Lesson Element
Unit 3: Analytical Techniques
LO1: Be able to use mathematical techniques
to analyse data
Variance and Standard Deviation
Instructions and answers for tutors
These instructions cover the learner activity section which can be found on page 4. This
Lesson Element supports Cambridge Technicals Level 3 in Science for Technicians.
When distributing the activity section to the learners either as a printed copy or as a
Word file you will need to remove the tutor instructions section.
The activity
Variance and standard deviation are statistical techniques that are used to determine how
spread out a set of data is from the mean value. They are often used to determine the quality
of a set of data.
In these two activities, learners will determine the variance and standard deviation for sets of
data. In Activity 1, the problem involves a sample set of data and in Activity 2 a set of data
representing the population.
Both activities provide the opportunity for developing ICT skills by solving the problems using
a spreadsheet.
Learners will require access to the Excel spreadsheet:
Standard Deviation Problems Data
Suggested timings
Activity 1: 45 minutes
Activity 2: 45 minutes
ABC – This activity offers an
opportunity for English skills
development.
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123 – This activity offers an
opportunity for maths skills
development.
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WORK – This activity offers
an opportunity for work
experience.
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Activity 1
To note – this problem involves sample data. The formula for variance and standard
deviation must take account of this with N  1 being used.
The solutions to this problem have been calculated with the aid of a spreadsheet. Learners
could calculate solutions manually, or could use ICT to develop a solution.
Part a
Mean Blood Pressure = 1464 / 18 = 81.3 mm Hg
Part b
Variance 2 = average or the squared differences from the mean = 1788 / (18  1)
Note: for sample data N = N – 1 i.e. 18 – 1
Variance 2 = 105.2 mm Hg
Part c
1
Standard deviation s = √N-1 ∑N
̅)2
i=1(xi -x
Standard deviation =  variance =  105.2 = 10.3 mm Hg
Part d
1SD higher than mean = 81.3 + 10.3 = 91.6 mm Hg
1SD lower than mean = 81.3 – 10.3= 71.1 mm Hg
In order to pass the blood pressure readings must be in the range 71.1 mm Hg to
91.6 mm Hg.
For the sample ... 4 out of 18 blood pressure readings fail.
Part e
4/18 × 100% = 22% blood pressure readings fail.
This means that a more detailed study should be conducted.
Tutors could develop similar example problems.
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Activity 2
To note – this problem involves population data. The formula for variance and standard
deviation must take account of this with N being used.
The solutions to this problem have been calculated with the aid of a spreadsheet. Again,
solutions could be determined manually or using ICT.
Part a
Mean concentration = 200 / 20 = 10.0 mol dm-3
Part b
Variance 2 = average or the squared differences from the mean = 1.32 / 20
Note: for population data N = N i.e. 20. Variance 2 = 0.07
Part c
1
2
Standard deviation 𝑠 = √𝑁 ∑𝑁
𝑖=1(𝑥𝑖 − 𝑥̅ )
Standard deviation =  variance =  0.07 = 0.26 mol dm-3
Part d
2SD higher than mean = 10.0 + (2 × 0.26) = 10.52 mol dm-3
2SD lower than mean = 10.0 – (2 × 0.26) = 9.48 mol dm-3
In order to pass quality control all concentration readings must be in the range 9.48 mol dm-3
to10.52 mol dm-3.
For the population data all values are in the range, and so the batch of sulfuric acid passes.
Tutors could develop similar example problems.
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© OCR 2016
Lesson Element
Unit 3: Analytical Techniques
LO1: Be able to use mathematical techniques
to analyse data
Learner Activity
Variance and Standard Deviation
Variance and standard deviation are statistical techniques that are used to determine how
spread out a set of data is from the mean value. They are often used to determine the quality
of a set of data.
In these two activities, you will determine the variance and Standard Deviation for sets of
data. In Activity 1, the problem involves a sample set of data and in Activity 2 a set of data
representing the population. You will need to carefully select the correct formula to use for
each activity.
Both activities provide you with the opportunity to develop your skills in using a spreadsheet
to solve problems.
Activity 1
A doctor’s practice carries out a small-scale study of the diastolic blood pressure of patients
in a particular age group.
During a one-week period they take a sample of the blood pressure readings of 18 patients.
The sample is believed to be representative of all of the patients in this age group registered
with the practice. Here are the measured diastolic blood pressures (mm Hg):
72
75
79
75
105
74
85
68
76
83
73
82
99
80
84
78
100
76
The lead nurse at the practice wishes to conduct a more detailed study if more than 10% of
the blood pressure readings are greater than one standard deviation (1SD) away from the
mean.
(a) Determine the mean for the sample.
(b) Determine the variance.
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(c) Determine the standard deviation.
(d) Identify all blood pressure readings in the sample that are greater than 1SD from the
mean.
Determine the percentage of blood pressure readings that are greater than 1SD from the
mean, and if a more detailed study should be conducted.
Activity 2
During the production of sulfuric acid in a chemical plant, the population data for 20 readings
of sulfuric acid concentration (mol dm-3) is as follows:
9.8
10.3
9.7
10.2
9.9
10.2
10.1
9.8
10.2
9.9
9.5
10.2
10.3
10
10.4
10
9.6
10.1
9.6
10.2
The processing of the sulfuric acid batch requires that none of the readings are more than
two standard deviations (2SD) from the mean.
(a) Determine the mean for the population.
(b) Determine the variance.
(c) Determine the standard deviation.
Identify if any of the readings are more than 2SD from the mean and if the batch passes or
fails.
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