Now you can measure center (mean, median) and spread (IQR, MAD), and you can draw box plots. To describe a data set well, say three things: its center, its spread and its shape.
Key idea
For data that are roughly symmetric, the mean and MAD describe it well. For data that are skewed or have outliers, use the median and IQR, because extreme values pull the mean and MAD but barely move the median and IQR.
Why it works
The mean uses the size of every value, so one value far away drags it toward itself. The median only uses the order, so moving the largest value farther away does not change which value is in the middle. Spread works the same way: the MAD averages distances, which an outlier makes bigger, while the IQR ignores the top and bottom quarters.
Worked example: An outlier
Phone-call times in minutes: 22, 24, 25, 25, 26, 28, 95. Which describes a typical call better, the mean or the median?
- Mean: $245 \div 7 = 35$.
- Median: the 4th value, 25.
- Six of the seven calls are 28 minutes or less, so 35 is not typical. The 95 pulled the mean up.
Answer: The median, 25 minutes.
Worked example: Describe the shape
Most students read 1 to 3 books, a few read 6, 8 or 10. Describe the shape.
- The values pile up on the left and trail off in a long tail to the right.
Answer: Skewed right; describe it with the median and IQR.
Common mistake
Reporting the mean for skewed data with an outlier. Quick check: compare the mean and median. If they are far apart, the data are skewed or have an outlier, and the median is the better description of a typical value.