Question

which measure of center must be equal to an actual data value? explain why.

choose the correct answer below.

a. since the mode is the most frequent observation that occurs in the data set, it must be an actual value from the data set.
b. since the median is the middle value when the data values are arranged in ascending order, the median must be one of the data values.
c. since the mean is considered the balancing point for the distribution, it must be an actual value from the data set.
d. the mean, median, and mode must all be values from the data set.

Explanation:

🆕 New Concept Discovered: Measures of Center
The middle, average, or most common value in a data set.

Step 1: Analyze the Mean

The mean (average) is calculated by adding all the data values together and dividing by the total number of values:

$$ \text{Mean} = \frac{\sum x_i}{n} $$

Because of this division, the resulting value does not have to be an integer or match any of the original data points. For example, in the data set \(\{1, 2\}\), the mean is:

$$ \frac{1 + 2}{2} = 1.5 $$

The value \(1.5\) is not in the original data set. Therefore, the mean does not have to be an actual data value.

Step 2: Analyze the Median

The median is the middle value when the data set is ordered from least to greatest.

• If there is an odd number of data values, the median is the single middle value (which is an actual data value).
• If there is an even number of data values, the median is the average of the two middle values. For example, in the data set \(\{1, 2, 3, 4\}\), the median is:

$$ \frac{2 + 3}{2} = 2.5 $$

The value \(2.5\) is not in the original data set. Therefore, the median does not have to be an actual data value.

Step 3: Analyze the Mode

The mode is defined as the value that appears most frequently in a data set. Because it is selected directly from the existing observations that occur most often, the mode must always be an actual value from the data set.

Answer:

A. Since the mode is the most frequent observation that occurs in the data set, it must be an actual value from the data set.