SAT® · Math: Problem-Solving and Data Analysis
Calculating Mean, Median, and Range
Mean is the total divided by the count, median is the middle value once numbers are sorted, and range is the biggest value minus the smallest. They measure different things, so they usually disagree.
The mean takes every value and spreads its weight evenly across the whole list. Add up all the numbers, divide by how many there are, and you get a balance point: a number where the total distance of values above it equals the total distance of values below it. Because every single value pulls on that balance point, one huge or tiny outlier can drag the mean far from where most of the data actually sits.
The median ignores size and cares only about position. Sort the values from least to greatest and find the middle one. With an odd count, that is a single actual data point. With an even count, there is no single middle value, so you average the two values sharing the middle spot. That average often is not a number in the original list at all, which trips people up. The range just measures spread: the largest value minus the smallest, with nothing in between counted.
These three numbers connect to the wider idea of distributions. When a data set is symmetric, mean and median sit close together. When it is skewed by a few extreme values, they pull apart, and that gap is itself a clue about shape. The balance-point idea behind the mean also previews weighted averages, where some values count more than others, and both ideas return whenever a data set needs a single number to represent it.
Key terms
- Mean
- The sum of all values divided by the number of values.
- Median
- The middle value of a sorted list, or the average of the two middle values if the count is even.
- Range
- The largest value in a data set minus the smallest value.
- Balance point
- A way to picture the mean as the spot where values above and below it offset each other exactly.
- Outlier
- An unusually large or small value that can pull the mean away from the rest of the data.
Watch it explained
- How it worksHow does the mean act like a balance point while the median acts like a headcount?
- Worked exampleFind the mean, median, and range of 12, 15, 15, 18, 22, 40.
- Worked exampleFind the sixth score when five scores are 82, 88, 91, 77, 95 and the six must average 87.
- Common mistakeWhy isn't the median always one of the numbers actually in the data set?
- Ties togetherHow does a balance point on a number line connect the mean to the idea of a weighted average?
Check yourself
End of topic.