Box plot

Edexcel S1 Mathematics 2003

(or box and whisker plot)

Introduction

Box plot diagrams:provide a diagrammatic representation

of the distribution use quartiles to divide the distribution

into intervals each containing ¼ of the data values

used to compare distributionsused to show skewness of distribution

Find the quartiles Use any given algorithm to calculate outliers find the values of the whiskers Draw a box plot to scale– on graph paper

Stages in drawing a box plot

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

(a) Find the median and inter-quartile range

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

(c) Comment on the skewness of these data.

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

(a) Find the median and inter-quartile range

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

(c) Comment on the skewness of these data.

This question uses a small number of datavalues for ease of calculations. The number of data values is usually larger. Very few data values can make the calculation of quartiles less meaningful.

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

(a) Find the median and inter-quartile range

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

(c) Comment on the skewness of these data.

Any rule to identify outliers will be specified inthe question. The rule provided here is a typicalone.

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

(a) Find the median and inter-quartile range

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

(c) Comment on the skewness of these data.

Make sure you use graph paper to draw a boxplot. Ask for graph paper in the moduleexam.

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

(a) Find the median and inter-quartile range

Answer re-order the data:

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

(a) Find the median and inter-quartile range

Answer re-order the data:

Possibly use a stem and leaf diagram tore-order the data

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

(a) Find the median and inter-quartile range

Answer re-order the data:

1 1 3 4 4 5 6 9 10 21

The median is the middle value:n/2 = 10/2 = 5 5.5th value =

Q2

Whole number - so

round up

to --.5th Find average of 5th and 6th value

2

54

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

(a) Find the median and inter-quartile range

Answer re-order the data:

1 1 3 4 4 5 6 9 10 21

The median is the middle value:n/2 = 10/2 = 5 5.5th value = = 4.5 mins

The lower quartile, Q1, is the 1/4th value:n/4 = 10/4 = 2.25 3rd value =

2

54

Q2

Not whole - so round

up

to whole

Find the 3rd value

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

(a) Find the median and inter-quartile range

Answer re-order the data:

1 1 3 4 4 5 6 9 10 21

The median is the middle value:n/2 = 10/2 = 5 5.5th value = = 4.5 mins

The lower quartile, Q1, is the 1/4th value:n/4 = 10/4 = 2.25 3 mins 3rd value =

2

54

Q1 Q2

Not whole - so round

up

to whole

Find the 3rd value

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

(a) Find the median and inter-quartile range

Answer re-order the data:

1 1 3 4 4 5 6 9 10 21

The median is the middle value:n/2 = 10/2 = 5 5.5th value = = 4.5 mins

The lower quartile, Q1, is the 1/4th value:n/4 = 10/4 = 2.25 3 mins

The upper quartile, Q3, is the 3/4th value:3n/4 = 3x10/4 = 7.75 8th value =

3rd value =

2

54

Q1 Q2

Not whole - so round

up

to whole

Find the 8th value

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

(a) Find the median and inter-quartile range

Answer re-order the data:

1 1 3 4 4 5 6 9 10 21

The median is the middle value:n/2 = 10/2 = 5 5.5th value = = 4.5 mins

The lower quartile, Q1, is the 1/4th value:n/4 = 10/4 = 2.25 3 mins

The upper quartile, Q3, is the 3/4th value:3n/4 = 3x10/4 = 7.75 8th value = 9 mins

3rd value =

2

54

Q1 Q3Q2

Not whole - so round

up

to whole

Find the 8th value

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

(a) Find the median and inter-quartile range

Answer re-order the data:

1 1 3 4 4 5 6 9 10 21

The median is the middle value:n/2 = 10/2 = 5 5.5th value = = 4.5 mins

The lower quartile, Q1, is the 1/4th value:n/4 = 10/4 = 2.25 3 mins

The upper quartile, Q3, is the 3/4th value:3n/4 = 3x10/4 = 7.75 8th value = 9 mins

The inter quartile range = Q3 – Q1 = 9 – 3 = 6 mins

3rd value =

2

54

Q1 Q3Q2

Answer calculate outliers:

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6Check below Q1 for outliers:

Answer calculate outliers:

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1Check below Q1 for outliers:

Answer calculate outliers:

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

Answer calculate outliers:

Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

3

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

Answer calculate outliers:

Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

3 – 1.5 x 6 =

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

Answer calculate outliers:

Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

3 – 1.5 x 6 = -6 This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1.

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

Answer calculate outliers:

Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

3 – 1.5 x 6 = -6 This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1.

Check above Q3 for outliers:

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

Answer calculate outliers:

Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

3 – 1.5 x 6 = -6 This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1.

Check above Q3 for outliers:

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2

Q3

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

Answer calculate outliers:

1 1 3 4 4 5 6 9 10 21

Q1 Q3Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

3 – 1.5 x 6 = -6 This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1.

Q3 + 1.5 x IQR =Check above Q3 for outliers:

9

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

Answer calculate outliers:

1 1 3 4 4 5 6 9 10 21

Q1 Q3Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

3 – 1.5 x 6 = -6 This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1.

Q3 + 1.5 x IQR =Check above Q3 for outliers:

9 + 1.5 x 6 =

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

Answer calculate outliers:

1 1 3 4 4 5 6 9 10 21

Q1 Q3Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

3 – 1.5 x 6 = -6 This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1.

Q3 + 1.5 x IQR =Check above Q3 for outliers:

9 + 1.5 x 6 = 18 This falls within the data range. So value 21 is an outlier. The right whisker is 18.

