Sure! Let's analyze the test scores and their frequencies to find the mean, median, mode, and range.
First, let's compile the complete dataset from the given scores and their frequencies.
Scores and their frequencies:
- 90 appears 1 time
- 80 appears 6 times
- 70 appears 9 times
- 60 appears 4 times
- 50 appears 3 times
- 40 appears 2 times
We'll start by writing down each score the number of times it appears and arranging them in ascending order:
[tex]\[
40, 40, 50, 50, 50, 60, 60, 60, 60, 70, 70, 70, 70, 70, 70, 70, 70, 70, 80, 80, 80, 80, 80, 80, 90
\][/tex]
Total number of scores (N) is:
[tex]\[ 1 + 6 + 9 + 4 + 3 + 2 = 25 \][/tex]
### Mean
The mean is the average of all the scores. To find the mean, we multiply each score by its frequency, sum these products, and then divide by the total number of scores.
[tex]\[
\text{Mean} = \frac{\sum (\text{Score} \times \text{Frequency})}{\text{Total number of scores}}
\][/tex]
Calculating the sum of products of scores and frequencies:
[tex]\[
(90 \times 1) + (80 \times 6) + (70 \times 9) + (60 \times 4) + (50 \times 3) + (40 \times 2)
\][/tex]
[tex]\[
= 90 + 480 + 630 + 240 + 150 + 80 = 1670
\][/tex]
Dividing by the total number of scores (25):
[tex]\[
\text{Mean} = \frac{1670}{25} = 66.8
\][/tex]
### Median
The median is the middle value when the scores are arranged in ascending order. Since there are 25 scores, the median will be the 13th score. Counting from the smallest:
[tex]\[
40, 40, 50, 50, 50, 60, 60, 60, 60, 70, 70, 70, \boxed{70}, 70, 70, 70, 70, 70, 80, 80, 80, 80, 80, 80, 90
\][/tex]
The 13th score is 70.
### Mode
The mode is the score that appears most frequently. From the frequencies given, the score with the highest frequency is 70, which appears 9 times.
### Range
The range is the difference between the highest and lowest scores.
[tex]\[
\text{Range} = \text{Highest score} - \text{Lowest score} = 90 - 40 = 50
\][/tex]
### Summary
So the final answers are:
- Mean [tex]\( = 66.8 \)[/tex]
- Median [tex]\( = 70 \)[/tex]
- Mode [tex]\( = 70 \)[/tex]
- Range [tex]\( = 50 \)[/tex]
If you have any more questions or need further explanation, let me know!
