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Standard Deviation (SQA National 5 Applications of Mathematics): Revision Note

Exam code: X844 75

Dan Finlay

Written by: Dan Finlay

Reviewed by: Roger B

Updated on

Standard deviation

What is the standard deviation of a data set?

  • The standard deviation is a measure of spread

    • It tells you how spread out the data set is around the mean

  • The larger the standard deviation, the more spread out the data is

  • The smaller the standard deviation, the less spread out the data is

How do I find the standard deviation of a data set?

  • The Formulae List in the exam paper gives you two different ways to calculate the standard deviation, s

    • s equals square root of fraction numerator straight capital sigma open parentheses x minus x with bar on top close parentheses squared over denominator n minus 1 end fraction end root

    • Error converting from MathML to accessible text.

Formula with the mean

  • s equals square root of fraction numerator straight capital sigma open parentheses x minus x with bar on top close parentheses squared over denominator n minus 1 end fraction end root

  • STEP 1
    Find the mean x with bar on top of the data

  • STEP 2
    Calculate capital sigma open parentheses x minus x with bar on top close parentheses squared

    • Subtract the mean from each value

    • Square these differences

    • Add the squares together

  • STEP 3
    Divide by n minus 1

    • This is one less than the number of values

  • STEP 4
    Take the square root of your answer

Formula without the mean

  • Error converting from MathML to accessible text.

  • STEP 1
    Calculate capital sigma x squared

    • Square each value

    • Add the squares together

  • STEP 2
    Calculate open parentheses capital sigma x close parentheses squared over n

    • Add the values together

    • Square the sum

    • Divide by the number of values

  • STEP 3
    Subtract open parentheses capital sigma x close parentheses squared over n from capital sigma x squared

  • STEP 4
    Divide by n minus 1

    • This is one less than the number of values

  • STEP 4
    Take the square root of your answer

Examiner Tips and Tricks

It can be useful to set up a table to work out these values and the sums.

If you have already worked out the mean, then you will have already worked out capital sigma x. You divide capital sigma x by the number of values to get the mean. So you can multiply the mean by the number of values to get capital sigma x.

Worked Example

A teacher recorded the number of correct answers achieved by a sample of seven students in a short mathematics test. The results for School A were:

19, 21, 16, 22, 17, 19, 26

Calculate the mean and standard deviation of the number of correct answers achieved by the students in School A.

Answer:

To calculate the mean

  • Find the sum of the data values

  • and divide it by the number of data values (7)

table row mean equals cell fraction numerator 19 plus 21 plus 16 plus 22 plus 17 plus 19 plus 26 over denominator 7 end fraction end cell row blank equals cell 140 over 7 end cell end table

mean = 20

To calculate the standard deviation, there are two different formulae you can use

Method 1

Using the formula s equals square root of fraction numerator straight capital sigma open parentheses x minus x with bar on top close parentheses squared over denominator n minus 1 end fraction end root

  • The mean, x with bar on top, is 20, as calculated above

  • n equals 7

Start by finding the value of straight capital sigma open parentheses x minus x with bar on top close parentheses squared

bold italic x

bold italic x bold minus bold italic x with bold bar on top

Error converting from MathML to accessible text.

19

19-20=-1

(-1)2=1

21

21-20=1

12=1

16

16-20=-4

(-4)2=16

22

22-20=2

22=4

17

17-20=-3

(-3)2=9

19

19-20=-1

(-1)2=1

26

26-20=6

62=36

sum = 68

Substitute the values into the formula

s equals square root of fraction numerator 68 over denominator 7 minus 1 end fraction end root equals 3.366501...

Round your answer to a sensible degree of accuracy

  • All the values in the question have two significant figures

standard deviation = 3.4

Method 2

Using the formula Error converting from MathML to accessible text.

  • n equals 7

Find the values of straight capital sigma x and straight capital sigma x squared

bold italic x

bold italic x to the power of bold 2

19

192=361

21

212=441

16

162=256

22

222=484

17

172=289

19

192=361

26

262=676

sum:  140

sum:  2868

Substitute the values into the formula

Error converting from MathML to accessible text.

Round your answer to a sensible degree of accuracy

  • All the values in the question have two significant figures

standard deviation = 3.4

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    Author
    Reviewer
    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Subject 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.

    Roger B

    Reviewer: Roger B

    Expertise: Maths Content Creator

    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.