Symbols usedSymbols usedx – x – the variable:the variable:

it may be height, weight, time etc.it may be height, weight, time etc.

x (x bar) – x (x bar) – the mean:the mean:

The sum of all the items The sum of all the items ÷ number of items÷ number of items

(sigma) (sigma) – – the sum of:the sum of:

x means add up all the items in the column x.x means add up all the items in the column x.

Symbols usedSymbols used

n – n – the number of items:the number of items:

ExampleExample x x : 7 , 8 , 8 , 10: 7 , 8 , 8 , 10 , 12 , 12

n

xx

becomes mean The

x x : 7 + 8 + 8 + 10 + 12 = 45: 7 + 8 + 8 + 10 + 12 = 45 nn = 5 = 5

95

45

n

xx

Standard Deviation is a measure of how values are spread out about the mean

Low Standard Low Standard DeviationDeviation

Higher Higher Standard Standard DeviationDeviation

High Standard High Standard DeviationDeviation

Standard Deviation for a Sample of DataStandard Deviation for a Sample of DataIn real life situations it is normal to work with a sample

of data ( survey / questionnaire ).

There are two formulae we can use to calculate the sample deviation.

Both are given in the formulae list.

s = standard deviation

n = number in sample

= The sum of

2

2

1

xx

nsn

x = sample mean

We will use this version We will use this version because it works for all because it works for all

types of questions.types of questions.

s(x x)2n 1

Standard DeviationStandard DeviationEight athletes have heart rates

70, 72, 73, 74, 75, 76, 76 and 76.

TotalsTotals

70

72

73

74

7576

76

76

592 x

Heart Rate x

4900

x2

5184

5329

5476

5625

5776

5776

5776

2x 43842

nx

x 8

592

74

1

)( 22

nnx

xs

78

)592(43842

2

s

734

s 202s

Heart rate x x2

80

81

83

90

94

96

96

100

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

x = 720

64006400

65616561

68896889

81008100

88368836

92169216

92169216

1000010000

6521865218 2x

nx

x

90x

1

)( 22

nnx

xs

78

72065218

2

s

7418

s

8720

x

737s

Who are fitter the Who are fitter the athletes or staff.athletes or staff.Compare meansCompare means

StaffAthletes

What does the What does the deviation tell us.deviation tell us.

Athletes are Athletes are fitterfitter

Staff data is more spread

out.

Mean = 74 bpm

Std Dev = 2·20 bpm

Mean = 90 bpm

Std Dev = 7·73 bpm

MathematicsMathematics EnglishEnglish ChemistryChemistry FrenchFrench

1717 1313 1515 88

1414 77 1313 1111

99 1212 1717 1313

55 1010 1919 77

1313 99 2020 1010

1111 77 1414 88

1212 1010 1616

1010 88

99

Calculate the mean and Standard Deviation Calculate the mean and Standard Deviation for each samplefor each sample

1.1. A sample of fifteen matchboxes was chosen from an A sample of fifteen matchboxes was chosen from an assembly line. The number of matches in each box was assembly line. The number of matches in each box was recorded. The results are shown below.recorded. The results are shown below.

6565 6363 6666 6262 5959 6161 6262 6262

(a)(a) Calculate, correct to 3 significant figures, the mean Calculate, correct to 3 significant figures, the mean and standard deviation of this sample.and standard deviation of this sample.

(b)(b) Boxes which contain less than one standard deviation Boxes which contain less than one standard deviation of matches below the mean are to be discarded and of matches below the mean are to be discarded and refilled. refilled. How many boxes in this sample would be discarded?How many boxes in this sample would be discarded?

2.2. A group of students took part in an end of year assessment. A group of students took part in an end of year assessment. The marks out of fifty for a sample of these students are listed The marks out of fifty for a sample of these students are listed below.below.

3838 2121 4444 4848 3535 3232 4545 4545 3939 3131

(a)(a) Calculate, correct to 3 significant figures, the mean and Calculate, correct to 3 significant figures, the mean and standard deviation of this sample.standard deviation of this sample.

(b)(b)A “highly commended” certificate is to be given to any A “highly commended” certificate is to be given to any student who has a mark that is more than one standard student who has a mark that is more than one standard deviation above the mean.deviation above the mean.

How many students in this sample would receive this How many students in this sample would receive this certificate? certificate?

s x x 2n 1

x 2 x 2

/n

n 1

Standard DeviationStandard Deviation

Using the formula when given Using the formula when given

∑ ∑ xx and and ∑ ∑ xx22

Using the formula :Using the formula :

1

)( 22

nnx

xs

Sometimes the values of Sometimes the values of xx and and xx22 are given are given and the above formula needs to be used to and the above formula needs to be used to

calculate calculate ss, the Standard Deviation., the Standard Deviation.

