To determine the most appropriate measure of center for a given data set, we can analyze the three common measures of central tendency: the mean, median, and mode. Let's calculate each of these for the data set: 20, 6, 42, 13, 15, 10, 9, 12, 12.
1. Mean:
The mean is the sum of all the numbers divided by the count of numbers.
[tex]\[
\text{Mean} = \frac{\sum_{i=1}^n x_i}{n} = \frac{20 + 6 + 42 + 13 + 15 + 10 + 9 + 12 + 12}{9}
= \frac{139}{9} \approx 15.44
\][/tex]
2. Median:
The median is the middle value when the numbers are sorted in ascending order. If the count of numbers is odd, the median is the middle number. If the count is even, it is the average of the two middle numbers.
Sorting the data set: [tex]\(6, 9, 10, 12, 12, 13, 15, 20, 42\)[/tex]
Since there are 9 numbers, the median is the 5th number:
[tex]\[
\text{Median} = 12
\][/tex]
3. Mode:
The mode is the number that appears most frequently in the data set.
Checking the frequencies:
- [tex]\(6: 1\)[/tex] time
- [tex]\(9: 1\)[/tex] time
- [tex]\(10: 1\)[/tex] time
- [tex]\(12: 2\)[/tex] times
- [tex]\(13: 1\)[/tex] time
- [tex]\(15: 1\)[/tex] time
- [tex]\(20: 1\)[/tex] time
- [tex]\(42: 1\)[/tex] time
The mode is:
[tex]\[
\text{Mode} = 12
\][/tex]
Now, let's determine which measure of center is most appropriate.
- Mean: Influenced by the outlier (42), which skews it higher.
- Median: Not affected by outliers and provides the middle value.
- Mode: Reflects the most frequent value but may not provide a central tendency effectively if there’s no significant repetition.
Since the mean is skewed by the outlier (42) and can be misleading, the median is often a better representation of the central tendency when the data set includes outliers.
Answer:
O B. median
