainab both average 13 seconds for the 100 m at inter-house sports. But Kemi's times are always close to 13 s, while Zainab's jump between 11 s and 15 s. The average alone can't show the difference — a measure of spread can.

Deviations from the mean

The range (highest − lowest) is quick, but it only uses two values.

A better idea is to see how far each value is from the mean x̄. This distance, x − x̄, is its deviation.

The deviations always add up to 0 (the positives and negatives cancel), so we must make them all positive — either by ignoring the sign or by squaring.

🔑 Key idea: Deviation = x − x̄. Their plain sum is always 0.

Mean deviation

Mean deviation = the average of the deviations ignoring signs: Σ|x − x̄| ÷ n.

For 2, 4, 6, 8, 10: x̄ = 30 ÷ 5 = 6. The deviations without signs are 4, 2, 0, 2, 4, which add to 12.

Mean deviation = 12 ÷ 5 = 2.4.

🔑 Key idea: Mean deviation = Σ|x − x̄| ÷ n.

Variance and standard deviation

Square each deviation instead: for 2, 4, 6, 8, 10 the squares are 16, 4, 0, 4, 16, which add to 40.

The variance is their mean: 40 ÷ 5 = 8. The standard deviation is its square root: √8 ≈ 2.83. Squaring and then square-rooting puts the answer back in the original units.

Adding the same number to every value does not change the SD (everything just shifts). Multiplying every value by k multiplies the SD by k.

🔑 Key idea: Variance = Σ(x − x̄)² ÷ n; SD = √variance.

Worked example

Find the mean deviation of 18, 10, 12, 22, 18, 16. (You may give a fraction or a decimal.)

  1. Step 1: Find the mean

    x̄ = (18 + 10 + 12 + 22 + 18 + 16) ÷ 6 = 96 ÷ 6 = 16

  2. Step 2: Distances from the mean |x − x̄|

    2, 6, 4, 6, 2, 0

    Why? Distance is never negative, so drop the minus signs.

  3. Step 3: Find their mean

    (2 + 6 + 4 + 6 + 2 + 0) ÷ 6 = 20 ÷ 6 = 103

Answer: 103

Summary

  • Deviation = x − x̄; the plain deviations always add to 0.
  • Mean deviation = Σ|x − x̄| ÷ n.
  • Variance = Σ(x − x̄)² ÷ n and standard deviation = √variance.
  • A small SD means the data is close to the mean — more consistent.
  • Adding a constant leaves the SD unchanged; multiplying by k multiplies it by k.

Ready to check you've got it? Scroll up and work through the lesson: the interactive visuals, practice questions and mastery check are all there.