How a New Value Changes Mean and Median

The mean uses every value, so it moves toward any value added. Adding a number above the current mean raises it, adding one below lowers it, and adding a number exactly equal to the mean leaves it unchanged. Think in terms of the total. If a set of 5 numbers has a mean of 12, the sum is 60. Adding a sixth value of 24 makes the sum 84 and the new mean 14.

The median depends only on position, so it barely notices extreme values. Take the set 2, 4, 6, 8, 10, where the median is 6. Replacing the 10 with 500 does not change the median at all, because 6 is still the middle value. The mean, meanwhile, jumps from 6 to 104. That contrast is the point of most of these questions.

This is why outlier questions favor the median. When a data set has one very large or very small value, the mean gets dragged toward it while the median stays near the bulk of the data. So a question asking which measure better represents a typical value in a skewed data set is pointing at the median.

For a data set with an even count, the median is the average of the two middle values once the list is ordered. In 3, 7, 9, 15, the median is 8, even though 8 is not in the list. Adding one more value makes the count odd and moves the median to an actual data point, which is why adding a single value can shift the median even when the new number is extreme.

Two habits prevent most errors. Order the list from least to greatest before you locate any median, since the question rarely presents the numbers in order. And when a frequency table is involved, expand it mentally into the full list, because the middle row of the table is usually not the middle value of the data.