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All the students of a class performed po...

All the students of a class performed poorly in physics. The teacher decided to give grace marks of 15 to the entire class. Which of the following statistical measures will not change even after the grace marks were given?

A

median

B

mode

C

variance

D

mean

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze how the grace marks of 15 affect different statistical measures: mean, median, mode, and variance. ### Step-by-Step Solution: 1. **Understanding the Effect of Grace Marks**: - When every student in the class receives 15 grace marks, it means that every student's score is increased by 15. 2. **Effect on Mean**: - The mean (average) of the scores will change because it is calculated by summing all the scores and dividing by the number of students. Since every score increases by 15, the total sum of scores increases, thus increasing the mean. - **Conclusion**: Mean will change. 3. **Effect on Median**: - The median is the middle value of the ordered data set. Since all scores are increased by the same amount (15), the order of the scores remains the same, and thus the median will also increase by 15. - **Conclusion**: Median will change. 4. **Effect on Mode**: - The mode is the value that appears most frequently in the data set. If all scores are increased by 15, the mode will also increase by 15, as the most frequent score will still be the same relative score. - **Conclusion**: Mode will change. 5. **Effect on Variance**: - Variance measures the spread of the scores around the mean. It is calculated using the formula: \[ \text{Variance} = \frac{1}{n} \sum (x_i - \bar{x})^2 \] - If we denote the original scores as \( y_i \) and the new scores after adding 15 as \( x_i = y_i + 15 \), the mean of the new scores will be \( \bar{x} = \bar{y} + 15 \). - When calculating the variance of the new scores: \[ \text{Variance} = \frac{1}{n} \sum ((y_i + 15) - (\bar{y} + 15))^2 = \frac{1}{n} \sum (y_i - \bar{y})^2 \] - The addition of 15 cancels out, and thus the variance remains unchanged. - **Conclusion**: Variance will not change. ### Final Answer: The statistical measure that will not change after the grace marks are given is **Variance**.
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