Year 8 · Unit 7 Averages and spread · Lesson 4 of 5

How spread out is it?

Two bus routes can have the same mean journey and still be very different to rely on. The range, the quartiles and the interquartile range say how far the data spreads, and how far its middle half spreads.

Statistics and ProbabilityNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. (Lesson 3) Which average does an outlier pull most: the mean, the median or the mode?
    AnswerThe mean. The median depends only on the middle value or values, and the mode on frequency.
  2. (Lesson 1) Find the median of 8, 3, 12, 5, 9.
    Answer8. In order: 3, 5, 8, 9, 12, and the middle one is 8.
  3. (Year 7, Unit 5, lesson 3) What number is halfway between 9 and 12?
    Answer10.5. (9 + 12) ÷ 2 = 21 ÷ 2.
Words used in this lesson
range
largest value minus smallest value. For example: in 4, 9, 15 the range is 11.
quartile
one of three values that cut ordered data into four parts of about equal size. For example: Q1, Q2 (the median), Q3.
lower half
the values below the median. For example: 2 and 4 in 2, 4, 7, 10, 13.
interquartile range (IQR)
Q3 minus Q1, the spread of the middle half. For example: 35 − 24 = 11.
spread
how far apart the values are. For example: marks from 20 to 95 are more spread than marks from 60 to 70.
five-number summary
smallest, Q1, median, Q3, largest. For example: 2, 5, 8, 15, 20 for 2, 5, 7, 8, 12, 15, 20.

Same mean, different journey

Two minibus routes both have a mean journey of 18 minutes. Route A took 17, 18, 18, 18 and 19 minutes on five days. Route B took 8, 12, 18, 24 and 28. Predict first: which route would you plan around if you must be at the station by a fixed time, and what number besides the mean would tell you? Then look at how spread out the data is in the figure below.

1
15range
9interquartile range
10.5median

Data set 1 of 4: 8 values. The 8 values split into two halves of 4. The median is 10.5. Q1 is 5.5 and Q3 is 14.5, so the IQR is 9 and the range is 15.

Show me the steps
  1. Set 1: 3, 5, 6, 9, 12, 14, 15, 18 (8 values). The median is (9 + 12) ÷ 2 = 10.5, and the halves are 3, 5, 6, 9 and 12, 14, 15, 18. Q1 = (5 + 6) ÷ 2 = 5.5, Q3 = (14 + 15) ÷ 2 = 14.5. IQR = 9, range = 15.
  2. Set 2: 4, 7, 8, 10, 11, 13, 15, 16, 19 (9 values). The median 11 is left out of both halves. Lower half 4, 7, 8, 10: Q1 = (7 + 8) ÷ 2 = 7.5. Upper half 13, 15, 16, 19: Q3 = (15 + 16) ÷ 2 = 15.5. IQR = 15.5 − 7.5 = 8, range = 19 − 4 = 15.
  3. Set 3: 12, 14, 14, 15, 15, 15, 16, 17, 21, 22, 30 (11 values). The median is 15. Lower half 12, 14, 14, 15, 15: Q1 = 14. Upper half 16, 17, 21, 22, 30: Q3 = 21. IQR = 7, range = 18.
  4. Set 4: 20, 21, 21, 22, 22, 22, 23, 23, 24 (9 values). The median is 22, Q1 = 21 and Q3 = 23. IQR = 2 and range = 4: this data is bunched, and both measures are small.

The range is the largest value minus the smallest. It says how far the whole data stretches, but it depends only on the two most extreme values.

Quartiles cut the ordered data into four parts with about a quarter of the values in each. The median is the middle one, Q2. Split the ordered data at the median into a lower half and an upper half: Q1 is the median of the lower half and Q3 is the median of the upper half. When the number of values is odd, the median itself goes in neither half.

The interquartile range is IQR = Q3 − Q1. It is the spread of the middle half of the data, and it ignores the extreme values at both ends. The five-number summary is the smallest value, Q1, the median, Q3 and the largest value.

