Interquartile Range (IQR)

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Interquartile Range (IQR): meaning, method and solved example (6th Grade) – Maths PDF

In short: Interquartile range measures how wide the central half of a sorted data set is, computed as the upper quartile minus the lower quartile.

6th GradeStatistics6.SP.B.5c

What is Interquartile Range?

The interquartile range, or IQR, tells how tightly the central half of the data is packed. Ignoring the lowest quarter and the highest quarter makes it steady even if a few values are far from the rest.

The formula is IQR \(= Q_3 – Q_1\). Close-packed middle values give a short stretch, scattered ones a long stretch. Visually, it equals the length of the rectangle in a box plot.

How to spot it

  • A question asks for the spread of the middle 50% of the values.
  • A box plot is given and you must measure the width of its box.
  • Two data sets are compared and one has a stray far-off value that would distort the range.
  • The words “Q1”, “Q3” or “middle half” appear.

The method

  1. Sort the values and mark the center.
  2. Get \(Q_1\) from the bottom group and \(Q_3\) from the top group.
  3. Subtract the lower cut point from the upper one.
  4. Give the interquartile range in the units of the data and read it as the width of the middle half.
05101520IQR = 14 – 5.5 = 8.5Q1 = 5.5Q3 = 14
The box from 5.5 to 14 holds the middle half and is 8.5 wide.

Interquartile Range: solved example

Problem

A coffee shop in Portland records daily cups sold over nine days: 3, 5, 6, 8, 9, 12, 13, 15, 20 (in dozens). Find the IQR.

Solution

The median is 9. The lower half 3, 5, 6, 8 has median \(\frac{5 + 6}{2} = 5.5\). The upper half 12, 13, 15, 20 has median \(\frac{13 + 15}{2} = 14\).

IQR \(= 14 – 5.5 = 8.5\). The middle half spans 8.5 dozen cups.

Try it

Question 1

A data set has \(Q_1 = 21\) and \(Q_3 = 34\). Find the interquartile range.

Show the answer

\(34 – 21 = 13\).

Question 2

Set A has IQR 4 and set B has IQR 11. Which is more consistent in the middle?

Show the answer

Set A, because its middle half is packed into a narrower stretch.

Interquartile Range vs. Range

The range subtracts the smallest from the largest and reacts to any extreme value. The IQR subtracts the quartiles and ignores the outer quarters, so it describes the typical spread and barely moves when one value is extreme.

Range (Statistics)

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