2.1.4 Coding

Coding is a technique used to simplify calculations, especially when dealing with large or complicated numbers. Coding transforms data values into a different set of values (called coded values) by applying a linear transformation. This makes it easier to calculate measures like the mean, variance, and standard deviation.

The Basic Idea of Coding

The transformation usually involves shifting and/or scaling the data, using a linear function:

$$y = \frac{x - a}{b}$$

Where:

• $y$ is the coded value.
• $x$ is the original data value.
• $a$ is a constant used to shift the data (often the mean of the dataset).
• $b$ is a constant used to scale the data (often the range or a standard value).

Purpose of Coding

• Simplification: Coding reduces complex numbers to simpler ones, making calculations less error-prone.
• Easier Computation: Coding can make it easier to compute the mean, variance, and standard deviation, especially if you later need to reverse the coding.
• No Change to Relative Position: Coding doesn't change the shape of the distribution or the relative positions of data points, so you can perform all the calculations in the coded form and then reverse them if needed.

Applying Coding to the Mean

Given the coding formula $y = \frac{x - a}{b}$, the mean of the coded data $\bar{y}$ can be related to the mean of the original data $\bar{x}$ :

$\bar{y} = \frac{\bar{x} - a}{b}$

If you know $\bar{y}$, you can find $\bar{x}$ by rearranging the equation:

$\bar{x} = b\bar{y} + a$

Applying Coding to Variance and Standard Deviation

For variance and standard deviation, the coding transformation affects only the scaling factor $b$ .

Variance:

$\text{Var}(y) = \frac{\text{Var}(x)}{b^2}$

If you know the variance of the coded data $\text{Var}(y)$ , you can find the variance of the original data $\text{Var}(x)$ by:

$\text{Var}(x) = b^2 \times \text{Var}(y)$

Standard Deviation:

$\text{SD}(y) = \frac{\text{SD}(x)}{b}$

Similarly, the standard deviation of the original data can be found by:

$\text{SD}(x) = b \times \text{SD}(y)$

Summary

Coding is a powerful technique that simplifies complex data, making calculations more straightforward. It's especially useful when dealing with large numbers or awkward decimals. By applying a linear transformation, coding allows for easier computation of statistical measures such as the mean, variance, and standard deviation, and results can be converted back to the original scale as needed.