Year 8: Data Handling 2

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

Last modified: 11th December 2014

Learning Outcomes: To understand stem and leaf diagrams, frequency polygons, box plots and cumulative frequency graphs.

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41 2 4 52 5 6 6 6 7 7 8 80 1 2 2 4 5 6 7 7 7 7 80 1 1 2

Key:2 | 1 means 2.1cm

Stem and Leaf Diagram - What is it?Suppose this “stem and leaf diagram” represents the lengths of beetles.

These numbers represent the first digit of the number.

These numbers represent the second.

The key tells us how two digits combine.

Value represented = 4.5cm?The numbers must be in order.

Example

Here are the weights of a group of cats. Draw a stem-and-leaf diagram to represent this data.

36kg 15kg 35kg 50kg 11kg 36kg 38kg 47kg 12kg 30kg 18kg 57kg

1

2

3

4

5

1 2 5 8

0 5 6 6 8

7

0 7

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Key:3 | 8 means 38kg?

What do you think are the advantages of displaying data in a stem-and-leaf diagram?

•Shows how the data is spread out.•Identifies gaps in the values.•All the original data is preserved (i.e. we don’t ‘summarise’ in any way).

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Your turn

Here is the brain diameter of a number of members of 8IW. Draw a stem and leaf diagram representing this data.

1.3cm 2.1cm 5.3cm 2.0cm 1.7cm 4.2cm 3.3cm 3.2cm 1.3cm 4.6cm 1.9cm

1

2

3

4

5

3 3 7 9

0 1

2 3

2 6

3

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Key:3 | 8 means 3.8cm?

Median width = 2.1cm ?

Lower Quartile = 1.7cmUpper Quartile = 4.2cm

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Exercises

Q1 on your provided worksheet.(Ref: Yr8-DataHandlingWorksheet.doc)

Suppose we wanted to plot the following data, where each value has a frequency.A suitable representation of this data would be a bar chart.

Frequency Diagram

Shoe Size Frequency

8 26

9 42

10 103

11 34

12 5

100

80

60

40

20

8 9 10 11 12

Shoe size

Freq

uenc

y

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When bar charts have frequency on the y-axis, they’re known as frequency diagrams.

But suppose that we had data grouped into ranges.What would be a sensible value to represent each range?

90 100 110 120 130 140

16

14

12

10

8

6

4

2

Frequency Polygons

IQ (x) Frequency

90 ≤ x < 100 2

100 ≤ x < 110 15

110 ≤ x < 120 8

120 ≤ x < 130 0

130 ≤ x < 140 4

Join the points up with straight lines.

This is known as a frequency polygon.

Modal class interval:100 ≤ x < 110?

Frequency Polygons – Exercises on sheet

b) 30 < x ≤ 40c) 16%

Q1 Q2

b) 20 < x ≤ 30c) 16%

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Median/LQ/UQ class interval

Estimate of Median/LQ/UQ/num

values in range

Determine Median/LQ/UQ

Width (cm) Frequency

0 < w < 10 4

10 < w < 25 6

25 < w < 60 2

Width (cm) Cum Freq

0 < w < 10 4

0 < w < 25 10

0 < w < 60 12

Widths (cm): 4, 4, 7, 9, 11, 12, 14, 15, 15, 18, 28, 42

Cumulative Frequency Table

Cumulative Frequency Graph

Box Plots

Histogram

Grouped Frequency Table

Frequency Polygon

The Whole Picture

Recap: Lower and Upper Quartile

Suppose that we line up everyone in the school according to height.

We already know that the median would be the middle person’s height.

50% of the people in the school would have a height less than them.

50%

The height of the person 25% along the line is known as the:lower quartile

The upper quartile is the height of the person 75% along the data.

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Check your understanding

50% of the data has a value more than the median.

75% of the data has a value less than the upper quartile.

25% of the data has a value more than the upper quartile.

75% of the data has a value more than the lower quartile.

0% 25% 50% 75% 100%

LQ UQMedian

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Here are the ages of 10 people at Pablo’s party. Choose the correct value.

12, 13, 14, 14, 15, 16, 16, 17, 19, 24

15 16Median: 15.5

13 14Lower: 13.5

17 19UQ: 18

(Click to vote)

InterquartileRange: 3 Range: 12??

Median/Quartile Revision

3

2.51.5 3.5

1.5 2 4.5

2 3.5 5

21? ? ?

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? ? ?

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Quickfire Quartiles

1, 2, 3

LQ Median UQ

1, 2, 3, 4

1, 2, 3, 4, 5

1, 2, 3, 4, 5, 6

Rule for lower quartile:•Even num of items: find median of bottom half.•Odd num of items: throw away middle item, find medium of remaining half.

What if there’s lots of items?

