Standardized Variables

Standardized variables are a crucial tool in statistics for comparing data that have different scales, making it possible to directly compare different variables.

Standardization involves transforming a variable into a new one with a mean of zero and a standard deviation of one.

The formula to calculate a standardized variable $ Z $ is:

$$ Z = \frac{X - \mu}{\sigma} $$

Where:

$ X $ is the original variable value.
$ \mu $ is the mean of the variable.
$ \sigma $ is the standard deviation of the variable.

This process converts any distribution into a standard normal distribution, with a mean of zero and a standard deviation of one.

Why is standardization useful?

Standardization allows for the comparison of different variables, like weight and height, each with its own units of measurement, by placing them on a common scale.

Additionally, in statistical models and machine learning methods, data often need to be normalized, meaning they should be on a consistent scale, to improve algorithm performance.

Standardized values indicate how many standard deviations a specific value is from the mean, making them easier to interpret.

For example, $ Z = 2 $ means the value is two standard deviations above the mean.

Note: Standardized variables are especially valuable in correlation and regression analyses, where the goal is to understand the relationship between variables. Using standardized variables allows regression coefficients to be interpreted in terms of standard deviations, making the results much easier to understand.

A Practical Example

Let’s consider a dataset of students' heights measured in centimeters, with a mean of $ \mu = 170 $ cm and a standard deviation of $ \sigma = 10 $ cm.

If a student has a height of 190 cm, we can standardize this height as follows:

$$ Z = \frac{190 - 170}{10} = \frac{20}{10} = 2 $$

This means that the student's height is two standard deviations above the mean.

Note: This is a very simple example, meant only to illustrate how to standardize a variable. In practice, standardization becomes especially useful when comparing data of different types, such as weight and height.

Example 2

In this example, we'll use the following data for a class of students:

Average height ($ \mu_{\text{height}} $): 170 cm, with a standard deviation ($ \sigma_{\text{height}} $) of 10 cm.
Average weight ($ \mu_{\text{weight}} $): 65 kg, with a standard deviation ($ \sigma_{\text{weight}} $) of 8 kg.

Now, let's consider the data for a particular student:

Height: 180 cm.
Weight: 75 kg.

To compare these two traits, we need to standardize them using the formula:

$$ Z = \frac{X - \mu}{\sigma} $$

Applying the formula to standardize the student's height:

$$ Z_{\text{height}} = \frac{180 - 170}{10} = \frac{10}{10} = 1 $$

The standardized height ($ Z_{\text{height}} $) is 1, indicating that the student's height is one standard deviation above the class average.

Now, let's standardize the student's weight:

$$ Z_{\text{weight}} = \frac{75 - 65}{8} = \frac{10}{8} = 1.25 $$

The standardized weight ($ Z_{\text{weight}} $) is 1.25, indicating that the student's weight is 1.25 standard deviations above the class average.

By comparing these standardized values, we can see that:

The student's height is 1 standard deviation above the average.
The student's weight is 1.25 standard deviations above the average.

This comparison shows that, relative to their classmates, the student is more "unusual" in terms of weight than height, since their weight deviates more from the average than their height does.

Standardization allows us to compare two different variables (height and weight) on the same scale.

This helps us determine which of the student's traits deviates more from the class average, making it easier to interpret the data.

And so on.