1

Chapter Two

Summarising numerical data: the median, range, interquartile range

and box plots

2

Summary Statistics for Numerical Data

The two most commonly used types of statistics are classified as: Measures of centre Measures of spread

3

The MedianA measure of

centre The median is the middle

value when the data is ordered. The median depends on position in the set of data and so is not distorted by extreme values and outliers.

4

Range & Interquartile Range:

Measures of Spread The Range is the difference

between the maximum value and the minimum value.

The Interquartile Range (IQR) is the spread of the middle 50% of the data.

5

Calculating the median

2 9 1 8 3 5 3 8 1 Place numbers in ascending order 1 1 2 3 3 5 8 8 9

Median = 3 (5th score)

6

Calculating the Range

1 1 2 3 3 5 8 89

Min = 1 Max = 9

Range = Max – Min

Range = 9 – 1

Range = 8

7

Calculating the Quartiles

1 1 2 3 3 5 8 89

Lower Quartile (Q1) Upper Quartile (Q3)

Q1 = 1.5 Q3 = 8

Median (Q2) =3

IQR = Q3 – Q1

IQR = 8 – 1.5

IQR = 6.5

8

Stem and Leaf Plots

1 2 3 3 5 6 2 3 4 5 7 8 8 3 0 1 3 3 7 4 0 0 5 5 1 6 6 0

There are 22 scores

The median will be between the 11th and 12th score.

Median = 29

Lower Quartile = 23

Upper Quartile = 40

IQR = 40 –23

IQR = 17Exercise 2A Pages 39 – 40 Questions 1-7

9

Five number summary

1 1 2 3 3 5 8 89

medianminimum maximum

Lower Quartile Q1Upper Quartile Q3

1

1.5

3

8

9

IQR = 6.5

10

Five Number Summary

Min = 1 Q1 = 1.5 Median = 3 Q3 = 8 Max = 9

11

Box and Whisker Plot

A Box and Whisker Plot illustrates the 5 number summary. It is often used as a visual comparison of similar data sets. The statistics median, IQR and Range can be easily observed from a boxplot.

12

Box and Whisker Plot

The box represents the middle 50% of the scores ie the IQR

Each whisker represents 25% of the scores

13

Box and whisker plot

Exercise 2B Page 44 Question 1 and 2

14

Box Plot with Outliers

Outliers are identified using the limits

Q1 – 1.5*IQR and

Q3 + 1.5*IQR Any data values outside

these boundaries are marked with an asterix on the boxplot.

15

Boxplot with outliers

1 2 3 3 2 3 4 5 7 8 3 0 1 3 8 4 5 6 0

Min =

Q1 =

Med =

Q3 =

Max =

IQR =

Q1-1.5*IQR =

Q3+1.5*IQR =

12

60

27

18

32

14

18 – 1.5*14 = -3

32 + 1.5*14 = 53

60 will be an outlier

Boxplot limits

16

Boxplot with outliers

17

Using a calculator

Calculator display Exercise 2B Questions 3

– 6

18

Relating the shape of a histogram to a box

plot

Symmetric

19

Relating the shape of a histogram to a

box plot

20

Relating the shape of a histogram to a box

plot

21

Questions

Exercise 2C Page 47 All

22

Comparing Boxplots

23

Questions

Exercise 2D Page 49 All