Egypt Preparatory 3 Mathematics, Week 14: Dispersion: range and interquartile range from lists, stem-and-leaf plots and box plots

Dispersion and standard deviation

Week 14 · Measures of dispersion: range, interquartile range and standard deviation for a list, a frequency table and a table with class intervals · 3 lessons

Dispersion: range and interquartile range from lists, stem-and-leaf plots and box plots

Mathematics Prep 3 Term 1, Unit 4, Lesson 1

GoalFind the range and the interquartile range, and say which one is affected by an extreme value.

New words

range · greatest value minus smallest valuequartiles · Q₁, Q₂ (the median) and Q₃ split ordered data into four equal partsinterquartile range · IQR = Q₃ − Q₁, the spread of the middle halfextreme value · a value far from all the others

Explanation

Two sets of data can have the same centre but be spread differently, so we measure dispersion. The range is the greatest value minus the smallest; it is easy, but it uses only two values, so one extreme value changes it a lot. For something steadier, order the data and find the median Q₂. If the number of values is odd, leave the median out; the median of the lower half is Q₁ and the median of the upper half is Q₃. The interquartile range is IQR = Q₃ − Q₁ and measures the middle half of the data. Example: 4, 7, 8, 11, 13, 15, 17, 20, 22, 26, 29. Range = 29 − 4 = 25. Q₂ = 15, Q₁ = 8, Q₃ = 22, so IQR = 14. Replace 29 by 80: the range jumps to 76, but the IQR is still 14. A box plot draws the five numbers: minimum, Q₁, median, Q₃, maximum.

Minimum	Q₁	Median	Q₃	Maximum
4	8	15	22	29

Worked examples

Marks 12, 15, 16, 20, 24, 30 (six values). Q₂ = (16 + 20) ÷ 2 = 18. Lower half 12, 15, 16 gives Q₁ = 15; upper half 20, 24, 30 gives Q₃ = 24. IQR = 24 − 15 = 9. Range = 30 − 12 = 18.

Stem	Leaves
2	1 4 7
3	2 5 5
4	1 8

Stem-and-leaf plot, key 2 | 1 means 21: the data are 21, 24, 27, 32, 35, 35, 41, 48. Range = 48 − 21 = 27.

Dot plot of goals in 12 matches: the smallest is 1 and the largest is 5, so the range is 4.

Class activity

Extreme value test. Write the heights of 7 friends in cm. Find range and IQR. Add one very tall (invented) height. Which of the two measures changes more?

Practice

Find the range of 7, 12, 5, 20, 9.

Find Q₁, Q₂, Q₃ and IQR of 6, 9, 10, 14, 15, 19, 25.

The range of 3, 8, k, 14 is 14, with k the largest. Find k.

Why is the IQR less affected than the range by one extreme value?

Read the range from the stem-and-leaf plot in the lesson.