Measures of center & variability
The idea
Summarizing data well takes two numbers, not one: a center that says what is typical, and a measure of variability that says how much the values stray from typical. For center you can use the mean — the even-share value — or the median, the middle of the ordered list. For variability, the range gives the rough span, while the mean absolute deviation, MAD, averages how far values sit from the mean. Your skills at adding, dividing, and reading data plots cover all the machinery.
Think of the MAD as the typical miss: if a team's mean is 10 points with a MAD of 1.6, scores usually land within a couple of points of 10, so the team is consistent. A MAD of 6 around the same mean would describe a streaky team. The standing misconception is that the mean alone tells the story — but two players can average identical points while one is steady and the other swings wildly, and only a variability measure separates them. Also remember the median resists outliers when the mean gets dragged.
Worked example
A basketball player scores 7, 9, 10, 10, and 14 points in five games. Find the mean and the mean absolute deviation (MAD), and interpret what they say about her consistency.
- Compute the mean as the even share: 7 + 9 + 10 + 10 + 14 = 50, and 50 ÷ 5 = 10 points per game.
- Find each game's distance from the mean: |7 − 10| = 3, |9 − 10| = 1, |10 − 10| = 0, |10 − 10| = 0, and |14 − 10| = 4.
- Average those distances for the MAD: 3 + 1 + 0 + 0 + 4 = 8, and 8 ÷ 5 = 1.6 points.
- Interpret the pair of numbers: she typically scores around 10, and a typical game lands within about 1.6 points of that — a steady scorer rather than a streaky one.
Answer. Her mean is 10 points per game with a MAD of 1.6 points, indicating consistent scoring close to her average.
Check your understanding
- Why are absolute values used when computing the MAD — what would go wrong if deviations kept their signs?
- When would you summarize a data set with the median instead of the mean, and what in the data signals that choice?
- How could two players share the same mean while one has a much larger MAD, and which would you rather have on your team?
- What does the range miss about a data set that the MAD captures?
Build the foundations first
Measures of center & variability builds on these concepts. If any feel shaky, start there.