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Stem and Leaf Diagram. People were asked their age as they entered a health centre. Their ages were: 25, 28, 21, 17, 33, 15, 26, 31, 20, 21 and 18. The data recorded can be shown in a stem and leaf diagram also referred to as a Stem Plot. . The title. Ages (Years). The Stem. 5 7 8. 1.

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stem and leaf diagram
Stem and Leaf Diagram

People were asked their age as they entered a health centre.

Their ages were: 25, 28, 21, 17, 33, 15, 26, 31, 20, 21 and 18.

The data recorded can be shown in a stem and leaf diagram also referred to as a Stem Plot.

The title

Ages (Years)

The Stem

5 7 8

1

The leaves

2

0 1 1 5 6 8

3

1 3

The Key

n = 11

1 5 represents 15 years of age

slide2

Ages (Years)

5 7 8

1

2

0 1 1 5 6 8

3

1 3

n = 11

1 5 represents 15 years of age

  • The figures on the left of the line form the stem.
  • Each figure on the right is called a leaf.
  • The leaves increase in value outwards from the stem.
  • Each row is called a level.
  • A title is needed at the top.
  • A key is needed at the bottom.

A stem and leaf diagram is easier to produce if you order the data first.

back to back stem and leaf diagram

Books borrowed from the library

Last Week This week

3

2

0 1

4 1

3

1 5 6

7 4 4 3

4

0 4 5

1 0

5

2

1

6

3

2 0 represents 20 books

n = 10

n = 10

Back to back stem and leaf diagram

Sometimes you want to compare one set of figures with another.

A back to back stem and leaf diagram is useful

The 2 level can be read as “last week there was an opening when 23 books were borrowed and this week there were openings with 20 and 21 books borrowed.” The library opens 10 times a week.

frequency tables
Frequency Tables

The receptionist at a vets surgery notes the types of animals as they are brought in.

This kind of table is referred to as a frequency table.

She decides to sort the data onto a table.

constructing a pie chart
Constructing A Pie Chart

Geologists carry out a survey on rocks. Here are their results.

slide9

1200

1800

slide10

Limestone

Granite

Sandstone

cumulative frequency
Cumulative Frequency

The cumulative frequency of age 5 is 24

This can be interpreted as 24 people in the sample aged 5 or less.

cumulative frequency diagram

Age In Years

(to nearest 5)

Frequency

Cumulative Frequency

10

5

5

15

7

12

20

12

24

25

32

56

cumulative

frequency

30

14

70

35

18

88

40

12

100

Age (years)

Cumulative Frequency Diagram

60 Patients are around 25 Years old

dotplots

13

10

11

12

Dotplots

It is sometimes useful to get a “feel” for the location of a data set on a number line. One way to do this is to construct a dotplot.

A group of athletes are timed in a 100m sprint. Their times are:

10.8, 10.9, 11.2, 11.5, 11.6, 11.6, 11.6, 11.9, 12.2, 12.2, 12.8.

the five figure summary
The Five Figure Summary

When a list of numbers is put in order it can be summarised by quoting five figures.

  • The highest number (H)
  • The lowest number (L)
  • The Median (Q2). This number halves the list and does not belong in either half.
  • The upper quartile (Q3). The median of the upper half.
  • The lower quartile (Q1). The median of the lower half.
slide19

Give a five figure summary of the following data.

3 5 6 6 7 8 8 8 9 10 11

Q1

Q2

H

L

Q3

L = 3

Q1 = 6

Q2 = 8

Q3 = 9

H = 11

slide20

Give a five figure summary of the following data.

3 5 6 6 7 8 8 9 10 11

Q1

Q2

H

L

Q3

L = 3

Q1 = 6

Q2 = ( 7 + 8 )  2 = 7.5

Q3 = 9

H = 11

slide21

Give a five figure summary of the following data.

3 5 6 6 7 8 9 10 11

Q1

Q2

H

L

Q3

L = 3

Q1 = ( 5 + 6 )  2 = 5.5

Q2 = 7

Q3 = ( 9 + 10 )  2 = 9.5

H = 11

boxplots
Boxplots

The five figure summary can be illustrated using a boxplot

A boxplot is drawn to a suitable scale and displays the five figure summary as follows.

H

Q3

Q2

Q1

L

A suitable scale

slide24

0

100

80

60

20

40

Example.

Lowest score = 12; highest score = 97; Q1 = 32, Q2 = 49, Q3 = 66.

For an exam out of 100, the boxplot is:

Note that: 25% of the candidates got between 12 and 32 (lower Whisker)

50% of the candidates got between 32 and 66 (in the box)

25% of the candidates got between 66 and 97 (upper whisker)

comparing distributions
Comparing Distributions

When comparing distributions it is useful to consider two things:

  • The typical score (the mean, the mode or the median)
  • The spread of the marks (the range can be useful, but more often the interquartile range or semi-interquartile range is used)
slide27

56

20

52

38

24

96

11

78

51

32

0

100

80

60

20

40

Boxplots can be used to help compare distributions.

January

June

Comparison of exam results by the same class.

On average the June results are better since the median is higher. But scores tended to be more variable. (larger interquartile range)

Note that the longer the box, the greater the interquartile range and hence the variability.

calculating the quartiles
Calculating the Quartiles

To find the quartiles of an ordered list we consider its length.

slide30

a) Where are the quartiles in a data list of 24 numbers.

13 TO 18

1 TO 6

19 TO 24

7 TO 12

1 TO 12

13 TO 24

24 numbers can be divided into 2 equal groups of 12 numbers.

The median will be between the 12th and 13th numbers

The lower quartile will be between the 6th and 7th numbers

The upper quartile will be between the 18th and 19th numbers

slide31

14 TO 19

1 TO 6

20 TO 25

7 TO 12

1 TO 12

14 TO 25

b) Where are the quartiles in a data list of 25 numbers.

25 numbers can be divided into 2 equal groups of 12 numbers.

The median will be the 13th number

The lower quartile will be between the 6th and 7th numbers

The upper quartile will be between the 19th and 20th numbers

slide32

14 TO 19

1 TO 6

21 TO 26

8 TO 13

1 TO 13

14 TO 26

c) Where are the quartiles in a data list of 26 numbers.

26 numbers can be divided into 2 equal groups of 13 numbers.

The median will be between the 13th and 14th numbers

The lower quartile will be the 7th number

The upper quartile will be the 20th number

slide33

15 TO 20

1 TO 6

22 TO 27

8 TO 13

1 TO 13

15 TO 27

d) Where are the quartiles in a data list of 27 numbers.

27 numbers can be divided into 2 equal groups of 13 numbers.

The median will be the 14th number

The lower quartile will be the 7th number

The upper quartile will be the 21st number

using a cumulative frequency column
Using a Cumulative Frequency Column

The frequency table shows the length of commercial breaks in minutes, broadcast on a TV channel one evening. Calculate the median and the quartiles of these times.

Data list is 22 numbers long

By adding a cumulative frequency column we can see the total data list

slide36

12 TO 16

1 TO 5

18 TO 22

7 TO 11

1 TO 11

12 TO 22

6th number is here

11th and 12th numbers are here

17th number is here

22 numbers can be split into two equal groups of 11

4 mins

4 mins

Q2 =

Q3 =

3 mins

Q1 =