Some calculators and spreadsheets use a different rule for quartiles and can give slightly different numbers. This unit always uses the halves method above.

Back to the routes: A's range is 2 minutes and B's is 20, with the same mean of 18.

A common mistake: splitting the list before it is in order. For 9, 3, 8, 5 the first half as written is 9 and 3, which would give Q1 = 6. In order the list is 3, 5, 8, 9, so the lower half is 3 and 5 and Q1 is 4. As with the median, order first.

Worked examples

Lengths of 11 beans in mm, already in order: 6, 8, 9, 11, 12, 14, 15, 15, 18, 20, 23. Find the five-number summary, the range and the IQR.

  1. 11 values, so the median is the 6th: 14. It goes in neither half.
  2. Lower half 6, 8, 9, 11, 12: its median is the 3rd, Q1 = 9.
  3. Upper half 15, 15, 18, 20, 23: its median is the 3rd, Q3 = 18.
  4. Summary: 6, 9, 14, 18, 23. Range = 23 − 6 = 17. IQR = 18 − 9 = 9.

Now an even number: 4, 6, 7, 10, 12, 13, 17, 19.

  1. 8 values, so the median is halfway between the 4th and 5th: (10 + 12) ÷ 2 = 11.
  2. The lower half is the first four, 4, 6, 7, 10: Q1 = (6 + 7) ÷ 2 = 6.5.
  3. The upper half is the last four, 12, 13, 17, 19: Q3 = (13 + 17) ÷ 2 = 15.
  4. Range = 19 − 4 = 15. IQR = 15 − 6.5 = 8.5.

Your go. The first step is given. Ten students' scores in a quiz: 31, 22, 40, 25, 27, 38, 22, 35, 29, 45. Find the median, Q1, Q3, the range and the IQR.

  1. In order: 22, 22, 25, 27, 29, 31, 35, 38, 40, 45.
  2. Now find the median and halve the list, then Q1, Q3, the range and the IQR, then check.
Check the answer

Median (29 + 31) ÷ 2 = 30. Lower half 22, 22, 25, 27, 29: Q1 = 25. Upper half 31, 35, 38, 40, 45: Q3 = 38. Range 45 − 22 = 23. IQR 38 − 25 = 13.

Check before you go on

1. The ordered list is 3, 5, 6, 8, 9, 11, 12. What is the lower quartile, Q1?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. The daily rainfall, in mm, on six days: 23, 8, 15, 31, 12, 27. What is the range?

3. A data set has Q1 = 12 and Q3 = 20. What is the interquartile range?

Level 2 · Building

4. Find the upper quartile, Q3, of the ordered list 2, 4, 7, 8, 11, 12, 15, 18, 19.

5. Find the interquartile range of the ordered list 7, 9, 10, 12, 14, 15, 19, 20.

6. Two classes' test marks have these five-number summaries (smallest, Q1, median, Q3, largest). Class A: 10, 20, 25, 30, 40. Class B: 5, 22, 25, 28, 45. Which statement is true?

Level 3 · Stretch

7. A data set of 11 values has median 14, upper quartile Q3 = 19 and interquartile range 9. What is the lower quartile, Q1?

8. Nine values in order are 3, 5, k, 9, 12, 14, 17, 18, 20. The lower quartile is 6.5. What is k?

Something to think about

A data set has 12 values. Add one value that is bigger than all of them. What happens to the range, and what do you expect to happen to the IQR? Try it with 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20 and a new value of 60.

One way to think about it

The range jumps from 17 to 57, because it depends on the largest value. The IQR changes only a little: it goes from 9 to 10.5, because adding a value moves the quartiles by at most a place. The spread of the middle half is much less sensitive to one extreme value.

Exit ticket

9. Two bakeries deliver bread with the same median delivery time. Bakery X has an IQR of 4 minutes and bakery Y has an IQR of 12 minutes. Which statement is correct?