Question

mean, median, and mode: comparisons
13, 17, 17, 17, 18, 21, 23, 26, 34, 35, 37, 49, 49, 54, 70, 78, 84, 90
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(a) for these data, which measures of central tendency take more than one value? choose all that apply.
(b) suppose that the measurement 90 (the largest measurement in the data set) were replaced by 96. which measures of central tendency would be affected by the change? choose all that apply.

Explanation:

Step1: Recall definitions

• Mean: The sum of all data values divided by the number of data values.
• Median: The middle - value when data is ordered.
• Mode: The most frequently occurring value.

Step2: Analyze part (a)

• Mean: There is only one formula $\text{Mean}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, so it has a single value.
• Median: For a set of $n = 18$ data values (even number of data points), $\text{Median}=\frac{x_{\frac{n}{2}}+x_{\frac{n}{2}+1}}{2}$. Since the data is ordered, there is a single calculation result.
• Mode: The value \(17\) occurs \(3\) times (more frequently than other values). So, there is a single mode.

Step3: Analyze part (b)

• Mean: The formula is $\text{Mean}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. If we change \(x_{n}=90\) to \(x_{n}=96\), the sum \(\sum_{i = 1}^{n}x_{i}\) changes. So, the mean changes.
• Median: The median depends on the middle - values. Since we are changing the largest value (not the middle - values for \(n = 18\), the 9th and 10th values in the ordered list), the median does not change.
• Mode: The mode depends on the most frequently occurring value. Changing the largest value does not affect the frequency of \(17\). So, the mode does not change.

Answer:

(a) None of these measures
(b) Mean