Mean and median tell you the center of a data set, but they say nothing about how spread out the values are. The range and the interquartile range fill that gap, and the second one is far more reliable. You can check your work with the free tools on Calc Solver.
What the Range Measures
The range is the simplest measure of spread: the largest value minus the smallest value. For 4, 7, 9, 12, 15, the range is 15 minus 4, which is 11. It is quick to find, but one extreme value can make it misleading, which is why statisticians usually prefer the interquartile range.
What Quartiles Are
Quartiles split an ordered data set into four equal parts. The first quartile (Q1) marks the 25 percent point, the second quartile is the median (50 percent), and the third quartile (Q3) marks the 75 percent point. The interquartile range (IQR) is Q3 minus Q1, so it covers the middle half of the data.
Step 1: Order the Data and Find the Median
Arrange the values from smallest to largest, then find the median. For 3, 5, 7, 8, 10, 12, 15, the median is 8 (the middle value of seven). With an even count, the median is the average of the two middle values.
Step 2: Find Q1 and Q3
Split the ordered data at the median, without including the median itself when the count is odd. The lower half (3, 5, 7) has median 5, so Q1 = 5. The upper half (10, 12, 15) has median 12, so Q3 = 12. For even counts, split the data into two equal halves and take each half’s median.
Step 3: Subtract to Get the IQR
IQR = Q3 – Q1 = 12 – 5 = 7. This says the middle half of the data spans 7 units. Compare two data sets by their IQRs: the larger IQR means the middle half is more spread out, even if both sets share the same median.
Why the IQR Beats the Range
The range uses only two values, so one outlier changes it completely. The IQR ignores the top and bottom quarters, so outliers barely move it. That robustness is exactly why box plots are drawn from quartiles: the box shows the IQR, and points far outside it are flagged as outliers (beyond 1.5 times the IQR from Q1 or Q3).
Practice Problems to Try
Find the range and IQR of 6, 9, 11, 14, 18, 22, 27, then of 2, 5, 8, 8, 10, 13, 40. Notice how the outlier 40 changes the range but not the IQR.