Standard Deviation (Edexcel GCSE Statistics)

Revision Note

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Standard Deviation

What is the standard deviation of a data set?

  • The standard deviation of a data set is a measure of dispersion (i.e. a measure of spread)

    • It measures how the data is spread out relative to the mean

      • If the standard deviation is small then most data values are close to the mean

      • If the standard deviation is large then many data values will be further away from the mean

    • If the data has units (seconds, cm, etc.), then the standard deviation has the same units as the values in the data set.

  • The Greek letter sigma (lower case sigma) is often used for standard deviation

How do I calculate the standard deviation for a data set?

  • There are two different formulas you can use to calculate the standard deviation

    • Usually the second formula will be the quickest one to use

      • But make sure you know how to use both of them

  • Standard space deviation equals square root of 1 over n sum open parentheses x minus x with bar on top close parentheses squared end root

    • In this formula:

      • n is the number of values in the data set

      • x with bar on top is the mean of the data set

      • x is 'any value' in the data set

  • Standard space deviation equals square root of fraction numerator sum x squared over denominator n end fraction minus open parentheses fraction numerator sum x over denominator n end fraction close parentheses squared end root

    • In this formula:

      • n is the number of values in the data set

      • sum x is the sum of all the data values

      • sum x squared is the sum of the squares of all the data values

    • Sometimes a question will give you the values of sum x and sum x squared for a data set

      • In that case definitely use this formula!

  • Both formulas are on the exam formula sheet

    • So you don't need to remember them

    • You just need to know how to use them

Examiner Tips and Tricks

  • Your calculator may be able to calculate the standard deviation for a list of data values

Worked Example

For the following set of data values

6       9       2       11       5

(a) Calculate the mean.

Add up the values and divide by the number of values (5)

fraction numerator 6 plus 9 plus 2 plus 11 plus 5 over denominator 5 end fraction equals 6.6

mean = 6.6


(b) Calculate the standard deviation using square root of 1 over n sum open parentheses x minus x with bar on top close parentheses squared end root.

It is easiest to set up a table to work out the different values

x

x minus x with bar on top

open parentheses x minus x with bar on top close parentheses squared

6

6 minus 6.6 equals negative 0.6

open parentheses negative 0.6 close parentheses squared equals 0.36

9

9 minus 6.6 equals 2.4

open parentheses 2.4 close parentheses squared equals 5.76

2

2 minus 6.6 equals negative 4.6

open parentheses negative 4.6 close parentheses squared equals 21.16

11

11 minus 6.6 equals 4.4

open parentheses 4.4 close parentheses squared equals 19.36

5

5 minus 6.6 equals negative 1.6

open parentheses negative 1.6 close parentheses squared equals 2.56

total

0.36 plus 5.76 plus 21.16 plus 19.36 plus 2.56 equals 49.2


Now we have all the values to put into the formula

square root of 1 fifth open parentheses 49.2 close parentheses end root equals square root of 9.84 end root equals 3.136877...

standard deviation = 3.14 (3 s.f.)

(c) Calculate the standard deviation using square root of fraction numerator sum x squared over denominator n end fraction minus open parentheses fraction numerator sum x over denominator n end fraction close parentheses squared end root.

It is easiest to set up a table to work out the different values

x

x squared

6

6 squared equals 36

9

9 squared equals 81

2

2 squared equals 4

11

11 squared equals 121

5

5 squared equals 25

total

6 plus 9 plus 2 plus 11 plus 5 equals 33

36 plus 81 plus 4 plus 121 plus 25 equals 267


Now we have all the values to put into the formula

square root of 267 over 5 minus open parentheses 33 over 5 close parentheses squared end root equals square root of 9.84 end root equals 3.136877...

standard deviation = 3.14 (3 s.f.)

Standard Deviation from a Table

How do I find the standard deviation for data in a table?

