Answer :
Certainly! Let's work through the problem step-by-step.
### 1. Total Number of Learners
First, we need to find the total number of learners. We have the number of learners for each score given in the table.
| Scores | Number of Learners |
|--------|--------------------|
| 45 | 5 |
| 39 | 4 |
| 35 | 3 |
| 25 | 2 |
| 18 | 1 |
| 15 | 2 |
| 5 | 1 |
The total number of learners can be calculated by summing up the number of learners for each score:
[tex]\[ 5 + 4 + 3 + 2 + 1 + 2 + 1 = 18 \text{ learners} \][/tex]
### 2. Total Score and Mean Score
Next, let's calculate the total score and the mean score.
- Total Score:
[tex]\[ (45 \times 5) + (39 \times 4) + (35 \times 3) + (25 \times 2) + (18 \times 1) + (15 \times 2) + (5 \times 1) = 589 \][/tex]
- Mean Score:
[tex]\[ \text{Mean Score} = \frac{\text{Total Score}}{\text{Total Number of Learners}} = \frac{589}{18} \approx 32.72 \][/tex]
### 3. Median Score
To find the median score, we first need to list all the scores in ascending order, taking into account the frequency of each score.
The sorted list of scores is:
[tex]\[ [5, 15, 15, 18, 25, 25, 35, 35, 35, 39, 39, 39, 39, 45, 45, 45, 45, 45] \][/tex]
There are 18 scores, so the median will be the average of the 9th and 10th values in this list:
[tex]\[ \text{Median Score} = \frac{35 + 39}{2} = 37.0 \][/tex]
### 4. Mode Score
The mode is the score that appears most frequently. In our data:
[tex]\[ \text{Score } 45 \text{ appears } 5 \text{ times, which is the highest frequency.} \][/tex]
So, the mode score is:
[tex]\[ 45 \][/tex]
### 5. Difference Between Median and Mode
The difference between the median score and the mode score is:
[tex]\[ 37.0 - 45 = -8.0 \][/tex]
### 6. Actual Pass Mark
The passing percentage is given as 40% of the total marks (which is 50).
[tex]\[ \text{Pass Mark} = 0.4 \times 50 = 20.0 \][/tex]
### 7. Probability of a Learner Scoring Less Than 40%
We need to find the number of learners who scored below the pass mark of 20:
| Scores | Number of Learners |
|--------|--------------------|
| 5 | 1 |
| 15 | 2 |
| 18 | 1 |
So:
[tex]\[ \text{Learners scoring below 20} = 1 + 2 + 1 = 4 \][/tex]
The probability:
[tex]\[ \text{Probability} = \frac{\text{Learners scoring below 20}}{\text{Total Number of Learners}} = \frac{4}{18} = \frac{2}{9} \approx 0.2222 \][/tex]
### Conclusion About the Performance
If the pass mark is 40% (which equates to a score of 20), we see that the probability of a learner scoring less than this pass mark is approximately 0.222 (or about 22.22%). This suggests that 22.22% of the learners scored less than the passing mark.
Given that the mean score is 32.72, which is significantly higher than the pass mark, we can conclude that the overall performance of the learners is above average, although a notable portion still scored below the passing threshold.
### 1. Total Number of Learners
First, we need to find the total number of learners. We have the number of learners for each score given in the table.
| Scores | Number of Learners |
|--------|--------------------|
| 45 | 5 |
| 39 | 4 |
| 35 | 3 |
| 25 | 2 |
| 18 | 1 |
| 15 | 2 |
| 5 | 1 |
The total number of learners can be calculated by summing up the number of learners for each score:
[tex]\[ 5 + 4 + 3 + 2 + 1 + 2 + 1 = 18 \text{ learners} \][/tex]
### 2. Total Score and Mean Score
Next, let's calculate the total score and the mean score.
- Total Score:
[tex]\[ (45 \times 5) + (39 \times 4) + (35 \times 3) + (25 \times 2) + (18 \times 1) + (15 \times 2) + (5 \times 1) = 589 \][/tex]
- Mean Score:
[tex]\[ \text{Mean Score} = \frac{\text{Total Score}}{\text{Total Number of Learners}} = \frac{589}{18} \approx 32.72 \][/tex]
### 3. Median Score
To find the median score, we first need to list all the scores in ascending order, taking into account the frequency of each score.
The sorted list of scores is:
[tex]\[ [5, 15, 15, 18, 25, 25, 35, 35, 35, 39, 39, 39, 39, 45, 45, 45, 45, 45] \][/tex]
There are 18 scores, so the median will be the average of the 9th and 10th values in this list:
[tex]\[ \text{Median Score} = \frac{35 + 39}{2} = 37.0 \][/tex]
### 4. Mode Score
The mode is the score that appears most frequently. In our data:
[tex]\[ \text{Score } 45 \text{ appears } 5 \text{ times, which is the highest frequency.} \][/tex]
So, the mode score is:
[tex]\[ 45 \][/tex]
### 5. Difference Between Median and Mode
The difference between the median score and the mode score is:
[tex]\[ 37.0 - 45 = -8.0 \][/tex]
### 6. Actual Pass Mark
The passing percentage is given as 40% of the total marks (which is 50).
[tex]\[ \text{Pass Mark} = 0.4 \times 50 = 20.0 \][/tex]
### 7. Probability of a Learner Scoring Less Than 40%
We need to find the number of learners who scored below the pass mark of 20:
| Scores | Number of Learners |
|--------|--------------------|
| 5 | 1 |
| 15 | 2 |
| 18 | 1 |
So:
[tex]\[ \text{Learners scoring below 20} = 1 + 2 + 1 = 4 \][/tex]
The probability:
[tex]\[ \text{Probability} = \frac{\text{Learners scoring below 20}}{\text{Total Number of Learners}} = \frac{4}{18} = \frac{2}{9} \approx 0.2222 \][/tex]
### Conclusion About the Performance
If the pass mark is 40% (which equates to a score of 20), we see that the probability of a learner scoring less than this pass mark is approximately 0.222 (or about 22.22%). This suggests that 22.22% of the learners scored less than the passing mark.
Given that the mean score is 32.72, which is significantly higher than the pass mark, we can conclude that the overall performance of the learners is above average, although a notable portion still scored below the passing threshold.
