Question

(a) which measures of central tendency do not exist for this data set? choose all that apply. (b) suppose that the measurement 24 (the smallest measurement in the data set) were replaced by 12. which measures of central tendency would be affected by the change? choose all that apply. (c) suppose that, starting with the original data set, the largest measurement were removed. which measures of central tendency would be changed from those of the original data set? choose all that apply. (d) which of the following best describes the distribution of the original data? choose only one.

Explanation:

• (a) The mean, median, and mode are all defined for numerical data. Since the data here is numerical (time in minutes), all measures exist.
• (b) The mean is calculated as \(\frac{\sum_{i = 1}^{n}x_{i}}{n}\). Changing a value (from 24 to 12) will change the sum \(\sum_{i=1}^{n}x_{i}\), thus affecting the mean. The median is the middle - value (when \(n\) is odd) or the average of two middle - values (when \(n\) is even). If the number of data points is large enough, changing the smallest value may not affect the position of the median. The mode is the most frequently occurring value. Changing the smallest value (which is not the mode) does not affect the mode.
• (c) The mean is \(\frac{\sum_{i = 1}^{n}x_{i}}{n}\). Removing the largest value will change the sum \(\sum_{i = 1}^{n}x_{i}\) and \(n\), so the mean changes. The median for \(n\) data points: if \(n\) was odd (\(n = 2k + 1\)), after removing one data point (\(n'=2k\)), but the relative position of the middle - two values (for even \(n'\)) may not be affected by just removing the largest value (if \(n\) is large). The mode is the most frequently occurring value. Removing the largest value (which is not the mode) does not affect the mode.
• (d) In a positively skewed distribution, the tail is on the right - hand side. Looking at the histogram (assuming the bar heights correspond to frequencies), the data has a tail on the right (towards larger time values).

Answer:

(a) All of these measures exist
(b) Mean
(c) Mean
(d) Positively skewed