National 5MathsSQAApplications of MathematicsRevision NotesGraphical Data, Probability & Statistics SkillsComparing DataStandard Deviation

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

Exam code: X844 75

Written by: Dan Finlay

Reviewed by: Roger B

Updated on 2 December 2025

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, $s$
- $s = \sqrt{\frac{Σ {(x - \bar{x})}^{2}}{n - 1}}$
- $s = \sqrt{\frac{Σ x^{2} - \frac{{(Σ x)}^{2}}{n}}{n - 1}}$

Formula with the mean

$s = \sqrt{\frac{Σ {(x - \bar{x})}^{2}}{n - 1}}$
STEP 1
Find the mean $\bar{x}$ of the data
STEP 2
Calculate $Σ {(x - \bar{x})}^{2}$
- Subtract the mean from each value
- Square these differences
- Add the squares together
STEP 3
Divide by $n - 1$
- This is one less than the number of values
STEP 4
Take the square root of your answer

Formula without the mean

$s = \sqrt{\frac{Σ x^{2} - \frac{{(Σ x)}^{2}}{n}}{n - 1}}$
STEP 1
Calculate $Σ x^{2}$
- Square each value
- Add the squares together
STEP 2
Calculate $\frac{{(Σ x)}^{2}}{n}$
- Add the values together
- Square the sum
- Divide by the number of values
STEP 3
Subtract $\frac{{(Σ x)}^{2}}{n}$ from $Σ x^{2}$
STEP 4
Divide by $n - 1$
- 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 $Σ x$ . You divide $Σ x$ by the number of values to get the mean. So you can multiply the mean by the number of values to get $Σ x$ .

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)

$\begin{array}{rcl} mean & = & \frac{19 + 21 + 16 + 22 + 17 + 19 + 26}{7} \\ = & \frac{140}{7} \end{array}$

mean = 20

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

Method 1

Using the formula $s = \sqrt{\frac{Σ {(x - \bar{x})}^{2}}{n - 1}}$

The mean, $\bar{x}$ , is 20, as calculated above
$n = 7$

Start by finding the value of $Σ {(x - \bar{x})}^{2}$

$x$	$x - \bar{x}$	${(x - \bar{x})}^{2}$
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

$s = \sqrt{\frac{68}{7 - 1}} = 3.366501 . . .$

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 $s = \sqrt{\frac{Σ x^{2} - \frac{{(Σ x)}^{2}}{n}}{n - 1}}$

$n = 7$

Find the values of $Σ x$ and $Σ x^{2}$

$x$	$x^{2}$
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

$s = \sqrt{\frac{2868 - \frac{{(140)}^{2}}{7}}{7 - 1}} = 3.366501 . . .$

Round your answer to a sensible degree of accuracy

All the values in the question have two significant figures

standard deviation = 3.4

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