  • A data set may be presented in a table of data values and associated frequencies

    • In this case the formulas to use are different

    • These formulas are not on the exam formula sheet

      • So you need to remember them

      • But note that they are closely related to the basic formulas

    • Usually the second formula will be the quickest one to use

      • But make sure you know how to use both of them

  • Standard space deviation equals square root of fraction numerator sum space f open parentheses x minus x with bar on top close parentheses squared over denominator sum space f end fraction end root

    • In this formula:

      • x with bar on top is the mean of the data set

      • x is 'any value' in the data set

      • f is the frequency associated with a particular data value x

      • sum space f is the sum of all the frequencies (this is the same as the total number of data values in the data set)

  • Standard space deviation equals square root of fraction numerator sum space f x squared over denominator sum space f end fraction minus open parentheses fraction numerator sum space f x over denominator sum space f end fraction close parentheses squared end root

    • In this formula:

      • x is 'any value' in the data set

      • f is the frequency associated with a particular data value x

      • sum space f x is the sum of frequency cross times data space value for all the data values in the set

      • sum space f x squared is the sum of frequency cross times open parentheses data space value close parentheses squared for all the data values in the set

      • sum space f is the sum of all the frequencies (this is the same as the total number of data values in the data set)

    • Sometimes a question will give you the values of sum space f x and sum space f x squared for a data set

      • In that case definitely use this formula!

Examiner Tips and Tricks

  • Your calculator may be able to calculate the standard deviation for a list of data values and their associated frequencies

Worked Example

Kira collected data about the numbers of pet rabbits owned by the members of her local house rabbits association. This data is shown in the following table:

Number of rabbits

1

2

3

4

5

Frequency

2

6

4

6

2

Work out the standard deviation of this data set.


Method 1: using square root of fraction numerator sum space f x squared over denominator sum space f end fraction minus open parentheses fraction numerator sum space f x over denominator sum space f end fraction close parentheses squared end root

It is easiest to set up a table to work out the different values

number, x

f

f x

f x squared

1

2

2 cross times 1 equals 2

2 cross times 1 squared equals 2

2

6

6 cross times 2 equals 12

6 cross times 2 squared equals 24

3

4

4 cross times 3 equals 12

4 cross times 3 squared equals 36

4

6

6 cross times 4 equals 24

6 cross times 4 squared equals 96

5

2

2 cross times 5 equals 10

2 cross times 5 squared equals 50

total

2 plus 6 plus 4 plus 6 plus 2 equals 20

2 plus 12 plus 12 plus 24 plus 10 equals 60

2 plus 24 plus 36 plus 96 plus 50 equals 208

So sum space f equals 20, sum space f x equals 60 and sum space f x squared equals 208
That gives us everything we need to put into the formula

square root of 208 over 20 minus open parentheses 60 over 20 close parentheses squared end root equals square root of 1.4 end root equals 1.183215...

standard deviation = 1.18 (3 s.f.)


Method 2: using square root of fraction numerator sum space f open parentheses x minus x with bar on top close parentheses squared over denominator sum space f end fraction end root

It is easiest to set up a table to work out the different values

number, x

f

f x

x minus x with bar on top

open parentheses x minus x with bar on top close parentheses squared

f open parentheses x minus x with bar on top close parentheses squared

1

2

2 cross times 1 equals 2

2

6

6 cross times 2 equals 12

3

4

4 cross times 3 equals 12

4

6

6 cross times 4 equals 24

5

2

2 cross times 5 equals 10

total

2 plus 6 plus 4 plus 6 plus 2 equals 20

2 plus 12 plus 12 plus 24 plus 10 equals 60

Now that we have the sum of the f and fx columns we can work out the mean
The sum of the fx column is the sum of all the data values
And the sum of the f column is the total number of data values

x with bar on top equals 60 over 20 equals 3

Now we can complete the rest of the table

number, x

f

f x

x minus x with bar on top

open parentheses x minus x with bar on top close parentheses squared

f open parentheses x minus x with bar on top close parentheses squared

1

2

2

1 minus 3 equals negative 2

open parentheses negative 2 close parentheses squared equals 4

2 cross times 4 equals 8

2

6

12

2 minus 3 equals negative 1

open parentheses negative 1 close parentheses squared equals 1

6 cross times 1 equals 6

3

4

12

3 minus 3 equals 0

open parentheses 0 close parentheses squared equals 0

4 cross times 0 equals 0

4

6

24

4 minus 3 equals 1

open parentheses 1 close parentheses squared equals 1

6 cross times 1 equals 6

5

2

10

5 minus 3 equals 2

open parentheses 2 close parentheses squared equals 4

2 cross times 4 equals 8

total

20

60

8 plus 6 plus 0 plus 6 plus 8 equals 28

So sum space f equals 20 and sum space f open parentheses x minus x with bar on top close parentheses squared equals 28
That gives us everything we need to put into the formula

square root of 28 over 20 end root equals square root of 1.4 end root equals 1.183215...

standard deviation = 1.18 (3 s.f.)

Standard Deviation for Grouped Data

How do I find the standard deviation for grouped data?

