What's the Difference Between Normalization and Standardization?

Normalization rescales numerical data to a fixed range, usually $0$ to $1$, whereas standardization transforms data so that it has a mean of $0$ and a standard deviation of $1$. Both are data-preprocessing techniques, but they use different calculations and produce different distributions.

How the transformations work

Min-max normalization is commonly calculated using:

$$x' = \frac{x-x_{\min}}{x_{\max}-x_{\min}}$$

The minimum value becomes $0$, the maximum becomes $1$, and all other values are placed proportionally between them.

Standardization calculates a z-score:

$$z = \frac{x-\mu}{\sigma}$$

Here, $\mu$ is the mean and $\sigma$ is the standard deviation. A positive z-score is above the mean, while a negative z-score is below it.

These transformations matter because features measured on larger numerical scales can dominate distance-based calculations and gradient-based learning. Scaling allows an algorithm to compare feature contributions more fairly without changing the order of observations.

When preparing training and test data, calculate the required parameters from the training set only, then apply those same parameters to the test set. Calculating them from all data causes data leakage because test-set information influences preprocessing.

Worked example

For the dataset $10, 20, 30$, normalize $20$:

$$\frac{20-10}{30-10}=0.5$$

Its standardized value is $0$ because $20$ is the mean:

$$z=\frac{20-20}{\sigma}=0$$

A common misconception is that normalization and standardization are interchangeable terms. In IB Computer Science A4.2, distinguish them by identifying the statistics used and the resulting scale.

Exam technique: For a command term such as distinguish, state both methods and make the difference explicit. For a calculation, show the formula, substitute the values, and interpret the transformed result.