Read the following paragraph carefully and answer the following five questions: In a ninth standard class of 50 students, the teacher administered test to measure achievement of students in Physics and Chemistry, respectively. The mean and SD of the class for the achievement scores in Physics were 50 and 20 , respectively whereas, the mean and SD of the class for the achievement scores in Chemistry were 20 and 5 respectively. One student Rekha scored 30 marks in Physics and 26 marks in Chemistry. Another student Reshma scored 56 marks in Physics and 22.5 marks in Chemistry. Another student Sameer scored 46 marks in Physics and 19 marks in Chemistry. The summary of these scores is given below:S ubject Mean of the class SD of the class Rekha Reshma Sameer Physics 50 20 30 56 46 Chemistry 20 5 26 22.5 19
Who had the most consistent scores on both tests?
Analyzing Student Score Consistency in Physics and Chemistry
To determine which student had the most consistent scores across both the Physics and Chemistry tests, we need to compare their performance relative to the class average and the spread of scores in each subject. Simply looking at the raw scores can be misleading because the tests have different means and standard deviations.
A good way to compare performance across different distributions is by using Z-scores (also known as standard scores). A Z-score measures how many standard deviations a particular score is away from the mean of the distribution. The formula for a Z-score is:
\( Z = \frac{X - \mu}{\sigma} \)
Where:
- \( X \) is the individual score
- \( \mu \) is the mean of the class scores
- \( \sigma \) is the standard deviation of the class scores
A score is considered more consistent if its Z-score (or rather, its absolute Z-score, \(|Z|\)) is closer to zero, meaning the score is closer to the class mean relative to the variability in the scores for that subject.
Let's first summarize the given data for the achievement scores:
| Subject | Mean of the class (\(\mu\)) | SD of the class (\(\sigma\)) | Rekha's Score | Reshma's Score | Sameer's Score |
|---|---|---|---|---|---|
| Physics | 50 | 20 | 30 | 56 | 46 |
| Chemistry | 20 | 5 | 26 | 22.5 | 19 |
Calculating Z-scores for Each Student
Now, let's calculate the Z-score for each student in both Physics and Chemistry.
Physics Z-scores: (\(\mu_P = 50\), \(\sigma_P = 20\))
- Rekha: \( Z_{Rekha, P} = \frac{30 - 50}{20} = \frac{-20}{20} = -1 \)
- Reshma: \( Z_{Reshma, P} = \frac{56 - 50}{20} = \frac{6}{20} = 0.3 \)
- Sameer: \( Z_{Sameer, P} = \frac{46 - 50}{20} = \frac{-4}{20} = -0.2 \)
Chemistry Z-scores: (\(\mu_C = 20\), \(\sigma_C = 5\))
- Rekha: \( Z_{Rekha, C} = \frac{26 - 20}{5} = \frac{6}{5} = 1.2 \)
- Reshma: \( Z_{Reshma, C} = \frac{22.5 - 20}{5} = \frac{2.5}{5} = 0.5 \)
- Sameer: \( Z_{Sameer, C} = \frac{19 - 20}{5} = \frac{-1}{5} = -0.2 \)
Comparing Consistency Using Absolute Z-scores
Consistency is about how close the score is to the mean relative to the standard deviation, regardless of whether it's above or below the mean. So, we look at the absolute value of the Z-scores.
| Student | Absolute Z-score (Physics) | Absolute Z-score (Chemistry) |
|---|---|---|
| Rekha | \( |Z_{Rekha, P}| = |-1| = 1 \) | \( |Z_{Rekha, C}| = |1.2| = 1.2 \) |
| Reshma | \( |Z_{Reshma, P}| = |0.3| = 0.3 \) | \( |Z_{Reshma, C}| = |0.5| = 0.5 \) |
| Sameer | \( |Z_{Sameer, P}| = |-0.2| = 0.2 \) | \( |Z_{Sameer, C}| = |-0.2| = 0.2 \) |
To assess overall consistency across both tests, we can sum the absolute Z-scores for each student. A smaller sum indicates greater overall consistency relative to the class performance in both subjects.
- Rekha: Sum of absolute Z-scores = \( 1 + 1.2 = 2.2 \)
- Reshma: Sum of absolute Z-scores = \( 0.3 + 0.5 = 0.8 \)
- Sameer: Sum of absolute Z-scores = \( 0.2 + 0.2 = 0.4 \)
Comparing the sums of absolute Z-scores, Sameer has the lowest value (0.4), followed by Reshma (0.8), and then Rekha (2.2). This means Sameer's scores in both Physics and Chemistry are the closest to their respective class means in terms of standard deviations, making Sameer's scores the most consistent.
Therefore, Sameer had the most consistent scores on both tests.
The final answer is Sameer.
Revision Table: Key Statistical Concepts
| Concept | Definition | Relevance to Score Analysis |
|---|---|---|
| Mean (\(\mu\)) | The average value in a dataset. | Indicates the central tendency of class scores. |
| Standard Deviation (\(\sigma\)) | A measure of the amount of variation or dispersion of a set of values. | Indicates the spread of scores around the mean. A larger SD means scores are more spread out. |
| Z-score (Standard Score) | Measures how many standard deviations away from the mean a data point is. | Allows comparison of scores from different distributions (tests) by standardizing them. Essential for comparing performance relative to the class average and spread. |
| Consistency of Scores | How close individual scores are to the average score in their respective distributions, relative to the variability. | Assessed using the absolute Z-scores; lower absolute Z-scores indicate higher consistency. |
Additional Information: Understanding Z-scores and Consistency
Z-scores are powerful tools in statistics because they allow us to understand where a particular data point stands within its distribution. 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 zero means the score is exactly at the mean.
When assessing consistency across different tests, simply comparing raw scores or even differences from the mean in raw units doesn't work well if the tests have different scales (means and standard deviations). For example, a difference of 10 marks might be significant on a test with an SD of 5, but not significant on a test with an SD of 20. The Z-score accounts for this variability.
In this specific problem, Sameer's Z-scores were -0.2 (Physics) and -0.2 (Chemistry). Both are very close to 0, indicating that in both subjects, Sameer's score was relatively close to the class mean when considering the spread of scores in each subject. Rekha, with Z-scores of -1 (Physics) and 1.2 (Chemistry), had scores that were further away from the mean (1 standard deviation below in Physics, 1.2 standard deviations above in Chemistry). Reshma had Z-scores of 0.3 (Physics) and 0.5 (Chemistry), which are closer to the mean than Rekha's but slightly further than Sameer's.
Therefore, by calculating and comparing the absolute Z-scores, we objectively determine that Sameer exhibited the highest level of consistency in performance relative to the class across both the Physics and Chemistry tests.
Important Questions from Data Interpretation
The table shows the monthly expenditure of Saumya. What percent of her monthly salary is she able to save?
Expense Amount Food 5,000 Rent 20,000 Travel 10,000 Clothing 3,000 Miscellaneous 15,000 Savings 7,500 Who had the best score on either test?
Who had the least consistent scores on both tests?
Who had the poorest score on either test?
Who had the best mean score on both tests?
