Center: Mean, Median, and Which One to Trust

In this lesson you will: Compute the mean and the median of a data set, including from a dot plot; Predict what happens to each one when a value is added, removed or changed; Decide which measure describes a given distribution honestly, and defend the choice.

The mean is where the data balances

Add the values, divide by how many there are. That is the arithmetic, and you have done it since fourth grade. The part worth adding now is the picture: the mean is the point where the data would balance if every value were a weight on a ruler.

That picture explains the mean's one weakness in a single sentence. A weight placed far out on one end tips the balance a long way, and it does not matter that it is only one weight.

The median only counts positions

Sort the list and take the middle value; with an even count, average the middle two. Notice what the median never looks at: how big the values are. It only cares about how many are on each side.

Change the largest value in a data set from 25 to 2,500 and the median does not move at all. The mean moves enormously. Neither reaction is wrong — they are answers to different questions.

Which one describes the typical case

Seven people work at a print shop. Their yearly pay, in thousands of dollars, is 31, 33, 34, 34, 36, 38 and 120. The mean is about 46.6; the median is 34.

Six of the seven people earn less than the mean. A number that almost nobody is near is a poor description of a typical worker, and the owner's $120,000 is what pushed it there. The median, 34, sits right in the middle of the six ordinary salaries.

The rule of thumb is short: when a distribution is roughly symmetric, use the mean; when it is skewed or has an extreme value, the median is the more honest summary.

Common mistake — Finding the median without sorting

Asked for the median of 27, 14, 33, 21, 46, a student counts to the middle of the list as written and answers 33.

The slip survives because the answer looks plausible. It is one of the data values, it sits inside the range, and nothing about it announces itself as wrong.

But the median is defined by position in the sorted list. Sorted, the values are 14, 21, 27, 33, 46 and the middle one is 27. The 33 was simply the number that happened to be typed third.

The same error hides inside quartile work, where students sort the list, find the median, and then forget to keep the halves in order before splitting them again.

How to catch it: Write the sorted list on its own line before you do anything else, then count in from both ends at the same time. Your two counts have to stop on the same value, or on the same neighboring pair.