Coding

In a nutshell

Coding is transforming each data value into a new set of data values, so that statistical calculations can be simplified.

Coding formula

You will usually use the following formula for coding:

Mean

The mean of the coded data is given by:

$$\boxed{\overline y=\dfrac{\overline x-a}{b}}$$​​

Standard deviation

The standard deviation of the coded data can be calculated with the formula:

$$\boxed{\sigma_y=\dfrac{\sigma_x}{b}}$$​​

Note: You will most likely be asked to find the statistics of the original data, given the statistics for the coded data. This way, the previous formulas can be simplified into:

$$\overline x=b\overline y+a$$

$$\sigma_x=b\sigma_y$$​​​

Example 1

The average temperature $$x$$​ of a city was measured every day for $$30$$ days. The data was coded using the formula $$y=\dfrac{x-3}{4}$$. Given that $$\overline y=3.6 \ ºC$$ and that $$S_{yy}=51.8$$, find the average daily temperature and the standard deviation for the data collected.

Start by finding the mean:

​$$\overline y=\dfrac{\overline x-3}{4}\Leftrightarrow \overline x=4\overline y+3\Leftrightarrow \overline x=4\times3.6+3=\underline{17.4 \ ºC}$$​​

Find the standard deviation for the coded value:

​$$\sigma_y=\sqrt{\dfrac{S_{yy}}{n}}=\sqrt{\dfrac{51.8}{30}}=1.31 \ ºC$$​​

​$$\sigma_x=b\sigma_y\Leftrightarrow\sigma_x=4\times1.31=\underline{5.24 \ ºC}$$​​

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