Five-Number Summary and Box Plots | IEB Grade 12 Maths

Q: How do I tell the skewness from a box plot?

Look at where the median sits in the box and which whisker is longer. A longer right whisker and a median near $Q_1$ means positive skew.

Q: Do I have to learn the 1,5 times IQR rule for outliers?

No. The IEB gives the outlier rule in the question when you need it. You must be able to apply it: work out the fences $Q_1 - 1{,}5 \times \text{IQR}$ and $Q_3 + 1{,}5 \times \text{IQR}$, then compare the data with them.

Q: What should I say when comparing two box plots?

Compare medians (centre), IQRs or ranges (spread) and skewness (shape), and always interpret the numbers in words.

The five-number summary and box-and-whisker plot show the centre, spread and shape of data in one picture. IEB questions ask you to draw them, compare two data sets, and describe skewness and outliers in words.

Quartiles and the interquartile range

The five-number summary is: minimum, $Q_{1}$ , median ( $Q_{2}$ ), $Q_{3}$ , maximum.

$Q_{1}$ is the median of the lower half of the data.
$Q_{3}$ is the median of the upper half.
IQR $= Q_{3} - Q_{1}$ , the spread of the middle 50%.

Example. $3; 5; 7; 8; 9; 11; 12; 14; 15; 18; 21; 35$ (12 values)

Median $= \frac{11 + 12}{2} = 11, 5$
$Q_{1} = \frac{7 + 8}{2} = 7, 5$ and $Q_{3} = \frac{15 + 18}{2} = 16, 5$
IQR $= 9$

Five-number summary: $(3; 7, 5; 11, 5; 16, 5; 35)$ .

Box-and-whisker diagrams

Draw a number line with a scale. Then:

Draw a box from $Q_{1}$ to $Q_{3}$ .
Draw a line inside the box at the median.
Draw whiskers from the box to the minimum and maximum.

The box shows the middle 50% of the data. Each whisker covers 25%.

Comparing two box plots. Comment on:

Centre: which median is higher.
Spread: which has the larger IQR or range. A smaller IQR means more consistent data.
Shape: skewness of each.

Skewness

Symmetric: the median is in the middle of the box, and the whiskers are similar. Mean $\approx$ median.
Positively skewed (skewed right): the median is closer to $Q_{1}$ , the right whisker is longer. Mean $>$ median.
Negatively skewed (skewed left): the median is closer to $Q_{3}$ , the left whisker is longer. Mean $<$ median.

Example. For the data above, median $- Q_{1} = 4$ and $Q_{3} -$ median $= 5$ , with a long right whisker to 35. The mean is $\frac{158}{12} \approx 13, 17$ , which is greater than the median. The data is positively skewed.

Outliers

An outlier is a value far from the rest of the data. A common rule:

$outlier if x < Q_{1} - 1, 5 \times IQR or x > Q_{3} + 1, 5 \times IQR$

Example. $Q_{3} + 1, 5 \times 9 = 30$ and $Q_{1} - 1, 5 \times 9 = - 6$ . The value 35 is an outlier.

In the IEB exam the outlier rule is given in the question when you need it, so you do not have to memorise it. On a box plot, an outlier can be shown as a separate dot beyond the whisker.

Key terms

Quartiles: Values that divide ordered data into four equal parts.
Interquartile range: $Q_{3} - Q_{1}$ , the range of the middle half of the data.
Skewness: The lack of symmetry in a distribution.
Outlier: A value that lies far from the rest of the data.

Common mistakes

Finding quartiles without ordering the data first.
Including the median in both halves when finding $Q_{1}$ and $Q_{3}$ for odd $n$ (follow your teacher's convention, but be consistent).
Describing skewness by the side the data bunches up on, instead of the side of the long tail.
Comparing data sets with numbers only, without words about centre and spread.

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

4 marks total

knowledge

Determine the five-number summary of:

$3; 5; 7; 8; 9; 11; 12; 14; 15; 18; 21; 35$

Worked solution

Hidden - try the question first

Solution

Minimum 3 and maximum 35
Median $= \frac{11 + 12}{2} = 11, 5$
$Q_{1} = \frac{7 + 8}{2} = 7, 5$
$Q_{3} = \frac{15 + 18}{2} = 16, 5$

Question 2

4 marks total

routine

For the same data:

(a) Calculate the interquartile range. (1)

(b) An outlier is a value greater than $Q_{3} + 1, 5 \times IQR$ or less than $Q_{1} - 1, 5 \times IQR$ . Show that 35 is an outlier. (3)

Worked solution

Hidden - try the question first

Solution

(a) IQR $= 16, 5 - 7, 5 = 9$
(b) $Q_{3} + 1, 5 \times IQR = 16, 5 + 13, 5$
$= 30$
$35 > 30$ , so 35 is an outlier

Question 3

6 marks total

complex

For the same data:

(a) Calculate the mean. (2)

(b) Draw a box-and-whisker diagram. (2)

Worked solution

Hidden - try the question first

Solution

(a) $= 16, 5 - 7, 5 = 9$
$\approx 13, 17$
(b) Box from 7,5 to 16,5 with median line at 11,5
Whiskers to 3 and 35 (or to 21 with 35 marked as an outlier)
(c) Positively skewed: mean $>$ median ( $13, 17 > 11, 5$ )
and the median is closer to $Q_{1}$ than to $Q_{3}$ ( $4 < 5$ ), with a high outlier at 35

Question 4

8 marks total

problem-solving

Two Grade 12 classes wrote the same test. Their five-number summaries are:

Class A: $(32; 48; 56; 65; 88)$
Class B: $(40; 52; 60; 63; 71)$

(a) Calculate the IQR and range of each class. (2)

(b) Which class performed more consistently? Give a reason. (2)

(d) The teacher says Class A did better because it had the highest mark. Comment on this claim. (2)

Worked solution

Hidden - try the question first

Solution

(a) Class A: IQR $= 16, 5 - 7, 5 = 9$ , range $= 56$
Class B: IQR $= 11$ , range $= 31$
(b) Class B
It has a smaller IQR and range, so its marks are closer together
(c) Median $- Q_{1} = 8$ and $Q_{3} -$ median $= 3$ , so the median is closer to $Q_{3}$
Class B is negatively skewed
(d) The claim is not supported: Class B has the higher median (60 compared to 56)
and a higher $Q_{1}$ , so most of Class B did better; one high mark in Class A does not describe the class

FAQs

How do I tell the skewness from a box plot?

Look at where the median sits in the box and which whisker is longer. A longer right whisker and a median near $Q_{1}$ means positive skew.

Do I have to learn the 1,5 times IQR rule for outliers?

No. The IEB gives the outlier rule in the question when you need it. You must be able to apply it: work out the fences $Q_{1} - 1, 5 \times IQR$ and $Q_{3} + 1, 5 \times IQR$ , then compare the data with them.

What should I say when comparing two box plots?

Compare medians (centre), IQRs or ranges (spread) and skewness (shape), and always interpret the numbers in words.

Five-number summary and box-and-whisker plots

Quartiles and the interquartile range

Box-and-whisker diagrams

Skewness

Outliers

Key terms

Common mistakes

Practice questions

FAQs

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