Changing the choice of classes can change the appearance of a histogram. Here is an example in which a small shift in the classes, with no change in the number of classes, has an important effect on the histogram. The data are the acidity levels (measured by pH) in 105 samples of rainwater. Distilled water has pH 7.00. As the water becomes more acidic, the pH goes down. The pH of rainwater is important to environmentalists because of the problem of acid rain.
4.33 4.75 4.94 5.10 5.19 5.41 5.55 5.68 5.81 6.03 6.43 4.38 4.76 4.96 5.12 5.23 5.43 5.56 5.69 5.81 6.04 6.62 4.48 4.82 5.00 5.15 5.29 5.46 5.62 5.75 5.81 6.05 6.65 4.50 5.82 5.01 5.15 5.32 5.46 5.64 5.75 5.85 6.06 6.81 4.55 4.83 5.02 5.15 5.33 5.47 5.65 5.75 5.85 6.07 4.59 4.86 5.05 5.16 5.35 5.50 5.65 5.76 5.90 6.09 4.59 4.93 5.06 5.16 5.37 5.51 5.66 5.76 5.90 6.13 4.61 4.94 5.08 5.16 5.37 5.53 5.67 5.79 6.00 6.21 4.61 4.94 5.09 5.18 5.39 5.55 5.67 5.80 6.03 6.34 4.48 4.78 4.97 5.16 5.24 5.44 5.61 5.70 5.81 6.04 6.62
(a) Make a histogram of pH with 14 classes, using class boundaries 4.2, 4.4,…, 7.0. How many modes
does your histogram show? More than one mode suggests that the data contain groups that have
different distributions.
(b) Make a second histogram, also with 14 classes, using class boundaries 4.14, 4.34,…, 6.94. The
classes are those from (a) moved 0.06 to the left. How many modes does the new histogram
show?
(c) Use your software’s histogram function to make a histogram without specifying the number of
classes or their boundaries. How does the software’s default histogram compare with those in (a)
and (b)?