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A transformation of data values by means of some mathematical function, such as\(\sqrt x \)or\(1/x\), can often yield a set of numbers that has 鈥渘icer鈥� 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 鈥淭ime Lapse Cinematographic Analysis of Beryllium鈥�Lung FibroblastInteractions鈥� (Environ. Research,1983: 34鈥�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 articleused 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,鈥� to construct a histogram of the original data. Use intervals1.1-<1.2, 1.2-<1.3,鈥� to do the same for the transformed data. What is the effect of the transformation?

Step by step solution

01

Given information

Theresults of the experiments were designed to study the behaviour of certain individual cells that had been exposed to beryllium.

The original data and the transformed data are provided.

02

Construct a histogram and state the features

For original data,

The table representing the frequency is given as,

Class	Frequency
10-<20	8
20-<30	14
30-<40	8
40-<50	4
50-<60	3
60-<70	2
70-<80	1
Total	40

The steps to construct a histogram are,

1) Determine the frequency or the relative frequency.

2) Mark the class boundaries on the horizontal axis.

3) Draw a rectangle on the horizontal axis corresponding to the frequency or relative frequency.

The histogram is represented as,

For tranformed data,

The table representing the frequency is given as,

Class	Frequency
1.1-<1.2	2
1.2-<1.3	6
1.3-<1.4	7
1.4-<1.5	9
1.5-<1.6	6
1.6-<1.7	4
1.7-<1.8	5
1.8-<1.9	1
Total	40

Using the above steps, the histogram is represented as,

It can be observed that the effect of the transformation is that from positively skewed data, the distribution is getting approximately symmetric.

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