Chapter 1.2, Problem 25E

A transformation of data values by means of some mathematical function, such as  or 1/x, can often yield a set of numbers that has “nicer" statistical properties than the original data. In particular, it may be possible to find a function for which the histogram of transformed values is more symmetric (or, even better, more like a bell-shaped curve) than the original data. As an example, the article “Time Lapse Cinematographic Analysis of Beryllium-Lung Fibroblast Interactions” (Environ. Research, 1983: 34-43) reported the results of experiments designed to study the behavior of certain individual cells that had been exposed to beryllium. An important characteristic of such an individual cell is its interdivision time (IDT). IDTs were determined for a large number of cells, both in exposed (treatment) and unexposed (control) conditions. The authors of the article used a logarithmic transformation, that is, transformed value = log(original value). Consider the following representative IDT data:

IDT	log10(IDT)	IDT log10(IDT)	IDT	log10(IDT)
28.1	1.45	60.1 1.78	21.0	1.32
31.2	1.49	23.7 1.37	22.3	1.35
13.7	1.14	18.6 1.27	15.5	1.19
46.0	1.66	21.4 1.33	36.3	1.56
25.8	1.41	26.6 1.42	19.1	1.28
16.8	1.23	26.2 1.42	38.4	1.58
34.8	1.54	32.0 1.51	72.8	1.86
62.3	1.79	43.5 1.64	48.9	1.69
28.0	1.45	17.4 1.24	21.4	1.33
17.9	1.25	38.8 1.59	20.7	1.32
19.5	1.29	30.6 1.49	57.3	1.76
21.1	1.32	55.6 1.75	40.9	1.61
31.9	1.50	25.5 1.41
28.9	1.46	52.1 1.72

Use class intervals 10−<20, 20−<30, ... to construct a histogram of the original data. Use intervals 1.1−<1.2, 1.2−<1.3, ... to do the same for the transformed data. What is the effect of the transformation?