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D4 Moving averages and cumulative frequency

Maths Age 14-16

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Contents

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D4.5 Box-and-whisker diagrams

D4 Moving averages and cumulative frequency

D4.3 Cumulative frequency

D4.2 Plotting moving averages

D4.1 Moving averages

D4.4 Using cumulative frequency graphs

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A box-and-whisker diagram

A box-and-whisker diagram, or boxplot, can be used to illustrate the spread of the data in a given distribution using the highest and lowest values, the median, the lower quartile and the upper quartile.These values can be found from a cumulative frequency graph.

Time in seconds

Cum

ulat

ive

freq

uenc

y

30 35 40 45 50 55 60

10

20

30

40

50

60

70

80

90

100

0

For example, for this cumulative frequency graph showing the results of 100 people holding their breath,

Minimum value = 30

Lower quartile = 42

Median = 47

Upper quartile = 51

Maximum value = 60

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A box-and-whisker diagram

The corresponding box-and-whisker diagram is as follows:

30

Minimum value

42

Lower quartile

47

Median

51

Upper quartile

60

Maximum value

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Lap times

James takes part in karting competitions and his Dad records his lap times on a spreadsheet.

The track is 1108 metres long. James’ fastest time in a race was 51.8 seconds.

In which position in the list would the median lap time be?

In 2004, 378 of James’ lap times were recorded.

There are 378 lap times and so the median lap time will be the

378 + 1

2

thvalue ≈ 190th value

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Lap times

In which position in the list would the lower quartile be?

There are 378 lap times and so the lower quartile will be the

378 + 1

4

thvalue ≈ 95th value

In which position in the list would the upper quartile be?

There are 378 lap times and so the upper quartile will be the

284th value378 + 1

4

thvalue ≈3 ×

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Lap times at Shenington karting circuit

James’ lap times are displayed in the following cumulative frequency graph.

Lap times in seconds

Cum

ulat

ive

freq

uenc

y

52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 920

50

100

150

200

250

300

350

400

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Box and whisker plot for James’ race times

What conclusions can you draw about James’ performance?

52

Minimum value

53

Lower quartile

54

Median

58

Upper quartile

91

Maximum value

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Comparing sets of data

Here are box-and-whisker diagrams representing James’ lap times and Shabnum’s lap times.

Who is better and why?

5253

54 58 91

James’ lap times

52 6054 65 86

Shabnum’s lap times