STANDARD DEVIATIONSTANDARD DEVIATION

Quartiles from Frequency Tables

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Reminder !

S5 Int2

Range : The difference between highest and Lowest values. It is a measure of spread.

Median : The middle value of a set of data.When they are two middle values the median is half way between them.

Mode : The value that occurs the most in a set of data. Can be more than one value.

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S5 Int2

Standard DeviationStandard DeviationFor a FULL set of DataFor a FULL set of Data

The range measures spread. Unfortunately any big change in either the largest value or smallest scorewill mean a big change in the range, even though

onlyone number may have changed.

The semi-interquartile range is less sensitive to a single number changing but again it is only really

based on two of the score.

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S5 Int2

Standard DeviationStandard DeviationFor a FULL set of DataFor a FULL set of Data

A measure of spread which uses all the data is the

Standard Deviation

The deviation of a score is how much the score differs from the mean.

Score Deviation(Deviation)2

70

72

75

78

80

Totals 375

Example 1 :Find the standard deviation of these fivescores 70, 72, 75, 78, 80.

S5 Int2

Standard DeviationStandard DeviationFor a FULL set of DataFor a FULL set of Data

Step 1 : Find the mean

375 ÷ 5 = 75Step 3 : (Deviation)2

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-5

-3

0

3

5

0

25

9

0

9

25

68

Step 2 : Score - MeanStep 4 : Mean square deviation

68 ÷ 5 = 13.6

Step 5 :

Take the square root of step 4

√13.6 = 3.7

Standard Deviation is 3.7 (to 1d.p.)

Example 2 :Find the standard deviation of these sixamounts of money £12, £18, £27, £36, £37, £50.

S5 Int2

Standard DeviationStandard DeviationFor a FULL set of DataFor a FULL set of Data

Step 1 : Find the mean

180 ÷ 6 = 30

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Step 2 : Score - Mean

Step 3 : (Deviation)2

Step 4 : Mean square deviation

962 ÷ 6 = 160.33

Score Deviation(Deviation)2

12

18

27

36

37

50

Totals 180

-18

-12

-3

6

7

20

324

144

9

36

49

400

0 962

Step 5 :

Take the square root of step 4

√160.33 = 12.7 (to 1d.p.)

Standard Deviation is £12.70

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S5 Int2

Standard DeviationStandard DeviationFor a FULL set of DataFor a FULL set of Data

When Standard Deviationis LOW it means the data values are close to the

MEAN.

When Standard Deviationis HIGH it means the data values are spread out from

the MEAN.

Mean Mean

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S5 Int2

Standard DeviationStandard DeviationFor a Sample of DataFor a Sample of Data

In real life situations it is normal to work with a sample of data ( survey / questionnaire ).

We can use two formulae to calculate the sample deviation.

2( )

1

x xs

n

s = standard deviationn = number in sample∑ = The sum of

22

1

xx

nsn

x = sample mean

We will use this version because it is easier to use in

practice !

Example 1a : Eight athletes have heart rates 70, 72, 73, 74, 75, 76, 76 and 76.

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S5 Int2

Standard DeviationStandard DeviationFor a Sample of DataFor a Sample of Data

Heart rate (x)

x2

70

72

73

74

75

76

76

76

Totals

4900

5184

5329

5476

5625

5776

5776

5776

∑x2 = 43842∑x = 592

Step 2 :

Square all the values and find the

total

Step 3 :

Use formula to calculate sample deviation

22

1

xx

nsn

259243842

88 1

s

43842 43808

7s

4.875s

2.2 ( 1 . .) s to d p

Step 1 :

Sum all the values

Q1a. Calculate the mean :

592 ÷ 8 = 74

Q1a. Calculate the sample deviation

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Heart rate (x)

x2

80

81

83

90

94

96

96

100

Totals

6400

6561

6889

8100

8836

9216

9216

10000

65218 64800

7s

418s

20.4 1 . .) ( s to d p

Example 1b : Eight office staff train as athletes. Their Pulse rates are 80, 81, 83, 90, 94, 96, 96 and 100 BPM

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S5 Int2

Standard DeviationStandard DeviationFor a Sample of DataFor a Sample of Data

∑x = 720

22

1

xx

nsn

272065218

88 1

s

Q1b(ii) Calculate the sample deviation

Q1b(i) Calculate the mean :

720 ÷ 8 = 90

∑x2 = 65218

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S5 Int2

Standard DeviationStandard DeviationFor a Sample of DataFor a Sample of Data

Q1b(iii) Who are fitter the athletes or staff.

Compare meansAthletes are fitter

StaffAthletes

2.2 1 . .) ( s to d p

74 Mean BPM 90 Mean BPM

20.4 1 . .) ( s to d p

Q1b(iv) What does the deviation tell us.Staff data is more

spread out.