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Answer :
We are given the following information:
- For the boys:
- Number of boys: $n_{\text{boys}} = 40$
- Mean mark: $\bar{x}_{\text{boys}} = 29.625$
- Standard deviation: $\sigma_{\text{boys}} = 10.897$
- For the girls:
- Number of girls: $n_{\text{girls}} = 40$
- Mean mark: $\bar{x}_{\text{girls}} = 30.0$
- Standard deviation: $\sigma_{\text{girls}} = 6.5$
We can follow these steps to compare their performance:
1. **Calculate the Total Marks (Optional):**
Although the problem does not require the total marks for comparison, it is useful to see the overall performance of each group.
- Total marks for boys:
$$\text{Total}_{\text{boys}} = n_{\text{boys}} \times \bar{x}_{\text{boys}} = 40 \times 29.625 = 1185.0$$
- Total marks for girls:
$$\text{Total}_{\text{girls}} = n_{\text{girls}} \times \bar{x}_{\text{girls}} = 40 \times 30.0 = 1200.0$$
2. **Compare the Mean Marks:**
The mean mark is a measure of the average score obtained by the students.
- The boys have a mean of $29.625$.
- The girls have a mean of $30.0$.
Since $30.0$ is slightly higher than $29.625$, we can say that on average, the girls scored a bit more than the boys.
3. **Compare the Standard Deviations:**
The standard deviation provides an idea about the spread or consistency of the marks.
- The boys have a standard deviation of $10.897$.
- The girls have a standard deviation of $6.5$.
A lower standard deviation indicates that the marks are more closely clustered around the mean. Hence, the girls were more consistent in their scores compared to the boys.
4. **Conclusion:**
Based on the analysis:
- The girls have a slightly higher mean mark of $30.0$ compared to the boys' mean of $29.625$.
- The girls also have a lower standard deviation ($6.5$ vs. $10.897$), meaning their scores were more consistent.
Therefore, we conclude that the girls performed slightly better and more consistently than the boys.
Final Answer: Girls performed slightly better than boys due to a higher mean ($30.0$ vs. $29.625$) and a lower standard deviation ($6.5$ vs. $10.897$).
- For the boys:
- Number of boys: $n_{\text{boys}} = 40$
- Mean mark: $\bar{x}_{\text{boys}} = 29.625$
- Standard deviation: $\sigma_{\text{boys}} = 10.897$
- For the girls:
- Number of girls: $n_{\text{girls}} = 40$
- Mean mark: $\bar{x}_{\text{girls}} = 30.0$
- Standard deviation: $\sigma_{\text{girls}} = 6.5$
We can follow these steps to compare their performance:
1. **Calculate the Total Marks (Optional):**
Although the problem does not require the total marks for comparison, it is useful to see the overall performance of each group.
- Total marks for boys:
$$\text{Total}_{\text{boys}} = n_{\text{boys}} \times \bar{x}_{\text{boys}} = 40 \times 29.625 = 1185.0$$
- Total marks for girls:
$$\text{Total}_{\text{girls}} = n_{\text{girls}} \times \bar{x}_{\text{girls}} = 40 \times 30.0 = 1200.0$$
2. **Compare the Mean Marks:**
The mean mark is a measure of the average score obtained by the students.
- The boys have a mean of $29.625$.
- The girls have a mean of $30.0$.
Since $30.0$ is slightly higher than $29.625$, we can say that on average, the girls scored a bit more than the boys.
3. **Compare the Standard Deviations:**
The standard deviation provides an idea about the spread or consistency of the marks.
- The boys have a standard deviation of $10.897$.
- The girls have a standard deviation of $6.5$.
A lower standard deviation indicates that the marks are more closely clustered around the mean. Hence, the girls were more consistent in their scores compared to the boys.
4. **Conclusion:**
Based on the analysis:
- The girls have a slightly higher mean mark of $30.0$ compared to the boys' mean of $29.625$.
- The girls also have a lower standard deviation ($6.5$ vs. $10.897$), meaning their scores were more consistent.
Therefore, we conclude that the girls performed slightly better and more consistently than the boys.
Final Answer: Girls performed slightly better than boys due to a higher mean ($30.0$ vs. $29.625$) and a lower standard deviation ($6.5$ vs. $10.897$).
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