Understanding Student Performance Comparison

The question asks us to identify the student who had the poorest performance on either the Physics or the Chemistry test among Rekha, Reshma, and Sameer. We are given their scores, along with the mean and standard deviation for the entire class in each subject. Simply comparing the raw scores might be misleading because the tests likely have different difficulty levels and scoring scales, as indicated by the different means and standard deviations.

To compare performance across different tests or subjects, it is often useful to standardize the scores. A common way to do this is by calculating a Z-score for each student's score in each subject. The Z-score tells us how many standard deviations a student's score is above or below the mean of the class for that specific test.

The formula for calculating a Z-score ($$z$$) is:

$$z = \frac{x - \mu}{\sigma}$$

Where:

• $$x$$ is the individual student's score.
• $$\mu$$ is the mean score of the class for that test.
• $$\sigma$$ is the standard deviation of the class scores for that test.

A lower (more negative) Z-score indicates a poorer performance relative to the class mean.

Calculating Z-Scores for Each Student

First, let's summarize the given data:

Now, let's calculate the Z-score for each student in each subject:

Rekha:

• Physics Z-score: $$z_{Physics} = \frac{30 - 50}{20} = \frac{-20}{20} = -1.0$$
• Chemistry Z-score: $$z_{Chemistry} = \frac{26 - 20}{5} = \frac{6}{5} = 1.2$$

Reshma:

• Physics Z-score: $$z_{Physics} = \frac{56 - 50}{20} = \frac{6}{20} = 0.3$$
• Chemistry Z-score: $$z_{Chemistry} = \frac{22.5 - 20}{5} = \frac{2.5}{5} = 0.5$$

Sameer:

• Physics Z-score: $$z_{Physics} = \frac{46 - 50}{20} = \frac{-4}{20} = -0.2$$
• Chemistry Z-score: $$z_{Chemistry} = \frac{19 - 20}{5} = \frac{-1}{5} = -0.2$$

Comparing Z-Scores to Determine Poorest Performance

Let's list the Z-scores for each student in each subject:

We are looking for the student who had the poorest score on *either* test. In terms of relative performance (measured by Z-scores), a more negative Z-score indicates poorer performance. Let's look at the lowest Z-score for each student across their two tests:

• Rekha's lowest Z-score: -1.0 (in Physics)
• Reshma's lowest Z-score: 0.3 (in Physics)
• Sameer's lowest Z-score: -0.2 (in both Physics and Chemistry)

Comparing these lowest Z-scores (-1.0, 0.3, -0.2), the lowest value is -1.0. This is Rekha's Z-score in Physics.

Therefore, based on the standardized scores (Z-scores), Rekha had the poorest performance relative to her class mean in Physics, and this relative performance is the lowest among the lowest relative performances of all three students in either test.

Although Sameer had the lowest raw score (19 in Chemistry), his performance relative to the Chemistry class (Z-score -0.2) was not as poor as Rekha's performance relative to the Physics class (Z-score -1.0).

Conclusion on Poorest Student Performance

By calculating and comparing the Z-scores, we can determine the student who performed worst relative to their class in either subject. Rekha's Z-score of -1.0 in Physics is the lowest among all calculated Z-scores, indicating her performance was 1 standard deviation below the mean in Physics, which is the poorest relative performance observed.

Revision Table: Key Concepts for Comparing Scores

Additional Information: Why Standardize Scores?

Comparing raw scores directly from different tests can be misleading because tests might have different maximum possible scores, different difficulty levels, and different distributions of scores. For example, scoring 60 on a test where the average is 70 is different from scoring 60 on a test where the average is 40. Standardizing scores, like calculating Z-scores, helps us understand a student's performance in the context of the group that took the test.

A Z-score tells us how far a particular score is from the mean, expressed in units of standard deviation. A positive Z-score means the score is above the mean, a negative Z-score means it's below the mean, and a Z-score of 0 means the score is exactly at the mean. The magnitude of the Z-score indicates how far the score is from the mean in standard deviation units. By using Z-scores, we can compare a student's relative standing in Physics to their relative standing in Chemistry, even though the tests have different scales and averages.