Compute and interpret mean, median, range, and interquartile range (IQR). | One-variable data: Distributions and measures of center and spread | SAT Math

How to compute the mean, median, range, and IQR from a small data set.

Core Idea

Mean is the arithmetic average; median is the middle value when data is sorted. Range is max minus min; IQR is Q3 minus Q1. The SAT expects you to compute these from small data sets and know which measure to use when.

Understanding

The mean treats every value equally:

$M e a n = s u m o f a l l v a l u e s 𝑛$

The median ignores how far values are from the center — it only cares about position. Sort the data, find the middle.

Range captures the full extent of the data: $m a x - m i n$ . IQR captures the spread of just the middle 50%: $𝑄 3 - 𝑄 1$ .

When to use which: The mean and range are sensitive to outliers. The median and IQR are resistant. If a data set has extreme values, the median and IQR give a more reliable picture of the "typical" value and spread.

On the SAT, expect to compute these from a list of 5–15 numbers or from a frequency table. The arithmetic is straightforward — the challenge is doing it accurately under time pressure.

Step by Step

To compute the mean: sum all values, divide by $𝑛$ .
To find the median: sort the data, locate the middle value (or average of two middle values).
To find quartiles: split the sorted data in half at the median, then find the median of each half. Q1 is the median of the lower half; Q3 is the median of the upper half.
Range = max − min. IQR = Q3 − Q1.

Misconceptions

Forgetting to sort the data before finding the median.
Including the overall median in both halves when computing Q1 and Q3 (conventions vary, but the SAT typically excludes it for odd $𝑛$ ).
Computing mean when the question asks for median, or vice versa — read the question carefully.

Question

Worked Example

A data set consists of the values: 4, 7, 9, 12, 15, 18, 22. What is the interquartile range?

Select an answer to see the explanation