Understanding Consistency in Physics and Chemistry Test Scores

To determine who had the least consistent scores on both tests, we need to compare the performance of each student relative to their class in both subjects. Since the mean and standard deviation are different for Physics and Chemistry, comparing raw scores directly is not appropriate. A better way to compare performance across different distributions is by using Z-scores.

What is a Z-Score?

A Z-score measures how many standard deviations a particular score is away from the mean of its distribution. It allows us to standardize scores from different tests or distributions, making them comparable. The formula for calculating a Z-score is:

\(Z = \frac{X - \mu}{\sigma}\)

Where:

• \(X\) is the individual score
• \(\mu\) is the mean of the class
• \(\sigma\) is the standard deviation of the class

Consistency in scores across the two tests for a student can be assessed by looking at how similar their Z-scores are in Physics and Chemistry. If the Z-scores are very different, it suggests that the student's performance relative to the class varied significantly between the two subjects, indicating less consistency.

Calculating Z-Scores for Each Student

Let's calculate the Z-scores for Rekha, Reshma, and Sameer in both Physics and Chemistry using the given data:

Physics Data: Mean (\(\mu_{Physics}\)) = 50, Standard Deviation (\(\sigma_{Physics}\)) = 20

Chemistry Data: Mean (\(\mu_{Chemistry}\)) = 20, Standard Deviation (\(\sigma_{Chemistry}\)) = 5

Z-Scores for Rekha:

• Physics Score = 30
• \(Z_{Rekha, Physics} = \frac{30 - 50}{20} = \frac{-20}{20} = -1\)
• Chemistry Score = 26
• \(Z_{Rekha, Chemistry} = \frac{26 - 20}{5} = \frac{6}{5} = 1.2\)

Z-Scores for Reshma:

• Physics Score = 56
• \(Z_{Reshma, Physics} = \frac{56 - 50}{20} = \frac{6}{20} = 0.3\)
• Chemistry Score = 22.5
• \(Z_{Reshma, Chemistry} = \frac{22.5 - 20}{5} = \frac{2.5}{5} = 0.5\)

Z-Scores for Sameer:

• Physics Score = 46
• \(Z_{Sameer, Physics} = \frac{46 - 50}{20} = \frac{-4}{20} = -0.2\)
• Chemistry Score = 19
• \(Z_{Sameer, Chemistry} = \frac{19 - 20}{5} = \frac{-1}{5} = -0.2\)

Comparing Z-Scores for Consistency

Let's summarize the calculated Z-scores:

To find the least consistent scores, we look for the student whose Z-scores in Physics and Chemistry are most different. We can calculate the absolute difference between the Z-scores for each student:

• Rekha: \(|1.2 - (-1.0)| = |1.2 + 1.0| = 2.2\)
• Reshma: \(|0.5 - 0.3| = 0.2\)
• Sameer: \(|-0.2 - (-0.2)| = |-0.2 + 0.2| = 0\)

The student with the largest absolute difference in Z-scores had the least consistent performance relative to the class across the two tests.

• Rekha's Z-score difference: 2.2
• Reshma's Z-score difference: 0.2
• Sameer's Z-score difference: 0.0

Rekha has the largest difference in Z-scores (2.2). This means her performance relative to the class was significantly different between Physics (1 standard deviation below the mean) and Chemistry (1.2 standard deviations above the mean). Reshma's scores were more consistent relative to the class, and Sameer's were perfectly consistent (same Z-score in both subjects).

Conclusion on Least Consistent Scores

Based on the Z-score analysis, Rekha had the least consistent scores on both the Physics and Chemistry tests because her performance relative to the class mean varied the most between the two subjects.

Revision Table: Key Concepts

Additional Information: Interpreting Z-Scores and Consistency

Interpreting Z-scores provides valuable insight into student performance beyond just the raw score. A positive Z-score means the student scored above the mean, while a negative Z-score means they scored below the mean. A Z-score of 0 means the student scored exactly at the mean.

When comparing consistency across multiple subjects, looking at the similarity of Z-scores is a standard method. A small difference between Z-scores indicates high consistency (the student performed similarly relative to their peers in both subjects). A large difference indicates low consistency (the student performed much better relative to peers in one subject compared to the other).

In this specific case:

• Rekha: Did poorly in Physics (1 SD below mean) but well in Chemistry (1.2 SD above mean) relative to her class. This wide swing shows low consistency.
• Reshma: Did slightly above average in both Physics (0.3 SD above mean) and Chemistry (0.5 SD above mean) relative to her class. Her performance was relatively consistent.
• Sameer: Did slightly below average in both Physics (0.2 SD below mean) and Chemistry (0.2 SD below mean) relative to his class. His performance was highly consistent.

Therefore, Rekha's scores were the least consistent when analyzed using Z-scores.