Answer :
Certainly! Let's break down each part of the question step by step.
### (a) How does the median divide the data in terms of percentage?
To determine how the median divides the given IQ scores into percentages, we need to consider the number of scores below the median and the number of scores at or above the median.
Given IQ scores: [tex]\(10, 8, 6, 5, 4, 1, 2\)[/tex]
The median of the data is [tex]\(5\)[/tex].
- Scores below the median: [tex]\(4, 1, 2\)[/tex] (3 scores)
- Scores at or above the median: [tex]\(10, 8, 6, 5\)[/tex] (4 scores)
The total number of scores is 7.
The percentage of scores below the median is:
[tex]\[ \frac{\text{Number of scores below the median}}{\text{Total number of scores}} \times 100\% = \frac{3}{7} \times 100\% \approx 42.86\% \][/tex]
The percentage of scores at or above the median is:
[tex]\[ \frac{\text{Number of scores at or above the median}}{\text{Total number of scores}} \times 100\% = \frac{4}{7} \times 100\% \approx 57.14\% \][/tex]
### (b) Find the median IQ score.
Given the dataset [tex]\(10, 8, 6, 5, 4, 1, 2\)[/tex], when sorted: [tex]\(1, 2, 4, 5, 6, 8, 10\)[/tex].
Since the number of scores is odd (7), the median is the middle value:
Median [tex]\(= 5\)[/tex].
### (c) Find the sum of scores below and above the median. Provide the sums separately.
First, identify the scores below and above the median:
- Scores below the median [tex]\(5\)[/tex]: [tex]\(4, 1, 2\)[/tex]
- Scores above the median [tex]\(5\)[/tex]: [tex]\(10, 8, 6\)[/tex]
Sum of scores below the median:
[tex]\[ 4 + 1 + 2 = 7 \][/tex]
Sum of scores above the median:
[tex]\[ 10 + 8 + 6 = 24 \][/tex]
### (d) Find the difference between the average scores below the median and the average scores above the median.
First, calculate the average scores below and above the median:
For the scores below the median:
[tex]\[ \text{Average score below the median} = \frac{\text{Sum of scores below the median}}{\text{Number of scores below the median}} = \frac{7}{3} \approx 2.33 \][/tex]
For the scores above the median:
[tex]\[ \text{Average score above the median} = \frac{\text{Sum of scores above the median}}{\text{Number of scores above the median}} = \frac{24}{3} = 8 \][/tex]
Finally, find the difference between the average scores above and below the median:
[tex]\[ \text{Difference} = 8 - 2.33 \approx 5.67 \][/tex]
### Summary
1. The median divides the data as approximately [tex]\(42.86\%\)[/tex] below and [tex]\(57.14\%\)[/tex] at or above the median.
2. Median IQ score is [tex]\(5\)[/tex].
3. Sum of scores below the median is [tex]\(7\)[/tex]; sum of scores above the median is [tex]\(24\)[/tex].
4. The difference between the average scores above and below the median is approximately [tex]\(5.67\)[/tex].
### (a) How does the median divide the data in terms of percentage?
To determine how the median divides the given IQ scores into percentages, we need to consider the number of scores below the median and the number of scores at or above the median.
Given IQ scores: [tex]\(10, 8, 6, 5, 4, 1, 2\)[/tex]
The median of the data is [tex]\(5\)[/tex].
- Scores below the median: [tex]\(4, 1, 2\)[/tex] (3 scores)
- Scores at or above the median: [tex]\(10, 8, 6, 5\)[/tex] (4 scores)
The total number of scores is 7.
The percentage of scores below the median is:
[tex]\[ \frac{\text{Number of scores below the median}}{\text{Total number of scores}} \times 100\% = \frac{3}{7} \times 100\% \approx 42.86\% \][/tex]
The percentage of scores at or above the median is:
[tex]\[ \frac{\text{Number of scores at or above the median}}{\text{Total number of scores}} \times 100\% = \frac{4}{7} \times 100\% \approx 57.14\% \][/tex]
### (b) Find the median IQ score.
Given the dataset [tex]\(10, 8, 6, 5, 4, 1, 2\)[/tex], when sorted: [tex]\(1, 2, 4, 5, 6, 8, 10\)[/tex].
Since the number of scores is odd (7), the median is the middle value:
Median [tex]\(= 5\)[/tex].
### (c) Find the sum of scores below and above the median. Provide the sums separately.
First, identify the scores below and above the median:
- Scores below the median [tex]\(5\)[/tex]: [tex]\(4, 1, 2\)[/tex]
- Scores above the median [tex]\(5\)[/tex]: [tex]\(10, 8, 6\)[/tex]
Sum of scores below the median:
[tex]\[ 4 + 1 + 2 = 7 \][/tex]
Sum of scores above the median:
[tex]\[ 10 + 8 + 6 = 24 \][/tex]
### (d) Find the difference between the average scores below the median and the average scores above the median.
First, calculate the average scores below and above the median:
For the scores below the median:
[tex]\[ \text{Average score below the median} = \frac{\text{Sum of scores below the median}}{\text{Number of scores below the median}} = \frac{7}{3} \approx 2.33 \][/tex]
For the scores above the median:
[tex]\[ \text{Average score above the median} = \frac{\text{Sum of scores above the median}}{\text{Number of scores above the median}} = \frac{24}{3} = 8 \][/tex]
Finally, find the difference between the average scores above and below the median:
[tex]\[ \text{Difference} = 8 - 2.33 \approx 5.67 \][/tex]
### Summary
1. The median divides the data as approximately [tex]\(42.86\%\)[/tex] below and [tex]\(57.14\%\)[/tex] at or above the median.
2. Median IQ score is [tex]\(5\)[/tex].
3. Sum of scores below the median is [tex]\(7\)[/tex]; sum of scores above the median is [tex]\(24\)[/tex].
4. The difference between the average scores above and below the median is approximately [tex]\(5.67\)[/tex].
