Sec 4 E-Math · Statistics

Standard deviation questions with answers

Practice questions on standard deviation, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

Find the standard deviation of $3, \ 6, \ 7, \ 9, \ 10, \ 13$ , correct to 3 significant figures.

Step 1 of 3

The formula (on the formula sheet)

\text{sd} = \sqrt{\dfrac{\Sigma x^2}{n} - \left(\dfrac{\Sigma x}{n}\right)^2}

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Questions

Question 1From a list
Find the standard deviation of $9, \ 13, \ 14, \ 18, \ 21, \ 27$ , correct to 3 significant figures.
Show the worked solution
1. Step 1: The formula (on the formula sheet) $\text{sd} = \sqrt{\dfrac{\Sigma x^2}{n} - \left(\dfrac{\Sigma x}{n}\right)^2}$
2. Step 2: Find n, Σx and Σx² $n = 6$ $\mathstrut \Sigma x = 9 + 13 + 14 + 18 + 21 + 27$ $\Sigma x = 102$ $\mathstrut \Sigma x^2 = 9^2 + 13^2 + 14^2 + 18^2 + 21^2 + 27^2$ $\Sigma x^2 = 1940$
3. Step 3: Substitute $\mathstrut \text{sd} = \sqrt{\dfrac{1940}{6} - \left(\dfrac{102}{6}\right)^2}$ $\text{sd} = 5.8594\ldots$ $\phantom{\text{sd}} = 5.86$ (3 s.f.)
Question 2From Σx and Σx²
For a set of $10$ values, $\Sigma x = 120$ and $\Sigma x^2 = 1484$ . Find the standard deviation, correct to 3 significant figures.
Show the worked solution
1. Step 1: The formula (on the formula sheet) $\text{sd} = \sqrt{\dfrac{\Sigma x^2}{n} - \left(\dfrac{\Sigma x}{n}\right)^2}$
2. Step 2: Substitute $\mathstrut \text{sd} = \sqrt{\dfrac{1484}{10} - \left(\dfrac{120}{10}\right)^2}$ $\text{sd} = 2.0976\ldots$ $\phantom{\text{sd}} = 2.10$ (3 s.f.)

Question 3Grouped data

The table shows the marks of $60$ students. Using the mid-values of the classes, estimate the standard deviation, correct to 3 significant figures.

Marks, $x$	Frequency
$0 < x \le 10$	3
$10 < x \le 20$	10
$20 < x \le 30$	15
$30 < x \le 40$	20
$40 < x \le 50$	8
$50 < x \le 60$	4

Show the worked solution

Step 1: Use the mid-values x with the frequencies f $x = 5, 15, 25, 35, 45, 55$ $\Sigma f = 60$ $\Sigma fx = 1820$ $\Sigma fx^2 = 64\,500$
Step 2: sd = √(Σfx² ÷ Σf − (Σfx ÷ Σf)²) $\mathstrut \text{sd} = \sqrt{\dfrac{64500}{60} - \left(\dfrac{1820}{60}\right)^2}$ $\text{sd} = 12.4454\ldots$ $\phantom{\text{sd}} = 12.4$ (3 s.f.)

Question 4Every value changed: the new mean and sd
The mean and the standard deviation of the marks of a class in a test are $27.5$ marks and $8.2$ marks. Every mark is then multiplied by $2$ so that the test is out of $100$ . Write down the new mean.
Show the worked solution
1. Step 1: Multiplying every value stretches the data: the mean and the spread are both multipliedThe mean is multiplied by $2$ .
2. Step 2: The new meanNew mean $= 27.5 \times 2$ $= 55$

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Worked example

Find the standard deviation of

3, \ 6, \ 7, \ 9, \ 10, \ 13

, correct to 3 significant figures.

Step 1 of 3

The formula (on the formula sheet)

\text{sd} = \sqrt{\dfrac{\Sigma x^2}{n} - \left(\dfrac{\Sigma x}{n}\right)^2}

Questions

Question 1From a list

Find the standard deviation of $9, \ 13, \ 14, \ 18, \ 21, \ 27$ , correct to 3 significant figures.

Show the worked solution

Step 1: The formula (on the formula sheet) $\text{sd} = \sqrt{\dfrac{\Sigma x^2}{n} - \left(\dfrac{\Sigma x}{n}\right)^2}$
Step 2: Find n, Σx and Σx² $n = 6$ $\mathstrut \Sigma x = 9 + 13 + 14 + 18 + 21 + 27$ $\Sigma x = 102$ $\mathstrut \Sigma x^2 = 9^2 + 13^2 + 14^2 + 18^2 + 21^2 + 27^2$ $\Sigma x^2 = 1940$
Step 3: Substitute $\mathstrut \text{sd} = \sqrt{\dfrac{1940}{6} - \left(\dfrac{102}{6}\right)^2}$ $\text{sd} = 5.8594\ldots$ $\phantom{\text{sd}} = 5.86$ (3 s.f.)

Question 2From Σx and Σx²

For a set of $10$ values, $\Sigma x = 120$ and $\Sigma x^2 = 1484$ . Find the standard deviation, correct to 3 significant figures.

Show the worked solution

Step 1: The formula (on the formula sheet) $\text{sd} = \sqrt{\dfrac{\Sigma x^2}{n} - \left(\dfrac{\Sigma x}{n}\right)^2}$
Step 2: Substitute $\mathstrut \text{sd} = \sqrt{\dfrac{1484}{10} - \left(\dfrac{120}{10}\right)^2}$ $\text{sd} = 2.0976\ldots$ $\phantom{\text{sd}} = 2.10$ (3 s.f.)

Question 3Grouped data

The table shows the marks of $60$ students. Using the mid-values of the classes, estimate the standard deviation, correct to 3 significant figures.

Marks, $x$	Frequency
$0 < x \le 10$	3
$10 < x \le 20$	10
$20 < x \le 30$	15
$30 < x \le 40$	20
$40 < x \le 50$	8
$50 < x \le 60$	4

Show the worked solution

Step 1: Use the mid-values x with the frequencies f $x = 5, 15, 25, 35, 45, 55$ $\Sigma f = 60$ $\Sigma fx = 1820$ $\Sigma fx^2 = 64\,500$
Step 2: sd = √(Σfx² ÷ Σf − (Σfx ÷ Σf)²) $\mathstrut \text{sd} = \sqrt{\dfrac{64500}{60} - \left(\dfrac{1820}{60}\right)^2}$ $\text{sd} = 12.4454\ldots$ $\phantom{\text{sd}} = 12.4$ (3 s.f.)

Question 4Every value changed: the new mean and sd

The mean and the standard deviation of the marks of a class in a test are $27.5$ marks and $8.2$ marks. Every mark is then multiplied by $2$ so that the test is out of $100$ . Write down the new mean.

Show the worked solution

Step 1: Multiplying every value stretches the data: the mean and the spread are both multipliedThe mean is multiplied by $2$ .
Step 2: The new meanNew mean $= 27.5 \times 2$ $= 55$