To analyze the performance of the two tests, we need to look at the implications of the standard deviation and the scores.
I. Which test was done well and why?
The test with the smaller standard deviation is typically considered to be "done better," assuming the mean scores are the same. A lower standard deviation indicates that the scores are closer to the mean, which implies greater consistency and less variability among the students' performances.
If one Chemistry test has a standard deviation of, say, 1.5 and the other has a standard deviation of 2.5, this means the scores in the first test are more tightly clustered around the average. This could suggest that the first test was more effective at differentiating between student understanding or ability, leading to more consistent performances.
II. If a student scored 19 in both tests, which test did he do better relative to the other students?
To determine which test the student performed better on relative to their peers, we need to calculate the z-scores for the student's score of 19 for each test.
The z-score formula is:
\[ z = \frac{{X - \mu}}{{\sigma}} \]
Where:
- \(X\) is the score.
- \(\mu\) is the mean score (which is the same for both tests).
- \(\sigma\) is the standard deviation for each test.
Since the mean score is the same and not provided, let's denote it as \( \mu \). We will calculate the z-scores for the score of 19:
- For the first test (with a standard deviation of 1.5):
\[ z_1 = \frac{{19 - \mu}}{{1.5}} \]
- For the second test (with a standard deviation of 2.5):
\[ z_2 = \frac{{19 - \mu}}{{2.5}} \]
To see which is better, note that a higher z-score indicates better relative performance:
- The z-score \(z_1\) will be larger than \(z_2\) if the score 19 is above the mean (assuming the mean is less than 19).
- If the mean is above 19, the z-scores will be negative, and the test with the smaller standard deviation will still give a higher z-score (in absolute terms) unless the score is very low.
Thus, if both tests have the same mean and one has a smaller standard deviation, a score of 19 will yield a higher z-score in the test with the smaller standard deviation. Therefore:
- The student did better relative to other students in the test with the standard deviation of 1.5 compared to the test with a standard deviation of 2.5.
In summary:
- Test with smaller standard deviation was done better.
- The student did better relative to peers on the test with smaller standard deviation (1.5).
1 answer