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

Answer calculate outliers:

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

1 1 3 4 4 5 6 9 10 21

Q1 Q3Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

3 – 1.5 x 6 = -6 This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1.

Q3 + 1.5 x IQR =Check above Q3 for outliers:

9 + 1.5 x 6 = 18 This falls within the data range. So value 21 is an outlier. The right whisker is 18.

Draw scale and boxplot:0 5 10 15 20 25

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2

Answer calculate outliers:

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

1 1 3 4 4 5 6 9 10 21

Q1 Q3Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

3 – 1.5 x 6 = -6 This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1.

Q3 + 1.5 x IQR =Check above Q3 for outliers:

9 + 1.5 x 6 = 18 This falls within the data range. So value 21 is an outlier. The right whisker is 18.

Draw scale and boxplot:0 5 10 15 20 25

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2

Answer calculate outliers:

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

1 1 3 4 4 5 6 9 10 21

Q1 Q3Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

3 – 1.5 x 6 = -6 This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1.

Q3 + 1.5 x IQR =Check above Q3 for outliers:

9 + 1.5 x 6 = 18 This falls within the data range. So value 21 is an outlier. The right whisker is 18.

Draw scale and boxplot:0 5 10 15 20 25

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2

Answer calculate outliers:

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

1 1 3 4 4 5 6 9 10 21

Q1 Q3Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

3 – 1.5 x 6 = -6 This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1.

Q3 + 1.5 x IQR =Check above Q3 for outliers:

9 + 1.5 x 6 = 18 This falls within the data range. So value 21 is an outlier. The right whisker is 18.

Draw scale and boxplot:0 5 10 15 20 25

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2

Answer calculate outliers:

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

1 1 3 4 4 5 6 9 10 21

Q1 Q3Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

3 – 1.5 x 6 = -6 This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1.

Q3 + 1.5 x IQR =Check above Q3 for outliers:

9 + 1.5 x 6 = 18 This falls within the data range. So value 21 is an outlier. The right whisker is 18.

Draw scale and boxplot:0 5 10 15 20 25

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2

Answer calculate outliers:

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

1 1 3 4 4 5 6 9 10 21

Q1 Q3Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

3 – 1.5 x 6 = -6 This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1.

Q3 + 1.5 x IQR =Check above Q3 for outliers:

9 + 1.5 x 6 = 18 This falls within the data range. So value 21 is an outlier. The right whisker is 18.

Draw scale and boxplot:0 5 10 15 20 25

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2

Answer calculate outliers:

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

1 1 3 4 4 5 6 9 10 21

Q1 Q3Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

3 – 1.5 x 6 = -6 This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1.

Q3 + 1.5 x IQR =Check above Q3 for outliers:

9 + 1.5 x 6 = 18 This falls within the data range. So value 21 is an outlier. The right whisker is 18.

Draw scale and boxplot:0 5 10 15 20 25

*

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2

Answer calculate outliers:

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

1 1 3 4 4 5 6 9 10 21

Q1 Q3Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

3 – 1.5 x 6 = -6 This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1.

Q3 + 1.5 x IQR =Check above Q3 for outliers:

9 + 1.5 x 6 = 18 This falls within the data range. So value 21 is an outlier. The right whisker is 18.

Draw scale and boxplot:0 5 10 15 20 25

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2

*

Answer calculate outliers:

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

1 1 3 4 4 5 6 9 10 21

Q1 Q3Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

3 – 1.5 x 6 = -6 This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1.

Q3 + 1.5 x IQR =Check above Q3 for outliers:

9 + 1.5 x 6 = 18 This falls within the data range. So value 21 is an outlier. The right whisker is 18.

Draw scale and boxplot:0 5 10 15 20 25

*

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2

Remember to use GRAPH

paper

Answer calculate outliers:

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

(c) Comment on the skewness of these data.

1 1 3 4 4 5 6 9 10 21

Q1 Q3Recall from (a) Q1 = 3, Q2 = 4.5, Q3 = 9, IQR = 6

Q1 – 1.5 x IQR =Check below Q1 for outliers:

3 – 1.5 x 6 = -6 This falls outside the data range, so there is no outlier below Q1. So left whisker is the least value 1.

Q3 + 1.5 x IQR =Check above Q3 for outliers:

9 + 1.5 x 6 = 18 This falls within the data range. So value 21 is an outlier. The right whisker is 18.

Draw scale and boxplot:0 5 10 15 20 25

*

1 1 3 4 4 5 6 9 10 21

Q1 Q3Q2

Q3 - Q2 = 9 – 4.5 = 4.5 Q2 -Q1 = 4.5 – 3 = 1.5

So Q3 – Q2 > Q2 – Q1 So distribution is right (positive) skewed

Answer (a) median = Q2 = 4.5, IQR = 6

(b) boxplot: 0 5 10 15 20 25

*

(c) Q3 – Q2 > Q2 – Q1 So distribution is right (positive) skewed

Example: Waiting times at a bus were recorded, to the nearest minute, on 10 occasions. The data collected were: 6, 1, 4, 9, 21, 3, 5, 10, 1, 4

(a) Find the median and inter-quartile range

An outlier is an observation that falls either 1.5 x (inter-quartile range) above the upper quartile or 1.5 x (inter-quartile range) below the lower quartile

(b) Draw a boxplot diagram to represent these data, indicating any outliers

(c) Comment on the skewness of these data.