Interquartile Range

Grade 6

Quartiles and IQR

The lower quartile (LQ) is the value one quarter of the way through the data. The upper quartile (UQ) is three quarters of the way. The interquartile range is the difference between them.

IQR = upper quartile − lower quartile

The IQR covers the middle 50% of the data — ignoring the top and bottom quarters entirely. That makes it a much more reliable measure of spread than the range.

Reading quartiles from a cumulative frequency curve

EXAMPLE
A cumulative frequency curve has n = 80 values. Read off the LQ, UQ and IQR. The curve gives: cf = 20 → 35 marks; cf = 60 → 58 marks.
LQ position: n ÷ 4 = 80 ÷ 4 = 20. Go across from cf = 20 → LQ = 35 marks. UQ position: 3n ÷ 4 = 60. Go across from cf = 60 → UQ = 58 marks. IQR = 58 − 35 = 23 marks.

Using IQR to compare distributions

A smaller IQR means the middle half of the data is packed more tightly — the distribution is more consistent. A larger IQR means more spread in the middle 50%.

When comparing two data sets using cumulative frequency curves, comment on both the median (who scored higher on average) and the IQR (who was more consistent). A complete comparison needs both points.
Nova
Try this one

A cumulative frequency graph is drawn for the heights of 60 plants. A student reads the following values from the graph:

  • lower quartile \( = 24 \) cm
  • median \( = 33 \) cm
  • upper quartile \( = 41 \) cm

(a) Work out the interquartile range. [2]

(b) Explain what the interquartile range tells you about the heights of the plants. [1]

Nova's hint:

This tests the meaning and calculation of the interquartile range.

Ask yourself: 'which two quartiles do I subtract, and which order?' The interquartile range describes how spread out the middle half of the data is — a smaller value means more consistent data.

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