Thilothammal et al. (A-19) designed a study to determine the efficacy of BCG (bacillus Calmette-Guerin) vaccine in preventing tuberculous meningitis. Among the data collected on each subject was a measure of nutritional status (actual weight expressed as a percentage of expected weight for actual height). The following table shows the nutritional status values of the 107 cases studied.

73.3 54.6 82.4 76.5 72.2 73.6 74.0 80.5 71.0 56.8 80.6 100.0 79.6 67.3 50.4 66.0 83.0 72.3 55.7 64.1 66.3 50.9 71.0 76.5 99.6 79.3 76.9 96.0 64.8 74.0 72.6 80.7 109.0 68.6 73.8 74.0 72.7 65.9 73.3 84.4 73.2 70.0 72.8 73.6 70.0 77.4 76.4 66.3 50.5 72.0 97.5 130.0 68.1 86.4 70.0 73.0 59.7 89.6 76.9 74.6 67.7 91.9 55.0 90.9 70.5 88.2 70.5 74.0 55.5 80.0 76.9 78.1 63.4 58.8 92.3 100.0 84.0 71.4 84.6 123.7 93.7 76.9 79.6 45.6 92.5 65.6 61.3 64.5 72.7 77.5 76.9 80.2 76.9 88.7 78.1 60.6 59.0 84.7 78.2 72.4 68.3 67.5 76.9 82.6 85.4 65.7 65.9 Source: Data provided courtesy of Dr. N. Thilothammal.

(a) For these data compute the following descriptive measures: mean, median, mode, variance, standard deviation, range, first quartile, third quartile, and IQR.

(b) Construct the following graphs for the data: histogram, frequency polygon, stem-and-leaf plot, and boxplot.

(c) Discuss the data in terms of variability. Compare the IQR with the range. What does the comparison tell you about the variability of the observations?

(d) What proportion of the measurements are within one standard deviation of the mean? Two standard deviations of the mean? Three standard deviations of the mean?

(e) What proportion of the measurements are less than 100?

(f) What proportion of the measurements are less than 50?