There are 31 items, in order of value. What items should we use for the median and lower/upper quartiles?

0 1 1 2 4 5 5 6 7 8 10 10 14 14 14 14 15 16 17 29 31 31 37 37 38 39 40 40 41 43 44

Use the 16th item Use itemMedian

LQ

UQ

Use the 8 th item Use item

Use the 24th item Use item

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Num items

15

23

39

4th 8th 12th

6th 12th 18th

10th 20th 30th

12th 24th 36th 47

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What if there’s lots of items?

LQ Median UQ

Box PlotsBox Plots allow us to visually represent the distribution of the data.

Minimum Maximum Median Lower Quartile Upper Quartile

3 27 17 15 22

0 5 10 15 20 25 30

Sketch Sketch Sketch Sketch Sketch

How is the IQR represented in this diagram?

How is the range represented in this diagram?

Sketch Sketch

IQR

range

Box Plots

0 4 8 12 16 20 24

Sketch a box plot to represent the given weights of cats:

5lb, 6lb, 7.5lb, 8lb, 8lb, 9lb, 12lb, 14lb, 20lb

Minimum Maximum Median Lower Quartile Upper Quartile

5 20 8 6.75 13? ? ? ? ?

Sketch

Box Plots

0 4 8 12 16 20 24

Sketch a box plot to represent the given ages of people at Dhruv’s party:

5, 12, 13, 13, 14, 16, 22

Minimum Maximum Median Lower Quartile Upper Quartile

5 22 13 12 16? ? ? ? ?

Sketch

£100k £150k £200k £250k £300k £350k £400k £450k

Kingston

Croydon

Box Plot comparing house prices of Croydon and Kingston-upon-Thames.

Comparing Box Plots

“Compare the prices of houses in Croydon with those in Kingston”. (2 marks)

For 1 mark, one of:•In interquartile range of house prices in Kingston is greater than Croydon.•The range of house prices in Kingston is greater than Croydon.

For 1 mark:•The median house price in Kingston was greater than that in Croydon.•(Note that in old mark schemes, comparing the minimum/maximum/quartiles would have been acceptable, but currently, you MUST compare the median)

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Time (s) Frequency Cum Freq

9.6 < t ≤ 9.7 1 1

9.7 < t ≤ 9.9 4 5

9.9 < t ≤ 10.05 10 15

10.05 < t ≤ 10.2 17 32

TOTAL

10.05 < t ≤ 10.2

Modal class interval

10.05 < t ≤ 10.2

Median class interval

Estimate of mean

10.02

100m times at the 2012 London Olympics

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Time (s) Frequency Cum Freq

9.6 < t ≤ 9.7 1 1

9.7 < t ≤ 9.9 4 5

9.9 < t ≤ 10.05 10 15

10.05 < t ≤ 10.2 17 32

9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 10.3Time (s)

Cum

ulati

ve F

requ

ency

32

28

24

20

16

12

8

4

0

Median = 10.07s

Lower Quartile = 9.95s

Upper Quartile = 10.13s

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Interquartile Range = 0.18s?

Cumulative Frequency Graphs

Plot

Plot

Plot

Plot

This graph tells us how many people had “this value or less”.

9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 10.3Time (s)

Cum

ulati

ve F

requ

ency

32

28

24

20

16

12

8

4

0

Cumulative Frequency Graphs

Estimate how many runners had a time less than 10.15s.

26 runners

Estimate how many runners had a time more than 9.95

32 – 8 = 24 runners

Estimate how many runners had a time between 9.8s and 10s

11 – 3 = 8 runners

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A Cumulative Frequency Graph is very useful for finding the number of values greater/smaller than some value, or within a range.

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Time (s) Frequency Cum Freq

9.6 < t ≤ 9.7 1 1

9.7 < t ≤ 9.9 4 5

9.9 < t ≤ 10.05 17 22

10.05 < t ≤ 10.2 10 32

9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 10.3

Time (s)

Cum

ulati

ve F

requ

ency

32

28

24

20

16

12

8

4

0

Cumulative Frequency Graph

Plot

Plot

Plot

Plot

Freq

uenc

y

18

16

14

12

10

8

4

2

09.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2

Time (s)

Frequency Polygon

Sketch Line

Worksheet

Cumulative Frequency GraphsPrinted handout. Q5, 6, 7, 8, 9, 10

Reference: GCSE-GroupedDataCumFreq

523353940

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179?

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34

Lower Quartile = 16

Upper Quartile = 44.5

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We previously found:Minimum = 9, Maximum = 57, LQ = 16, Median = 34, UQ = 44.5

1 mark: Range/interquartile range of boys’ times is greater.1 mark: Median of boys’ times is greater.

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44100134153160

25< 𝐴≤35

30

40.9−24.1=16.8?

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B

C

D

A

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