Measuring spread

Two sections of the same class both average 72 on the midterm. In one, nearly every score sits between 68 and 76. In the other, scores run from 44 to 98. The mean can't tell them apart. This lesson gives you the number that can.

Why the range isn't enough

The quickest measure of spread is the range: the largest value minus the smallest. For the second section that's 98 − 44 = 54 points. Quick, but it listens to only two values. Add one student who scored 12 and the range jumps to 86, even though the other 29 scores didn't move.

We want a measure that listens to every value. The idea is simple: measure how far each value sits from the mean, then take a kind of average of those distances.

Standard deviation, step by step

Most of the time a spreadsheet function computes it for you. Doing it once by hand shows you what the function does, which makes its answer easier to trust.

s= nΣi=1(x_i − x)²n − 1 s equals the square root of the sum of squared distances from the mean, divided by n minus 1.

s

the sample standard deviation

x_i

each value in the sample

x

the sample mean

n

how many values there are

Notice step 3. Without squaring, the distances above and below the mean cancel out and always add up to zero. Squaring makes every distance positive; the square root in step 5 undoes it, so the answer is back in points, not points squared.

Extreme values pull the standard deviation up with them. In lesson 1.5 you'll meet the interquartile rangeThe distance between the first and third quartiles: the spread of the middle half of the data., which ignores the extremes.

Practice 1 of 3Two classes both average 72. Which one has the larger standard deviation?

AScores 70, 71, 72, 73, 74 BScores 44, 60, 72, 86, 98 CNeither: equal means give equal spread DYou can't tell without the medians

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B. Its scores sit 12 to 28 points from the mean of 72; A's sit 0 to 2 points away. Its standard deviation is about 21.2 points, against A's 1.6. Equal means say nothing about spread, which is why C tempts people.

Sample or population?

The formula divides by n − 1, not n. You almost always work with a sample, and a sample's values sit a little closer to their own mean than to the true mean of everyone. Dividing by a slightly smaller number nudges the answer up to make up for it.

If you really have every value you care about (every order this year, every employee), divide by n. Spreadsheets make the choice explicit: STDEV.S for a sample, STDEV.P for a population.

Practice 2 of 3"These are the heights of 12 plants in our sample." What do you divide the sum of squared distances by?

A12 B11 C13 DThe mean height

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B. The word "sample" decides it: divide by n − 1 = 11. Dividing by 12 would be right only if these 12 plants were every plant you care about.

Reading a standard deviation

A standard deviation on its own is just a number. It becomes useful next to the mean. For data with one hump and no strong skew, about two thirds of values fall within one standard deviation of the mean, and about 95% fall within two.

So if deliveries average 30 minutes with a standard deviation of 4, most take 26 to 34 minutes, and a 42-minute delivery is unusual: it's three standard deviations out.

Practice 3 of 3Deliveries average 30 minutes with a standard deviation of 4. Which delivery time is unusual?

A27 minutes B33 minutes C42 minutes DNone of them

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C. 42 minutes is 12 minutes above the mean: three standard deviations. 27 and 33 are each within one standard deviation, where about two thirds of deliveries land.

Review the terms

Turn each card over only after you've said the meaning out loud. Getting it wrong first is part of how it sticks.

Lesson 1.4 terms

4 cards. Select a card to turn it over.

• Term Standard deviation Turn overMeaningThe typical distance between a value and the mean, in the data's own units. Back to the term
• Term Variance Turn overMeaningThe average squared distance from the mean. Its units are squared, so we report its square root. Back to the term
• Term n − 1 Turn overMeaningThe divisor for a sample's variance. It corrects a sample's habit of guessing the spread too small. Back to the term
• Term Range Turn overMeaningLargest value minus smallest. Quick, but one extreme value can double it. Back to the term