Statistics

Range and IQR

Lesson

Mean and median tell you where the middle is. To describe how spread out a data set is, two simple measures:

Range = largest value − smallest value. Tells you the full spread.

\text{Range} = \max - \min

Interquartile range (IQR) = spread of the middle 50% of the data. More robust than range because it ignores the extremes.

\text{IQR} = Q_3 - Q_1

Here $Q_1$ is the value one-quarterof the way through the sorted data (“first quartile”) and $Q_3$ is the value three-quartersof the way through (“third quartile”).

To find quartiles:

Sort the data.
Find the median (this is $Q_2$ ). If the count is odd, exclude the median from both halves.
$Q_1$ is the median of the lower half.
$Q_3$ is the median of the upper half.

Worked example 1 — range

Data: 15, 22, 18, 25, 10, 30.

\text{Range} = 30 - 10 = 20

Worked example 2 — IQR (odd count)

Data: 2, 5, 8, 11, 14, 17, 20, 23, 26. Already sorted. Nine values, so median is the 5th: $Q_2 = 14$ .

Lower half (before the median): $2, 5, 8, 11$ . Median of those: $Q_1 = (5 + 8)/2 = 6.5$ .

Upper half: $17, 20, 23, 26$ . Median: $Q_3 = (20 + 23)/2 = 21.5$ .

\text{IQR} = 21.5 - 6.5 = 15

How to type your answer

Type a single number — the range, IQR, or specific quartile the question asks for. Use a decimal point if needed. Examples: 12, 22.5, 15, 6.5.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

\text{Range of } 2,\ 5,\ 8,\ 10,\ 14

Problem 2

\text{Range of } 7,\ 3,\ 9,\ 1,\ 5

Problem 3

\text{IQR of } 1,\ 3,\ 5,\ 7,\ 9,\ 11,\ 13

Problem 4

\text{IQR of } 2,\ 4,\ 6,\ 8,\ 10,\ 12,\ 14

Practice

Standard problems matching the lesson.

Problem 5

\text{Range of } 15,\ 22,\ 18,\ 25,\ 10,\ 30

Problem 6

\text{Range of } -5,\ 0,\ 3,\ 8,\ -2,\ 6

Problem 7

\text{IQR of } 2,\ 5,\ 8,\ 11,\ 14,\ 17,\ 20,\ 23,\ 26

Problem 8

\text{IQR of } 1,\ 4,\ 7,\ 10,\ 13,\ 16,\ 19,\ 22

Problem 9

\text{Range of } 3.5,\ 7.2,\ 5.8,\ 9.1,\ 4.3

Problem 10

\text{IQR of } 10,\ 15,\ 20,\ 25,\ 30

Problem 11

\text{Range of } 100,\ 250,\ 175,\ 320,\ 90,\ 410

Problem 12

\text{IQR of } 6,\ 8,\ 10,\ 12,\ 14

Problem 13

\text{Range of } 0,\ 0,\ 5,\ 10,\ 15,\ 20

Problem 14

\text{IQR of } 3,\ 7,\ 11,\ 15,\ 19,\ 23,\ 27,\ 31,\ 35

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

\text{IQR of } 12,\ 15,\ 18,\ 22,\ 24,\ 27,\ 30,\ 35,\ 40,\ 42,\ 48

Problem 16

\text{IQR of } 2,\ 4,\ 6,\ 8,\ 10,\ 12,\ 14,\ 16,\ 18,\ 20

Problem 17

\text{Range of } -12,\ 8,\ -5,\ 14,\ 2,\ -8,\ 7

Problem 18

\text{IQR of } 5,\ 10,\ 15,\ 20,\ 25,\ 30,\ 35

Problem 19

\text{IQR of } 4,\ 8,\ 12,\ 15,\ 18,\ 22,\ 27,\ 31

Problem 20

\text{Range of cereal-box weights (oz): } 15.2,\ 15.5,\ 15.8,\ 16.1,\ 16.4

Problem 21

Q_1 \text{ of } 3,\ 9,\ 12,\ 15,\ 18,\ 21,\ 25,\ 28,\ 33

Problem 22

Q_3 \text{ of } 6,\ 11,\ 17,\ 23,\ 30,\ 36,\ 41,\ 47,\ 52,\ 58

Practice

Standard problems matching the lesson.

Problem 23

Daily high temperatures (°F) for a week: 72, 68, 75, 80, 73, 76, 71. Find the range.

Problem 24

Quiz scores: 70, 75, 80, 85, 90, 95, 100. Find the IQR.

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

Student heights (in): 58, 60, 62, 64, 66, 68, 70, 72, 74. Find the IQR.

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