Standard deviation and coding

The idea

Variance $= \frac{\sum x^2}{n} - \bar{x}^2$ and the standard deviation is its square root. Coding $y = \frac{x - a}{b}$ changes the mean the same way but the standard deviation only by the division: $\sigma_x = b\,\sigma_y$ .

Find $\bar{x} = \frac{\sum x}{n}$ .
Variance $= \frac{\sum x^2}{n} - \bar{x}^2$ ; take the square root for the standard deviation.
To undo coding: $\bar{x} = a + b\bar{y}$ and $\sigma_x = b\sigma_y$ .

Watch out Adding or subtracting a constant does not change the standard deviation.

See it worked

Worked example

$n = 10$ , $\sum x = 60$ and $\sum x^2 = 420$ . Find the mean and standard deviation.

$\bar{x} = \frac{60}{10} = 6$ .
Variance $= \frac{420}{10} - 6^2 = 42 - 36 = 6$ .
Standard deviation $= \sqrt{6} = 2.45$ (3 s.f.).

Answer Mean $6$ , standard deviation $2.45$

Your turn

Question 1Calculator allowed3 marks

$n = 8$ , $\sum x = 56$ and $\sum x^2 = 440$ . Find the standard deviation.

Question 2Non-calculator3 marks

Data are coded with $y = x - 50$ . The coded data have mean $3.2$ and standard deviation $1.5$ . Find the mean of $x$ .

Question 3Non-calculator3 marks

Data are coded with $y = \frac{x - 100}{5}$ . The coded data have mean $2$ and standard deviation $1.2$ . Find the standard deviation of $x$ .

In past papers

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