Measure of dispersion for ungrouped data

Question

A set of data is arranged in ascending order; 7, w, x, 10, y, z, 12. 7 It is given that the range and standard deviation of the set of data are 5 and 1.67 respectively. Determine the effect on

a) the range when a number 20 is added,

b) the interquartile range when the number 12 is removed,

c) the standard deviation when a number 9 is added.

Answer 1

Unfortunately, as the mean of the dataset is not given, there's not enough information to solve part c, but there is enough info to solve parts a and b.

For part a, the range of a dataset is the difference between the largest number and the smallest number. When you add 20 to the set, as it is larger than the previous largest number, 12, it's now the largest number in the set, and subtracting the smallest number 7 from the largest number 20 yields a range of 13.

For part b, recall that the interquartile range of a dataset is the difference between the third quartile and the first quartile. The dataset with 12 removed consists of six elements: 7, w, x, 10, y, z. The first quartile is the median of the lower half of the dataset (7, w, x), and the third quartile is the median of the upper half of the dataset (10, y, z). The respective medians are w and y, so the interquartile range will be y-w.