Box Cox Transformation

TL;DR

Transforms skewed, non-normal data toward a more normal (bell-shaped) distribution.
Uses a power transformation controlled by a parameter λ chosen to maximize normality.
Commonly applied when skewness or outliers make analysis and interpretation difficult.

Definition

The Box-Cox transformation is a statistical method used to transform data that is non-normal into a more normal distribution by applying a power transformation determined by a parameter, lambda (λ).

Explanation

Normal distributions are symmetric and bell-shaped, with most data clustered around the mean. Skewed data deviates from this shape and can be harder to analyze.
The Box-Cox transformation applies a mathematical function to each data point to reduce skewness and make the distribution more symmetric.
The specific function applied depends on a parameter λ (lambda). The optimal λ for a dataset is found through a process described as power transformation, which evaluates how well different λ values make the data fit a normal distribution.
Once the optimal λ is determined, the transformation is applied to the original data to produce transformed values that are closer to normal.

Examples

Student heights

A dataset of student heights that is skewed by a few significantly taller students can be transformed by first finding an optimal λ via power transformation. If the optimal λ is 0.5, the Box-Cox transformation is applied using:

y = \frac{x^{\lambda} - 1}{\lambda}

where y is the transformed data, x is the original data, and λ = 0.5. After applying this transformation, the heights become more symmetrical and follow a bell-shaped curve.

Financial data (daily stock prices)

Daily stock-price data that is skewed by unusually high or low days can be transformed similarly. If power transformation identifies an optimal λ of 0.3, apply:

y = \frac{x^{\lambda} - 1}{\lambda}

with λ = 0.3. The transformed stock prices become more symmetrical and easier to analyze for trends and prediction.

Use cases

Finance (e.g., stock prices)
Healthcare
Education

Notes or pitfalls

Data skewness can arise from outliers or a lack of data, both of which make analyses based on normality assumptions more difficult.
The Box-Cox transformation requires selecting an appropriate λ; this selection is performed via power transformation tests that evaluate fit to normality.

Power transformation