Mean, median and spread

Most statistics questions are not asking you to compute an average. They are asking what happens to the average when one value changes, or which measure survives an extreme value. Both are answerable without adding a long list of numbers.

The mean is a total in disguise

The mean times the count equals the sum, and that single rearrangement answers most of these questions. If five numbers have a mean of 12, their sum is 60. Four of them are 8, 10, 14 and 15, which total 47, so the fifth value is 13. Questions that give a mean before and after adding a value are the same idea done twice: compute both totals, then subtract.

The median depends on order, not on size

Sort the values first, then take the middle one. With an even count there are two middles, and the median is their average. Because it is a position rather than a sum, one extreme value does not move it. That is exactly why it is the better summary of a typical value when a data set has a few very large or very small entries, and why a question that mentions an outlier is usually steering you toward the median.

Reading the shape from the two measures

Compare the mean and the median and you can describe the shape without a graph. If the mean is larger than the median, a few large values are pulling it up, so the data is skewed to the right. If the mean is smaller, the data is skewed to the left. If they are close, the data is roughly symmetric, and either measure is fair.

Spread: range, interquartile range, standard deviation

Spread describes how far the values sit from each other, not where they sit. Range is the largest minus the smallest. The interquartile range is the third quartile minus the first, which is the width of the middle half of the data. Standard deviation is a typical distance from the mean, and on this test it is compared rather than computed: a set with values bunched near the mean has a smaller standard deviation than a set spread wide, even when both have the same mean.

• Add the same number to every value: the mean and median move, the range, interquartile range and standard deviation do not
• Multiply every value by a positive number: the mean, median and spread all scale by that number
• Remove the largest value: the mean falls, and the median may not move at all

Comparing two data sets

Questions that show two sets usually want one of two sentences. Same center and different spread, which means one set is more consistent. Same spread and different center, which means one set is uniformly higher. Read the two summaries in that order: center first, then spread. Saying which set is larger is usually the wrong question, because the interesting comparison is which one is more predictable.

How do I find a missing value when I know the mean?

Multiply the mean by the number of values to get the total, add the values you know, and subtract. The difference is the missing value.

Which measure is affected by an outlier?

The mean, and along with it the range and standard deviation. The median and the interquartile range are built from position, so an extreme value barely moves them.

Does adding 10 to every value change the standard deviation?

No. Shifting every value by the same amount moves the whole set and its mean together, so the distances between values, and therefore the standard deviation, stay the same.

What does the interquartile range measure?

The width of the middle half of the data, from the first quartile to the third. It ignores the extreme quarters entirely, which is why it is more stable than the range.

Related guides

• Probability without replacement
• Margin of error and statistical claims
• Percent change with multipliers

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