Spread & standard deviation

Two bus routes take 10 minutes to school on average. Route A is on the road for about 10 minutes every day, route B sometimes takes 4 minutes and sometimes 16. The mean is the same for both, yet only one of them is something you can rely on in the morning. What the mean is missing is a measure of how widely the values scatter around it.

Same mean, different picture

Write down the travel time on five days, in minutes. For route A those are 7, 9, 10, 11, 13 and for route B 4, 8, 10, 12, 16. The mean is the sum of all values divided by how many there are, and both data sets add up to 50, so both have the same mean 50 : 5 = 10 minutes. Even so, the values of A crowd tightly around 10, while those of B fan out widely. So one single number as a summary is not enough, you need a second one for the spread.

The range

The simplest measure of spread is the range: largest value minus smallest value. For route A that is 13 − 7 = 6 minutes, for route B it is 16 − 4 = 12 minutes. The range is quick to compute but has a weakness: it uses only two values, everything in between stays invisible. The lists 5, 10, 10, 10, 15 and 5, 5, 10, 15, 15 both have mean 10 and range 10, even though the first crowds around the middle and the second sits at the edges. A single outlier also inflates the range straight away.

The typical distance from the mean

A better measure looks at every single value and asks: how far is it from the mean? For route A those deviations are −3, −1, 0, 1 and 3 minutes. Simply adding them up gets you nowhere, because the sum is always 0: whatever lies above the mean exactly balances whatever lies below it. That is why you square each deviation, since squares are never negative. The mean of the squared deviations is called the variance, and because squaring also squared the unit, you take the square root at the end. The result is the standard deviation.

Work this through once for route A. The mean is 10, the deviations are −3, −1, 0, 1, 3 and their squares are 9, 1, 0, 1, 9. Together that makes 20, divided by the 5 values that is 4, and the square root of 4 is 2. So the travel times of A typically lie about 2 minutes away from the mean. The same calculation gives B a standard deviation of 4 minutes, exactly twice the spread, even though both data sets share the same mean.