A box and whisker diagram illustrates the spread of a set of data. It also displays the upper quartile, lower quartile and inter-quartile range of the data set.
A quartile is any one of the values which divide the data set into four equal parts, so each part represents a quarter of the sample. The upper quartile represents the highest 25% of the data. It can be considered as the median of the upper half of the values in the set. The lower quartile represents the lowest 25% of the data. It can be considered as the median of the lower half of all the values in the set. The inter-quartile range is the difference in value between the upper quartile and the lower quartile values. The median is the middle value, half of the data set is below and half is above.
Example
You can draw a box and whisker diagram for the results below which were obtained by a year 6 class in an English test marked out of 20.
Pupil A - 14
Pupil B - 13
Pupil C - 3
Pupil D - 7
Pupil E - 9
Pupil F - 12
Pupil G - 17
Pupil H - 4
Pupil I - 9
Pupil J - 10
Pupil K - 18
Pupil L - 16
There are 12 pupils' scores. You must place these in order as follows 3, 4, 7, 9, 9, 10, 12, 13, 14, 16, 17, 18. The range is 18 - 3 = 15.
There are an even number of values so the median is the midway between 10 and 12, ie the median has the value 11.
The upper quartile is the median value for 12, 13, 14, 16 17, 18 and so is midway between 14 and 16, ie the upper quartile is 15. 25% of all the pupils score above 15 and 75% score less than 15.
The lower quartile is the median value for 3, 4, 7, 9, 9, 10 and so is midway between 7 and 9, ie the lower quartile is 8. 25% of all the pupils score less than 8 and 75% score more than 8.
Had there been an odd number of values, the median would be the middle value, but in determining the lower and upper quartiles the median value is then ignored. For example, if the values are:
2, 3, 5, 7, 9, 13, 15, 18, 19, 20, 22, 23
the median value is clearly 14.
A box and whisker diagram can be used to display this information. The table below shows the results of an English test, constructed for presentation as a box and whisker diagram.
| English test scores out of 20 |
|---|
| Lowest score |
3 |
| Highest score |
18 |
| Lower quartile |
8 |
| Upper quartile |
15 |
The data in the table is represented by a box and whisker diagram as follows:
A box and whisker diagram shows at a glance the range of scores of the middle 50% of pupils (the box) and the total range of all the scores (the whiskers).
The two whiskers show the highest and lowest values from which the range of the data set can be calculated. For this data set, the range is given by 18 - 3 = 15.
The box gives the range of the middle 50% normally called the inter-quartile range. It represents the range of the scores between 25% (the lower quartile) and 75% (the upper quartile), hence the middle 50%. In this example, the interquartile range is given by 15 - 8 = 7. Comparisons between different sets of data (eg. different test scores) can be made by plotting a box and whisker diagram for each set of scores on the same graph.
Box and whisker plots can be constructed using cumulative frequency curves or cumulative percentage curves as these curves can be used to show the quartiles and the median. You are advised to read the section on cumulative frequency graphs as well.
Worked examples
Example one
A year 9 class completed three science tests. The scores for the three tests are presented in the table below. Display this data using a box and whisker diagram.
| Year 9 science test scores - marks out of 20 for each test |
|---|
| Test 1 | Test 2 | Test 3 |
|---|
| Lowest score |
5 |
7 |
2 |
| Highest score |
19 |
20 |
20 |
| Lower quartile |
9 |
9 |
6 |
| Upper quartile |
15 |
16 |
17 |
The box and whisker diagrams for each test are shown below:
The diagrams show that in test 3 pupils achieved a wider spread of results (from 2 to 20 with a range of 18) than was the case in tests 1 and 2. This could lead teachers to ask questions about why pupils' performance ranged more widely on this test than on the other two. In the diagram for test 1, the two ‘whiskers’ are the same length, so the upper and lower quartile ranges are the same.
Example two
The scores for three tests undertaken by a year 5 group are presented in the table below:
| Test 1
(score out of 20) | Test 2
(score out of 50) | Test 3
(score out of 100) |
|---|
| Lowest score |
5 |
12 |
20 |
| Highest score |
19 |
43 |
75 |
| Lower quartile |
9 |
19 |
35 |
| Upper quartile |
15 |
35 |
70 |
Which of the following statements are true?
- The highest and lowest marks for the year group declined over the three tests
- There was no change in the year group's performance over the three tests
- Marks for the year group improved over the three tests
All the tests have different maximum scores and will be easier to compare if converted into percentages. Box and whisker diagrams can be drawn on the same scale and comparisons made between the tests.
Test 1 scores are out of 20 and need to be converted so that they are out of 100, by multiplying top and bottom of the fraction by 5 (because 5 x 20 = 100).
Test 2 scores are out of 50, so can be converted by doubling each score.
Test 3 scores are already in percentages, because they are already given out of 100.
The figures show that both the highest and lowest scores have fallen from test one to test two and from test two to test three. As statement one is true, statements two and three must be false.
Example three
National relationship between school 1996 KS3 test results and their 1998 GCSE equivalent results
Note: for clarity, the median for each test is not indicated.)
1996 KS3 average level
Which of the following statements are true for schools in the band 'scoring less than level 3.5'?
- The lowest average GCSE point score was 5
- The highest GCSE point score achieved by the bottom 25% of schools was 27
- The lowest average GCSE point score achieved by the top 25% of schools was 15
To answer the question, consider if each statement is true. We only need to look at the first box and whisker diagram, which is for an average level of less than 3.5.
Statement one:
Look at the diagram and select the lower whisker.
Read horizontally from the lower whisker to the vertical scale.
The average GCSE score is 5.
The statement is therefore true.
Statement two:
Look at the diagram and look at the bottom of the box.
Read across to the vertical scale.
The average GCSE point score is about 8.
The statement is therefore not true.
Statement three:
Look at the graph and look at the top of the box.
Read across to the vertical scale.
The average GCSE score is 15.
The statement is therefore true.
Therefore statements one and three are true.