If data are transformed to a log-scale, how would you interpret percentiles on the log-scale relative to the original scale?

• A. Percentiles on the log-scale correspond to multiplicative factors on the original scale; back-transform by exponentiating.
• B. Percentiles on the log-scale correspond to additive shifts on the original scale.
• C. Percentiles are not affected by transformation.
• D. Back-transforming is unnecessary.

Answer: (A)

When data are on a log scale, equal steps on that scale reflect equal multiplicative changes on the original scale. A percentile of the log-transformed data, when you apply the inverse transform, back to the original scale, becomes a percentile of the original data. Since the log transform turns multiplicative relationships into additive ones, the differences between percentiles on the log scale correspond to ratios (multiplicative factors) between the original-scale values.

So the pth percentile on the log scale is interpreted by back-transforming, i.e., exponentiating, to get the corresponding value on the original scale. This is why back-transforming is the natural way to interpret those percentiles in the original units. If you use natural log, you exponentiate with e; if you use log base 10, you use 10^, but the principle is the same. Note that this requires positive data, since the log is not defined for zero or negative values.