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Standard Deviation. Symbols used. x (x bar) – the mean: The sum of all the items ÷ number of items. x – the variable: it may be height, weight, time etc. (sigma) – the sum of: x means add up all the items in the column x. Symbols used. n – the number of items:. - PowerPoint PPT Presentation
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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