Which of the following four data sets has the most dispersion?

Concept. Dispersion measures how far the observations of a data set lie from their own centre; it describes spread, not the size of the numbers. Two standard measures are the range, largest value minus smallest value, and the standard deviation σ = √(Σ(x − x̄)2 / n), the root-mean-square distance of the observations from their mean x̄. The larger the range and the standard deviation, the greater the dispersion.

Application. Each set here has n = 6 observations, so find the mean, the range and the standard deviation of every set in turn.

1. 88, 91, 90, 92, 89, 91 — mean = 541/6 ≈ 90.17; range = 92 − 88 = 4; the squared deviations from the mean add up to about 10.83, so σ = √(10.83/6) ≈ 1.34.

2. 0, 1, 1, 0, −1, −2 — mean = −1/6 ≈ −0.17; range = 1 − (−2) = 3; the squared deviations from the mean add up to about 6.83, so σ = √(6.83/6) ≈ 1.07.

3. 3, 5, 2, 4, 1, 5 — mean = 20/6 ≈ 3.33; range = 5 − 1 = 4; the squared deviations from the mean add up to about 13.33, so σ = √(13.33/6) ≈ 1.49.

4. 0, 5, 8, 10, −2, −8 — mean = 13/6 ≈ 2.17; range = 10 − (−8) = 18; the squared deviations from the mean add up to about 228.83, so σ = √(228.83/6) ≈ 6.18.

Comparison. The four sets side by side:

Cross-check. Both measures point the same way: the set 0, 5, 8, 10, −2, −8 spans 18 units where the other three span 4, 3 and 4, and its standard deviation of about 6.18 is roughly four times the next largest. Notice also that 88, 91, 90, 92, 89, 91 holds by far the biggest numbers yet no reading of it lies more than about 2.2 units from its own mean, which is why large values do not by themselves mean large dispersion.

Result. The data set 0, 5, 8, 10, −2, −8 has the greatest dispersion.