Use the formula to work out the standard deviationUse the formula to work out the standard deviation in each of the examples below. in each of the examples below.

1. In a race the 8 finalists gave the following time data ∑ x = 360∑ x = 360··22 ∑ x∑ x22 = 16223 = 16223··22

Calculate the mean and standard deviation giving your answers correct to 1 decimal place.

1

)( 22

nnx

xs

78

2360216223

2s

ss = 0·9 = 0·9

MeanMean = 360·2 = 360·2 ÷ 8 = ÷ 8 = 4545·0·0

2.2. In a class of 10 pupils, the heights data gave the following In a class of 10 pupils, the heights data gave the following resultsresults

∑ ∑ x = 618 ∑ xx = 618 ∑ x22 = 42475 = 42475 Calculate the mean and standard deviation giving your Calculate the mean and standard deviation giving your

answers correct to 1 decimal place.answers correct to 1 decimal place.

3. In a class of 12 pupils the marks in a test gave the data3. In a class of 12 pupils the marks in a test gave the data

∑ ∑ x = 192 ∑ xx = 192 ∑ x22 = 3678 = 3678 Calculate the mean and standard deviation giving your Calculate the mean and standard deviation giving your answers correct to 1 decimal place.answers correct to 1 decimal place.

4.4. In a survey to see how many books were carried to school by In a survey to see how many books were carried to school by 10 pupils the data obtained gave the results 10 pupils the data obtained gave the results

∑ ∑ x = 48 ∑ xx = 48 ∑ x2 2 = 269= 269 Calculate the mean and standard deviation giving your Calculate the mean and standard deviation giving your

answers correct to 1 decimal placeanswers correct to 1 decimal place.

5. In a 400m race the 7 finalists gave the following time (seconds) data ∑ times = 384 ∑ times∑ times = 384 ∑ times22 = 21082 = 21082 Calculate the mean and standard deviation giving your

answers correct to 1 decimal place.

6.6. The heights ‘h’ in cm of 10 plants gave the following The heights ‘h’ in cm of 10 plants gave the following results results ∑ h = 66 ∑ h∑ h = 66 ∑ h22 = 456.16 = 456.16 Calculate the mean and standard deviation of the plant Calculate the mean and standard deviation of the plant

heights giving your answers correct to 1 decimal place.heights giving your answers correct to 1 decimal place.

7.7. In a class of 6 pupils doing a sixth year maths exam the In a class of 6 pupils doing a sixth year maths exam the results data showed thatresults data showed that

∑ ∑ marks = 404 ∑ marksmarks = 404 ∑ marks22 = 29686 = 29686 Calculate the mean and standard deviation of the marks Calculate the mean and standard deviation of the marks

giving your answers correct to the nearest whole number.giving your answers correct to the nearest whole number.

8. In a survey to see how much television is watched by pupils in In a survey to see how much television is watched by pupils in a single night, the data produced by 12 pupils gave the a single night, the data produced by 12 pupils gave the following results following results ∑ hours = 46.9 ∑ hours ∑ hours = 46.9 ∑ hours22 = 207.31 = 207.31

Calculate the mean and standard deviation giving your Calculate the mean and standard deviation giving your answers correct to 2 decimal places.answers correct to 2 decimal places.

9.9. The heights ‘h’ in metres of 8 pupils gave the following The heights ‘h’ in metres of 8 pupils gave the following results results ∑ heights = 14.76 ∑ heights ∑ heights = 14.76 ∑ heights22 = 30.144 = 30.144 Calculate the mean and standard deviation of the heights Calculate the mean and standard deviation of the heights

giving your answers correct to 1 decimal place.giving your answers correct to 1 decimal place.

10.10.The weights ‘w’ in kg of the same 8 pupils gave the The weights ‘w’ in kg of the same 8 pupils gave the following results following results ∑ weights = 2259 ∑ weights∑ weights = 2259 ∑ weights22 = 641047 = 641047 Calculate the mean and standard deviation of the weights Calculate the mean and standard deviation of the weights

giving your answers correct to 1 decimal place.giving your answers correct to 1 decimal place.

QuestionQuestion MeanMean Standard Standard DeviationDeviation

11 45.045.0 0.90.9

22 61.861.8 21.821.8

33 1616 7.47.4

44 4.84.8 2.12.1

55 54.954.9 1.71.7

66 6.66.6 1.51.5

77 6767 2222

88 3.913.91 1.481.48

99 1.91.9 0.60.6

1010 282.4282.4 21.321.3

Answers to Using the Formula