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Learn Index · Data Management and Statistics Grade 6–8

Range and interquartile range

Range is max − min; IQR = Q3 − Q1 and measures the spread of the middle half.

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Centre isn’t everything

Two classes both have a mean test score of 70. In Class A, scores range from 65 to 75. In Class B, they range from 30 to 100. The averages match, but the spread is very different. Measures of spread describe how scattered the data is.

The range

Range = highest value − lowest value

Example: 12, 15, 18, 22, 40

Range = 40 − 12 = 28

The range is quick but uses only two values, so one outlier can make it huge.

Quartiles

Quartiles divide ordered data into four equal parts.

Quartile Meaning
Q1 (lower quartile) median of the lower half; 25% of data is below it
Q2 the median; 50% below
Q3 (upper quartile) median of the upper half; 75% below

Finding quartiles

Example: 3, 5, 7, 8, 10, 12, 14, 18 (8 values)

  1. Median (Q2): (8 + 10) ÷ 2 = 9
  2. Lower half: 3, 5, 7, 8 → Q1 = (5 + 7) ÷ 2 = 6
  3. Upper half: 10, 12, 14, 18 → Q3 = (12 + 14) ÷ 2 = 13

With an odd number of values, leave the median out of both halves.

Example: 2, 4, 6, 8, 10, 12, 14 → Q2 = 8; lower half 2, 4, 6 → Q1 = 4; upper half 10, 12, 14 → Q3 = 12.

(Some textbooks and calculators use slightly different methods for quartiles, so answers can differ a little.)

The interquartile range (IQR)

IQR = Q3 − Q1

From the first example: IQR = 13 − 6 = 7.

The interquartile range measures the spread of the middle 50% of the data. It ignores the extremes, so it isn’t thrown off by outliers.

Range vs IQR

Data: 10, 11, 12, 13, 14, 15, 90

  • Range = 90 − 10 = 80 (huge because of 90)
  • Q1 = 11, Q3 = 15 → IQR = 4 (shows most data is tightly grouped)

Spotting outliers with the IQR

A common rule: a value is an outlier if it is more than 1.5 × IQR below Q1 or above Q3.

From the data above: 1.5 × 4 = 6. Upper fence = 15 + 6 = 21. 90 > 21, so 90 is an outlier.

Worked examples

Example 1: Range of 45, 32, 58, 41, 50?

58 − 32 = 26.

Example 2: Find Q1, Q3 and IQR for 1, 3, 4, 6, 7, 9, 10, 12.

Q1 = 3.5, Q3 = 9.5, IQR = 6.

Example 3: Which is less affected by an outlier: range or IQR?

IQR.

Common mistakes

  • Not ordering the data before finding quartiles.
  • Including the median in the halves for odd-sized data (method-dependent; follow your class method).
  • Giving the range as "12 to 40" instead of a single number (28).

Try it yourself

  1. Range of 7, 19, 3, 25, 11
  2. Q1 and Q3 of 2, 4, 6, 8, 10, 12
  3. IQR of 5, 7, 8, 10, 12, 15, 20
  4. Is 50 an outlier if Q1 = 20, Q3 = 30?

Answers: 1) 22 2) Q1 = 4, Q3 = 10 3) 15 − 7 = 8 4) Yes: IQR = 10, upper fence = 45.

Quick recap

  • Range = highest − lowest.
  • Quartiles split data into quarters; IQR = Q3 − Q1.
  • The IQR measures spread without being skewed by outliers.