Mean & Standard Deviation (SQA National 5 Maths): Revision Note

Finding the mean & standard deviation

What are the mean and standard deviation of a data set?

• The mean of a data set is a type of average

  • Its value gives a 'typical value' for a number in the data set

  • The median is another type of average

• 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

      • If the standard deviation is zero, then all the data values are equal to the mean!

  • The interquartile range is another measure of spread

• The mean and standard deviation are used together to analyse a set of data

  • The median and interquartile range may also be used together to analyse a data set

  • But you should not use the mean with the interquartile range, or the median with the standard deviation

How do I find the mean of a data set?

• The mean is the sum of the data values divided by the number of values

  • The mean of 1, 2, 6 is (1 + 2 + 6) ÷ 3 = 3

• The mean can be fraction or a decimal

  • It may need rounding

  • You do not need to force it to be a whole number

    • You can have a mean of 7.5 people, for example!

• The mean of a data set is sometimes represented by $\overline{x}$

  • Then the mean can be written as a formula

    • $\overline{x}=\frac{\mathrm{\Sigma }x}{n}$

  • where

    • $\mathrm{\Sigma }x$ is the sum of all the data values

    • $n$ is the number of data values

  • This formula is not given to you in the exam

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, $s$

  • $s=\sqrt{\frac{\mathrm{\Sigma }{\left(x-\overline{x}\right)}^2}{n-1}}$

  • $s=\sqrt{\frac{\mathrm{\Sigma }{x}^2\mathit{-}{\displaystyle \frac{{\mathit{\left(}\mathrm{\Sigma }x\mathit{\right)}}^2}{n}}}{n-1}}$

• In those formulae:

  • $n$ is the sample size (the number of values in the data set)

  • $\overline{x}$ is the mean of the data set

  • $x$ stands for 'a value in the data set'

  • $\mathrm{\Sigma }{x}^2$ is the sum of the squares of the data values

    • Square each data value, then add those squared values together

  • ${\mathit{\left(}\mathrm{\Sigma }x\mathit{\right)}}^2$ is the square of the sum of the data values

    • Find the sum of all the data values, then square the sum

  • $\mathrm{\Sigma }{\left(x-\overline{x}\right)}^2$ is the sum of the squares of the differences between the data values and the mean

    • Find the difference between each data value and the mean

    • Square each one of those differences

    • Then add those squared differences together

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

  • See the Worked Example