Measures of Spread

Algebra 2 · Axiom Academy

Range, variance, and standard deviation — three ways to measure how far a data set spreads from its mean. 1. Range: the crudest measure of spread The range is the simplest way to measure spread: subtract the smallest value from the largest. It captures the total width of the data in a single subtraction. Our data set: 10, 15, 20, 25, 30 The largest value is 30 and the smallest is 10 , so the range is 30 - 10 = 20 . To use every value, measure how far each one sits from the mean . That signed distance, , is a deviation — positive above the mean, negative below it. Deviations for 10, 15, 20, 25, 30 (mean ) 10 - 20 = -10 , 15 - 20 = -5 , 20 - 20 = 0 , 25 - 20 = 5 , 30 - 20 = 10 . Add them up: -10 - 5 + 0 + 5 + 10 = 0 . 3. Variance: the mean of the squared deviations Square every deviation so nothing cancels, then take their average . That average of the squared deviations is the variance , written . Squared deviations for 10, 15, 20, 25, 30 (-10)^2 = 100 , (-5)^2 = 25 , 0^2 = 0 , 5^2 = 25 , 10^2 = 100 . Their total is 100 + 25 + 0 + 25 + 100 = 250 ; dividing by N = 5 gives . 4. Standard deviation: back to the original units Squaring changed the units — variance is measured in square units. Take the square root to return to the data's own scale. That is the standard deviation , . , in the same units as the data. Most values sit within about 7.07 of the mean — the band .

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