Year 11 Standard · Unit 1 Describing data · Lesson 3 of 6

How far from the rest does a value have to be to count as an outlier?

A box plot shows five numbers at a glance. The box is the middle half of the data, and a fence a set distance beyond it decides which values are drawn as separate points.

Statistics and ProbabilityGDC optional

Warm-up

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

  1. From Lesson 2: a curve is drawn for 60 values. At what count is the upper quartile read?
    Answer45. Three quarters of 60 is 3 × 60 ÷ 4.
  2. From Year 8: the median of 7, 2, 9, 4, 10 is ?
    Answer7. Order them: 2, 4, 7, 9, 10, and take the middle one.
  3. From Year 7: 1.5 × 12 = ?
    Answer18. 12 + 6 = 18, since half of 12 is 6.
Words used in this lesson
five-number summary
the minimum, lower quartile, median, upper quartile and maximum. For example: the five values that a box plot is drawn from.
quartile
a value that divides ordered data into quarters. For example: the median of the upper half of a list.
interquartile range
the upper quartile minus the lower quartile. For example: the width of the box.
outlier
a value beyond a fence, drawn as a separate point. For example: a finishing time far from the rest.
fence
a boundary at 1.5 times the IQR beyond a quartile. For example: the line Q3 + 1.5 × IQR.
whisker
the line from the box to the most extreme value that is not an outlier. For example: the line on each end of a box plot.

The slow runner

A trail-running club in Kathmandu records the finishing time of every member in one race. All but one finished within a few minutes of each other, and one runner took far longer after a fall. The club wants a picture that shows the typical spread, without letting that single long time stretch the whole display.

Predict first: how far from the rest should a value be before it is drawn as a separate point? Should the answer depend on how spread out the other values are? Use the first figure to test your ideas, then use the second figure to see what each part of a box plot holds.

24
32.5upper fence
noneoutliers

Largest value 24: Q1 = 15, median = 18, Q3 = 22, IQR = 7, upper fence 22 + 1.5 × 7 = 32.5. No outliers.

Show me the steps
  1. Largest value 24: the median is 18, the quartiles are 15 and 22, the IQR is 7 and the upper fence is 22 + 1.5 × 7 = 32.5. Nothing is beyond the fence.
  2. Slide to 32: it is still not above 32.5, so it is not an outlier. The whisker stretches to 32 and the box has not changed.
  3. Slide to 34: it is above 32.5, so it is drawn as a separate point and the whisker stops at 24. The quartiles, and so the fence, have not moved.

A box plot is five numbers and a fence.

  • The five-number summary is the minimum, lower quartile, median, upper quartile and maximum.
  • For an ordered list, the median splits it into two halves (leave the middle value out if there is an odd number). The lower quartile is the median of the lower half and the upper quartile is the median of the upper half.
  • The fences are Q1 − 1.5 × IQR and Q3 + 1.5 × IQR. A value beyond a fence is an outlier and is drawn as a separate point.
  • Each whisker stops at the most extreme value that is not an outlier.

Each of the four sections of a box plot holds about a quarter of the values, whatever its length.

30
50
70
40interquartile range
20length of the shortest section

Quartiles 30, 50, 70: the sections are 20, 20, 20, 20 long, each holding 10 values. IQR = 40.

Show me the steps
  1. Quartiles 30, 50 and 70: the four sections are each 20 long and hold 10 values each, so the values are evenly spread. The interquartile range is 40.
  2. Move the lower quartile to 20: the lowest section is only 10 long but still holds 10 values, so they are crowded together, and the next section grows to 30 long.
  3. Move the upper quartile to 80: the top section is now 10 long, and the interquartile range is 80 − 20 = 60. Each section still holds a quarter of the values.

The whisker does not always reach the maximum. If the largest value is an outlier, the whisker ends at the next value in. And a long section does not hold more values than a short one: each holds about a quarter. A long section means the values there are spread out, a short one means they are close together. Find the quartiles from the halves of the list, not from the position of the value in the whole list.

Worked example: the trail race

Finishing times in minutes for 11 runners: 41, 44, 45, 47, 49, 50, 52, 55, 58, 61, 90. Find the five-number summary and say whether any value is an outlier.

  1. The median is the 6th value, 50. Leave it out. The lower half is 41, 44, 45, 47, 49 and its median is 45.
  2. The upper half is 52, 55, 58, 61, 90 and its median is 58. So Q1 = 45 and Q3 = 58, and the IQR is 13.
  3. The upper fence is 58 + 1.5 × 13 = 77.5 and the lower fence is 45 − 19.5 = 25.5.
  4. 90 is above 77.5, so it is an outlier. The upper whisker ends at 61. The summary is 41, 45, 50, 58, 90.

Your go. The first step is given. The heights in cm of nine seedlings: 21, 23, 24, 26, 27, 29, 30, 33, 52. Find the quartiles and say whether 52 is an outlier.

  1. The median is the 5th value, 27. The lower half is 21, 23, 24, 26.
  2. Find the median of each half, then the IQR, then the upper fence.
Check the answer

Q1 = (23 + 24) ÷ 2 = 23.5 and Q3 = (30 + 33) ÷ 2 = 31.5, so the IQR is 8. The upper fence is 31.5 + 1.5 × 8 = 43.5, and 52 is beyond it, so 52 is an outlier. A common slip is to compare 52 with the IQR, or with 31.5 + 8, instead of with 1.5 times the IQR.

With a GDC (optional)

Enter the data as a list and open the one-variable statistics screen (often under a “stats” or “calc” menu). It gives the minimum, the quartiles, the median and the maximum. A box-plot display can also be drawn from the list, and some GDCs have a setting that shows outliers as separate points.

Different models use slightly different rules for the quartiles of a short list, for example whether the median is included in each half. Work out the halves by hand first and compare. Everything here can be done by hand.

Check before you go on

1. The ordered list is 2, 4, 5, 7, 9, 10, 12, 15. What is the lower quartile, found as the median of the lower half?

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 ordered list is 3, 5, 8, 9, 12, 14, 17, 20, 23. The median is 12. Leave it out and find the lower quartile.

3. For the same list, 3, 5, 8, 9, 12, 14, 17, 20, 23, the upper quartile is 18.5. What is the interquartile range?

Level 2 · Building

4. A data set has lower quartile 24 and upper quartile 38. What is the upper fence, Q3 + 1.5 × IQR?

5. A data set has lower quartile 15 and upper quartile 27. A value counts as an outlier if it is more than 1.5 × IQR beyond a quartile. Which of these values is an outlier?

Level 3 · Stretch

6. The upper quartile is 41 and the IQR is 18. Values are whole numbers. What is the largest value that is not an outlier?

7. These 12 values are ordered: 6, 8, 9, 11, 12, 14, 15, 17, 18, 21, 22, 40. By the 1.5 × IQR rule exactly one is an outlier. It is removed. What is the median of the remaining values?

Something to think about

Two box plots have the same median and the same interquartile range. One has a long whisker on the right and the other does not. What can you say about the two sets of data, and what can you not say?

One way to think about it

You can say the middle half of each set sits in the same place and has the same width, and that one set has some values much further to the right. You cannot say how many values are in the long whisker, or whether the two sets are the same size. A box plot hides the shape inside each quarter.

Exit ticket

8. A box plot for 80 values has minimum 4, lower quartile 12, median 20, upper quartile 36 and maximum 60. Which statement is true?