Question

mean, median, and mode: comparisons
24, 26, 31, 32, 34, 34, 36, 37, 37, 38, 38, 39, 39, 41, 42, 43, 43, 43, 43, 43, 44
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(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.

Explanation:

Step1: Recall definitions

The mean is the average of the data set ($\frac{\sum_{i = 1}^{n}x_{i}}{n}$), the median is the middle - value when data is ordered, and the mode is the most frequently occurring value. For any non - empty data set of real numbers, these measures exist.

Step2: Analyze effect of changing a value

The mean is calculated as $\frac{\sum_{i = 1}^{n}x_{i}}{n}$. If we change a value (from 24 to 12), the sum $\sum_{i = 1}^{n}x_{i}$ changes, so the mean changes.
The median is the middle value. Since we are only changing the smallest value and the number of data points ($n = 19$, odd) remains the same, the position of the median (the 10th value) does not change.
The mode is the most frequently occurring value. Changing the smallest value (24 to 12) does not affect the frequency of the most common value (43).

Answer:

(a) All of these measures exist
(b) Mean