  • For grouped data we no longer have access to the original data values

    • Therefore we can only find an estimate for the standard deviation

  • To calculate an estimate for the standard deviation for a set of grouped data:

    • Use the same formulas as used for data in a table

      • See the 'Standard Deviation from a Table' spec point

    • But use the midpoints of the class intervals as the data values

      • i.e. as the values for x in the formulas

      • The mean x with bar on top will also be an estimate where it appears in a formula

Examiner Tips and Tricks

  • Your calculator may be able to calculate an estimate for the standard deviation from a list of midpoints and their associated frequencies

Worked Example

Kira collected data about how long the pet rabbits, owned by the members of her local house rabbits association, took to eat their lunch. This data is shown in the following table:

Time, t (minutes)

0 ≤ t < 3

3 ≤ t < 6

6 ≤ t < 9

9 ≤ t < 12

Frequency

1

5

8

6

Work out an estimate for the standard deviation of this data set.


Method 1: using square root of fraction numerator sum space f x squared over denominator sum space f end fraction minus open parentheses fraction numerator sum space f x over denominator sum space f end fraction close parentheses squared end root

It is easiest to set up a table to work out the different values
Remember to use the class interval midpoints as the x values

midpoint, x

f

f x

f x squared

1.5

1

1 cross times 1.5 equals 1.5

1 cross times 1.5 squared equals 2.25

4.5

5

5 cross times 4.5 equals 22.5

5 cross times 4.5 squared equals 101.25

7.5

8

8 cross times 7.5 equals 60

8 cross times 7.5 squared equals 450

10.5

6

6 cross times 10.5 equals 63

6 cross times 10.5 squared equals 661.5

total

1 plus 5 plus 8 plus 6 equals 20

1.5 plus 22.5 plus 60 plus 63 equals 147

2.25 plus 101.25 plus 450 plus 661.5 equals 1215

So sum space f equals 20, sum space f x equals 147 and sum space f x squared equals 1215
That gives us everything we need to put into the formula

square root of 1215 over 20 minus open parentheses 147 over 20 close parentheses squared end root equals square root of 6.7275 end root equals 2.593742...

standard deviation = 2.59 (3 s.f.)


Method 2: using square root of fraction numerator sum space f open parentheses x minus x with bar on top close parentheses squared over denominator sum space f end fraction end root

It is easiest to set up a table to work out the different values
Remember to use the class interval midpoints as the x values

midpoint, x

f

f x

x minus x with bar on top

open parentheses x minus x with bar on top close parentheses squared

f open parentheses x minus x with bar on top close parentheses squared

1.5

1

1 cross times 1.5 equals 1.5

4.5

5

5 cross times 4.5 equals 22.5

7.5

8

8 cross times 7.5 equals 60

10.5

6

6 cross times 10.5 equals 63

total

1 plus 5 plus 8 plus 6 equals 20

1.5 plus 22.5 plus 60 plus 63 equals 147

Now that we have the sum of the f and fx columns we can work out the estimated mean
The sum of the fx column is the estimated sum of all the data values
And the sum of the f column is the total number of data values

x with bar on top equals 147 over 20 equals 7.35

Now we can complete the rest of the table

midpoint, x

f

f x

x minus x with bar on top

open parentheses x minus x with bar on top close parentheses squared

f open parentheses x minus x with bar on top close parentheses squared

1.5

1

1.5

1.5 minus 7.35 equals negative 5.85

open parentheses negative 5.85 close parentheses squared equals 34.2225

1 cross times 34.2225 equals 34.2225

4.5

5

22.5

4.5 minus 7.35 equals negative 2.85

open parentheses negative 2.85 close parentheses squared equals 8.1225

5 cross times 8.1225 equals 40.6125

7.5

8

60

7.5 minus 7.35 equals 0.15

open parentheses 0.15 close parentheses squared equals 0.0225

8 cross times 0.0225 equals 0.18

10.5

6

63

10.5 minus 7.35 equals 3.15

open parentheses 3.15 close parentheses squared equals 9.9225

6 cross times 9.9225 equals 59.535

total

20

147

34.2225 plus 40.6125 plus 0.18 plus 59.535equals 134.55

So sum space f equals 20 and sum space f open parentheses x minus x with bar on top close parentheses squared equals 134.55
That gives us everything we need to put into the formula

square root of fraction numerator 134.55 over denominator 20 end fraction end root equals square root of 6.7275 end root equals 2.593742...

standard deviation = 2.59 (3 s.f.)

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Roger B

Author: Roger B

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.