One item on a questionnaire asks, "How many siblings (brothers and sisters) did you have when you were a child?" A researcher computes the mean, the median, and the mode for a set of n = 50 responses to this question. Which of the following statements accurately describes the measures of central tendency?
a. Because the scores are all whole numbers, the mean will be a whole number.
b. Because the scores are all whole numbers, the median will be a whole number.
c. Because the scores are all whole numbers, the mode will be a whole number.
d. All of the other options are correct descriptions.
(c)
To see why, select a set of, say, 6 numbers and check it out.
b. Because the scores are all whole numbers, the median will be a whole number.
To determine which of the statements accurately describes the measures of central tendency, let's define what each of the measures means and then analyze each statement in turn.
1. Mean: This is calculated by adding up all the values and dividing the sum by the total number of values. It is the average value of the dataset.
2. Median: This is the middle value of a sorted dataset. In case the dataset has an even number of values, the median is the average of the two middle values.
3. Mode: This is the value or values that appear most frequently in the dataset.
Now let's analyze each statement:
a. "Because the scores are all whole numbers, the mean will be a whole number."
This statement is not necessarily true. The mean can be a whole number regardless of the type of data. It depends on the specific values in the dataset, not just the fact that they are whole numbers. Therefore, statement a is not an accurate description.
b. "Because the scores are all whole numbers, the median will be a whole number."
This statement is also not necessarily true. The median is determined by the position of the values in the dataset, not their type. Therefore, statement b is not an accurate description.
c. "Because the scores are all whole numbers, the mode will be a whole number."
This statement is correct. Since the scores are whole numbers, it is likely that the mode will also be a whole number since it represents the most frequently occurring value(s) in the dataset. Therefore, statement c is an accurate description.
d. "All of the other options are correct descriptions."
This statement is not correct since statement a and statement b are not accurate descriptions.
In conclusion, the accurate description of the measures of central tendency in this scenario is that because the scores are all whole numbers, the mode will be a whole number (statement c).