Exploring One-Variable Data

Measures of Center & Spread

Calculating mean, median, range, and interquartile range.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Describing Distributions

Making 'Typical' and 'Spread Out' Precise

The noon temperature (°F) was recorded on 10 days: 68, 70, 71, 72, 73, 74, 75, 76, 78, 83. Before reading on: 'the temperature was typically in the low 70s' is a reasonable informal description — but what exact number would you compute to back that up precisely?

Definition — Mean, Median, Range, and IQR

The mean, x̄ = (Σx)/n, is the arithmetic average. The median is the middle value of the sorted data (or the average of the two middle values, for an even n). The range is the maximum minus the minimum. The interquartile range, IQR = Q3 − Q1, measures the spread of the middle 50% of the data.

\bar{x} = \frac{\sum x}{n}

Worked Example — Computing Mean, Median, Range, and IQR

For the 10 temperatures (sum = 740): x̄ = 740/10 = 74°F. Sorted, the median is the average of the 5th and 6th values: (73+74)/2 = 73.5°F. Range = 83 − 68 = 15 degrees. Q1 (median of the lower 5 values, 68,70,71,72,73) = 71. Q3 (median of the upper 5 values, 74,75,76,78,83) = 76. IQR = 76 − 71 = 5 degrees. The mean (74) and median (73.5) land close together, consistent with a fairly symmetric distribution with no dramatic outlier.

Worked Example — Why IQR Resists an Extreme Value and Range Doesn't

Suppose the warmest day's reading had actually been 95°F instead of 83°F. The range would jump to 95 − 68 = 27 degrees — nearly double. But Q1 and Q3 are computed from the middle values only, not the extremes: the upper half becomes {74,75,76,78,95}, whose median (Q3) is still 76, and Q1 is still 71 — so the IQR stays exactly 5, completely unaffected by the more extreme value. Range uses only the two most extreme values, so a single unusual measurement can swing it dramatically; IQR ignores the most extreme values entirely, which is exactly why it resists that same swing.

Tip

Range is fast to compute but fragile — a single unusually high or low value can distort it completely. IQR takes a little more work but stays reliable even when the data includes an extreme value.

Common Mistakes

Computing Q1 and Q3 by including the overall median in both halves of the data.
For the standard method used here, the median itself is excluded when splitting an odd-sized dataset into lower and upper halves — for an even-sized dataset like this 10-value example, the data splits cleanly into two equal halves with nothing to exclude.

Key Takeaways

Mean and median both describe a 'typical' value, but respond differently to skewed data.
Range uses only the two most extreme values; IQR uses only the middle 50%, making it resistant to a single extreme value.
A dataset's mean and median landing close together is itself a sign of a roughly symmetric distribution.

Summary

Mean, median, range, and IQR summarize center and spread using specific pieces of the data. The next lesson introduces a measure of spread that uses every single value at once.

Track your progress

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