Range and standard deviation

Measures of variability (or dispersion) show how spread out values are around the mean, that is, how much the group differs within itself. Two data sets can have the same mean, median or mode but very different spread. The common ones are range and standard deviation.

Range = maximum − minimum (the largest value minus the smallest).

• For the book's heights: 115 − 85 = 30 cm.
• It can be found only for numerical data, and it shows the coverage of values (salaries, marks, toy prices).
• It uses only the two extreme values, so an outlier badly distorts it.

Standard deviation (written σ, the Greek letter sigma) also measures spread, but uses all the values. It is the positive square root of the average of the squared differences of each value from the mean.

• Steps: find the mean; subtract it from each value; square each difference; find the average of the squares; take the square root.
• For the heights (mean 101.33 cm), the squared differences add up to 938; 938 ÷ 9 = 104.22; √104.22 = 10.2 cm.
• A smaller σ means values are close together; a larger σ means they are more spread out.

Choosing a technique (from the book's table):

• disparity in all employees' salaries: standard deviation;
• a class's average performance in a test: mean;
• comparing heights or incomes of residents of two cities: mean and spread together;
• the dominant value in a set, or the most popular car colour in a survey: mode.

Example (new): two batters each average 40 runs. One scored 35, 40, 45; the other 0, 40, 80. The ranges are 10 and 80. For the first, the squared differences from 40 are 25, 0, 25, average 50 ÷ 3 ≈ 16.67, so σ ≈ 4.08; for the second they are 1600, 0, 1600, average 3200 ÷ 3 ≈ 1066.7, so σ ≈ 32.7. Same mean, very different consistency.