##### Box-and-whisker plots

A box-and-whicker plot is a diagram that shows the distribution of a set of data at a glance. They are drawn using five summary statistics: the lowest and highest values( the range), the first and third quartiles (The interquartile range) and the median.

Drawing box-and-whisker plots

All box-and-whisker plots have the same basic features. You can see these on the diagram.

Note! Box-and-whisker plots ( are also called boxplots) are a standardized way of showing the range and a typical value ( the median). These five summary statistics are also called the 5-number summary.

Example 1: The masses in kilograms of 20 students were rounded to the nearest kilogram and listed in order:

48, 52, 54, 55, 55, 58, 58, 61, 62, 63, 63, 64, 65, 66, 67, 69, 70, 72, 79.

Draw a box-and-whisker plot to represent this data.

The minimum and maximum values can be read from the data set

Minimum=48 kg

Maximum=79 kg

Calculate the median.

There are 20 data values, so the median will lie halfway between the 10th and 11th values. In this data set they are both 63, so the median is 63 kg.

Next, calculate the lower and upper quartiles (Q1 and Q3). Q1 is the ** mean** of the 5th and 6th values and Q3 is the mean of the 15th and 16th values.

$\displaystyle {{Q}_{1}}=\frac{{55+58}}{2}=56.5kg$

$\displaystyle {{Q}_{3}}=\frac{{66+67}}{2}=66.5kg$

To draw the box-and-whisker plot:

- Draw a scale with equal intervals that allows for the minimum and maximum values.
- Mark the position of the median and the lower and upper quartiles against the scale.
- Draw a rectangular box with Q1 at one end and Q3 at the other. Draw a line parallel to Q1 and Q3 inside the box to show the position of the median.
- Extend lines (the whiskers) from Q1 and Q3 sides of the box to the lowest and highest values.

Note! Box-and-whisker plots are very useful for comparing two or more sets of data. When you want to compare two sets of data, you plot the diagrams next to each other on the same scale.

Example 2: The heights of ten 13-year old boys and ten old girls ( to the nearest cm) are given in the table.

Draw a box-and-whisker plot for both sets of data and compare the interquartile range.

First arrange the data sets in order. Then work out the five number summary for each data set:

Draw a scale that allows for a minimum and maximum values.

Plot both diagrams and label them to show which is which.

The IQR for girls(10cm) is wider than that for boys (7cm) showing that the data for girls is more spread out and varied.

Interpreting box-and-whisker plots

To interpret a box-and-whisker plot, you need to think about what information the diagram gives you about the data set.

This box-and-whisker plot shows the result of a survey in which a group of teenagers wore a fitness tracker to record the number of steps they took each day.

The box-and-whicker plot shows that:

- The number of steps ranged from 4000 to 9000 per day.
- The median number of steps was 6000 steps per day.
- 50% of the teenagers took 6000 or fewer steps per day( the data below the median value)
- 25% of the teenagers took 5000 or fewer steps per day( the lower `whiscker` represents the lower 25% of the data)
- 25% of the teenagers took more than 7000 steps per day ( the upper `whisker` shows the top 25% of the data)
- The data is fairly regularly distributed because the median line is in the middle of the box (in other words, equally far from Q1 and Q3)

Example 3: The box-and-whisker plots below show the test results that the same group of students achieved for two tests. Test 2 was taken two weeks after Test 1.

Comment or how the students performed in the two tests.

The highest and lowest marks were the same for both tests. The marks ranged from 7 to 27, a difference of 20 marks.

Q3 is the same for both tests. This means that 75% of the students scored $\displaystyle \frac{{22}}{{30}}$ or less on both tests. Only 25% of the students scored 22 or more.

For the first test Q1 was 12, so 75% of the students scored 12 or more marks. In the second test, Q1 increased from 12 to 15. This means that 75% of the students scored 15 or more marks in the second test, suggesting that the group did slightly better in the second test.