1. Standard
Deviation

2. it may be height, weight, time etc.
Symbols used
x – the variable:
it may be height, weight, time etc.
x (x bar) – the mean:
The sum of all the items ÷ number of items
 (sigma) – the sum of:
x means add up all the items in the column x.

3. Symbols used n – the number of items: Example x : 7 , 8 , 8 , 10 , 12

4. Standard Deviation is a measure of how values are spread out about the mean
Higher Standard Deviation
Low Standard Deviation
High Standard Deviation

5. Standard Deviation for a Sample of Data
In real life situations it is normal to work with a sample
of data ( survey / questionnaire ).
We will use this version because it works for all types of questions.
There are two formulae we can use to calculate the sample deviation.
Both are given in the formulae list.
s = standard deviation
 = The sum of
x = sample mean
n = number in sample

6. Eight athletes have heart rates
Standard Deviation
Eight athletes have heart rates
70, 72, 73, 74, 75, 76, 76 and 76.
Totals
Heart Rate x x2 70
4900 72
5184 73
5329 74
5476 75
5625 76
5776 76
5776 76
5776
43842

7. Example 1b : Eight office staff train as athletes.
Their Pulse rates are 80, 81, 83, 90, 94, 96, 96 and 100 BPM
Heart rate x x2 80 81 83 90 94 96
100
6400
6561
6889
8100
8836
9216
9216
10000
x = 720
65218

8. Who are fitter the athletes or staff. Staff data is more spread out.
Compare means
What does the deviation tell us.
Staff data is more spread out.
Athletes are fitter
Athletes
Staff
Mean = 74 bpm
Mean = 90 bpm
Std Dev = 2·20 bpm
Std Dev = 7·73 bpm

9. Calculate the mean and Standard Deviation for each sample
Mathematics
English
Chemistry
French 17 13 15 8 14 7 11 9 12 5 10 19 20 16

10. 1. A sample of fifteen matchboxes was chosen from an. assembly line
1. A sample of fifteen matchboxes was chosen from an assembly line. The number of matches in each box was recorded. The results are shown below.
(a) Calculate, correct to 3 significant figures, the mean and standard deviation of this sample.
(b) Boxes which contain less than one standard deviation of matches below the mean are to be discarded and refilled.
How many boxes in this sample would be discarded?

11. 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 below.
(a) Calculate, correct to 3 significant figures, the mean and standard deviation of this sample.
(b) A “highly commended” certificate is to be given to any student who has a mark that is more than one standard deviation above the mean.
How many students in this sample would receive this certificate?

12. Using the formula when given ∑ x and ∑ x2
Standard Deviation
Using the formula when given
∑ x and ∑ x2

13. Using the formula :
Sometimes the values of x and x2 are given and the above formula needs to be used to calculate s, the Standard Deviation.

14.  Use the formula to work out the standard deviation
in each of the examples below.
In a race the 8 finalists gave the following time data ∑ x = 360· ∑ x2 = 16223·2
Calculate the mean and standard deviation giving your answers correct to 1 decimal place. 
s = 0·9
Mean = 360·2 ÷ 8 = 45·0

15. In a class of 10 pupils, the heights data gave the following
results
∑ x = ∑ x2 = 42475
Calculate the mean and standard deviation giving your answers correct to 1 decimal place.
3. In a class of 12 pupils the marks in a test gave the data
∑ x = ∑ x2 = 3678
Calculate the mean and standard deviation giving your
answers correct to 1 decimal place.
In a survey to see how many books were carried to school by
10 pupils the data obtained gave the results
∑ x = ∑ x2 = 269
Calculate the mean and standard deviation giving your answers correct to 1 decimal place.

16. In a 400m race the 7 finalists gave the following time
(seconds) data ∑ times = ∑ times2 = 21082
Calculate the mean and standard deviation giving your answers correct to 1 decimal place.
The heights ‘h’ in cm of 10 plants gave the following
results ∑ h = ∑ h2 =
Calculate the mean and standard deviation of the plant heights giving your answers correct to 1 decimal place.
In a class of 6 pupils doing a sixth year maths exam the results data showed that
∑ marks = ∑ marks2 = 29686
Calculate the mean and standard deviation of the marks giving your answers correct to the nearest whole number.

17. 8. In a survey to see how much television is watched by pupils in a single night, the data produced by 12 pupils gave the following results ∑ hours = ∑ hours2 =
Calculate the mean and standard deviation giving your answers correct to 2 decimal places.
The heights ‘h’ in metres of 8 pupils gave the following
results ∑ heights = ∑ heights2 =
Calculate the mean and standard deviation of the heights giving your answers correct to 1 decimal place.
The weights ‘w’ in kg of the same 8 pupils gave the
following results ∑ weights = ∑ weights2 =
Calculate the mean and standard deviation of the weights giving your answers correct to 1 decimal place.

18. Question Mean Standard Deviation 1 45.0 0.9 2 61.8 21.8 3 16 7.4 4 4.8
Answers to Using the Formula
Question
Mean
Standard Deviation 1
45.0
0.9 2
61.8
21.8 3 16
7.4 4
4.8
2.1 5
54.9
1.7 6
6.6
1.5 7 67 22 8
3.91
1.48 9
1.9
0.6 10
282.4
21.3