Data Transformation (statistics) - Reasons For Transforming Data

Reasons For Transforming Data

Guidance for how data should be transformed, or whether a transform should be applied at all, should come from the particular statistical analysis to be performed. For example, a simple way to construct an approximate 95% confidence interval for the population mean is to take the sample mean plus or minus two standard error units. However, the constant factor 2 used here is particular to the normal distribution, and is only applicable if the sample mean varies approximately normally. The central limit theorem states that in many situations, the sample mean does vary normally if the sample size is reasonably large. However if the population is substantially skewed and the sample size is at most moderate, the approximation provided by the central limit theorem can be poor, and the resulting confidence interval will likely have the wrong coverage probability. Thus, when there is evidence of substantial skew in the data, it is common to transform the data to a symmetric distribution before constructing a confidence interval. If desired, the confidence interval can then be transformed back to the original scale using the inverse of the transformation that was applied to the data.

Data can also be transformed to make it easier to visualize them. For example, suppose we have a scatterplot in which the points are the countries of the world, and the data values being plotted are the land area and population of each country. If the plot is made using untransformed data (e.g. square kilometers for area and the number of people for population), most of the countries would be plotted in tight cluster of points in the lower left corner of the graph. The few countries with very large areas and/or populations would be spread thinly around most of the graph's area. Simply rescaling units (e.g. to thousand square kilometers, or to millions of people) will not change this. However, following logarithmic transformations of both area and population, the points will be spread more uniformly in the graph.

A final reason that data can be transformed is to improve interpretability, even if no formal statistical analysis or visualization is to be performed. For example, suppose we are comparing cars in terms of their fuel economy. These data are usually presented as "kilometers per liter" or "miles per gallon." However if the goal is to assess how much additional fuel a person would use in one year when driving one car compared to another, it is more natural to work with the data transformed by the reciprocal function, yielding liters per kilometer, or gallons per mile.