Question

A set of data consists of the numbers: 5, 8, 12, 15, 20.
Part 1: The median of the data set is equal to the mean of the data set.
Part 2: If the number 10 is added to the data set, the new median will be less than the original median.
Evaluation: Part 1: False. Part 2: False.

Hint:

Always ensure your data set is sorted before finding the median. When adding a new number to a set, remember to re-sort the data before calculating the new median.

Answer 1

Step 1: Understanding the Concept:
This question requires us to evaluate two statements about a set of data.
Mean: The sum of the values divided by the number of values.
Median: The middle value in an ordered set of data. If the set has an even number of values, the median is the average of the two middle values.
Step 2: Detailed Explanation:
Evaluation of Part 1:
The given data set is already ordered: \{5, 8, 12, 15, 20\}.
Median: There are 5 numbers, so the median is the middle (3rd) number.
\[ \text{Median} = 12 \]
Mean: First, find the sum of the numbers.
\[ \text{Sum} = 5 + 8 + 12 + 15 + 20 = 60 \]
There are 5 numbers in the set.
\[ \text{Mean} = \frac{\text{Sum}}{\text{Count}} = \frac{60}{5} = 12 \]
The statement says the median (12) is equal to the mean (12). This is correct. Therefore, Part 1 is True.
Evaluation of Part 2:
A new number, 10, is added to the data set.
The original median was 12.
The new data set is \{5, 8, 12, 15, 20, 10\}. Let's put it in order:
\[ \{5, 8, 10, 12, 15, 20\} \]
New Median: The set now has 6 numbers. The median is the average of the two middle numbers (the 3rd and 4th).
\[ \text{New Median} = \frac{10 + 12}{2} = \frac{22}{2} = 11 \]
The statement says the new median (11) will be less than the original median (12). This is correct, as \(11<12\). Therefore, Part 2 is True.
Step 3: Final Answer:
Both statements are factually true based on mathematical calculation. The provided answer key in the document ("Evaluation: Part 1: False. Part 2: False.") is incorrect. The correct evaluation is that both parts are true.