The mean score in a test of 12 boys in a class was 16 while the overall mean of 30 students in the class was 18.1. What was the mean score of the girls?
19.5
Calculating Mean Score for Girls in the Class Test
The problem asks us to find the mean score of the girls in a class test, given information about the boys' scores and the overall class scores. We have the number of boys, their mean score, the total number of students, and the overall mean score for the class.
| Group | Number of Students | Mean Score | Total Score |
|---|---|---|---|
| Boys | 12 | 16 | ? |
| Girls | ? | ? | ? |
| Total Class | 30 | 18.1 | ? |
Step-by-Step Solution for Mean Score Calculation
To find the mean score of the girls, we first need to determine the total score of all students and the total score of the boys. Once we have these, we can find the total score of the girls and then calculate their mean score.
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Calculate the Total Score of the Boys:
The total score for a group is the number of students in the group multiplied by their mean score.
Total score of boys = Number of boys $\times$ Mean score of boys
Total score of boys = $12 \times 16 = 192$
-
Calculate the Total Score of All Students:
Similarly, the total score for the entire class is the total number of students multiplied by the overall mean score.
Total score of all students = Total number of students $\times$ Overall mean score
Total score of all students = $30 \times 18.1 = 543$
-
Determine the Number of Girls:
The total number of students in the class is the sum of the number of boys and the number of girls.
Number of girls = Total number of students - Number of boys
Number of girls = $30 - 12 = 18$
-
Calculate the Total Score of the Girls:
The total score of the girls is the difference between the total score of all students and the total score of the boys.
Total score of girls = Total score of all students - Total score of boys
Total score of girls = $543 - 192 = 351$
-
Calculate the Mean Score of the Girls:
The mean score of the girls is their total score divided by the number of girls.
Mean score of girls = Total score of girls / Number of girls
Mean score of girls = $351 / 18 = 19.5$
Thus, the mean score of the girls in the class test was 19.5.
| Group | Number of Students | Mean Score | Total Score |
|---|---|---|---|
| Boys | 12 | 16 | 192 |
| Girls | 18 | 19.5 | 351 |
| Total Class | 30 | 18.1 | 543 |
Revision Table: Key Concepts
| Concept | Definition | Formula |
|---|---|---|
| Mean (Average) | The sum of all values in a dataset divided by the number of values. | $\text{Mean} = \frac{\text{Sum of values}}{\text{Number of values}}$ |
| Total Score | The sum of scores for all individuals in a group. | $\text{Total Score} = \text{Number of individuals} \times \text{Mean Score}$ |
Additional Information: Understanding Weighted Averages
This problem is an example of how mean values work when dealing with different groups within a larger set. The overall mean score of the class is a weighted average of the mean score of the boys and the mean score of the girls.
A weighted average takes into account the size or 'weight' of each group. The formula for a weighted average is:
Overall Mean $= \frac{(\text{Mean}_1 \times \text{Weight}_1) + (\text{Mean}_2 \times \text{Weight}_2) + \dots}{\text{Weight}_1 + \text{Weight}_2 + \dots}$
In this case, the 'weights' are the number of students in each group (boys and girls).
Overall Mean $= \frac{(\text{Mean Score of Boys} \times \text{Number of Boys}) + (\text{Mean Score of Girls} \times \text{Number of Girls})}{\text{Number of Boys} + \text{Number of Girls}}$
Using the total scores method as shown in the solution is often the most straightforward way to solve these kinds of problems, as it directly uses the fundamental definition of the mean.
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