Question

The following data are the ages (in years) at diagnosis for 20 patients under treatment for meningitis:

18 18 25 19 23 20 69 18 21 18 20 18 18 20 18 19 28 17 18 18

(a) Calculate and interpret the values of the sample mean, variance, and standard deviation.

(b) Compute the sample median. Why might you recommend the sample median as a measure of center rather than the sample mean?

(c) Compute the upper quartile, the lower quartile, and the interquartile range.

(d) Illustrate the center, spread, and symmetry or skewness of this data using a horizontal modified boxplot.

Answer 1

(a) Calculate and interpret the values of the sample mean, variance, and standard deviation Using the Minitab, the obtained output for the descriptive statistics is, Statistics Total Variable Count Mean StDev Variance Q1 Median IQR Skewness Q3 20 22.15 11.37 18.50 20.75 2.75 4.07 Age 129.19 18.00 From the obtained Minitab output, the sample mean is r 22.15, the sample variance is s2129.19 and the sample standard deviation is s 11.37 (b) Compute the sample median. Why might you recommend the sample median as a measure of center rather than the sample mean? Referring to the Minitab output part (a), the sample median is f 18.50. It can be observed that the sample mean of 22.15 is greater than the median value of 18.50. So, the distribution of ages is said to be positively skewed. For skewed distributions, median is the best measure of central tendency than the sample mean. Therefore, we strongly recommend the median for the distribution of ages. (c) Compute the upper quartile, the lower quartile, and the interquartile range Referring to the Minitab output in part (a), the upper quartile is Q 20.75, the lower quartile is Q 18 and the Interquartile range is IQR 2.75

(d) Illustrate the center, spread, and symmetry or skewness of this data using a horizontal modified boxplot Using the Minitab, the obtained output for the boxplot is, Boxplot of Age 10 20 30 40 Age Look at the boxplot, median is the center of the distribution. IQR refers to more spread in the data, the skewness is 4.07, which represents the positive skewness. 70 60 50