I chose to scale the results as the variables use different units, with different ranges.

It sounds like you are dealing with a situation where you need to normalize or scale different variables that have different units and ranges. This is often necessary in data analysis to ensure that the contributions of each variable are comparable. Here’s a brief overview of scaling methods you might consider.

Scaling Methods:

1. Min-Max Scaling: This technique scales each variable to a fixed range, typically [0, 1]. The formula is: $X' = \frac{X - X_{\text{min}}}{X_{\text{max}} - X_{\text{min}}}$ where $X'$ is the scaled value, $X$ is the original value, $X_{\text{min}}$ and $X_{\text{max}}$ are the minimum and maximum values of the variable.

2. Z-Score Standardization: This method scales the data based on the mean and standard deviation, resulting in a distribution with a mean of 0 and a standard deviation of 1. The formula is: $Z = \frac{X - \mu}{\sigma}$ where $Z$ is the Z-score, $X$ is the original value, $\mu$ is the mean of the variable, and $\sigma$ is the standard deviation.

3. Robust Scaling: This technique scales features using statistics that are robust to outliers, specifically the median and the interquartile range (IQR). The formula is: $X' = \frac{X - \text{median}(X)}{\text{IQR}(X)}$

Conclusion

Scaling your data is an important step when the variables are on different scales or have different units. Choose the scaling method that best suits your data characteristics and analysis needs. Let me know if you need further assistance on this topic!