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(D) Let the original observations be $x_1, x_2, \dots, x_n$. The new observations after adding $10$ to each are $x_i' = x_i + 10$.Mean,median,and mode

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Variance

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(D) Let the original observations be $x_1, x_2, \dots, x_n$. The new observations after adding $10$ to each are $x_i' = x_i + 10$.Mean,median,and mode are measures of central tendency and they increase by $10$ when each observation is increased by $10$.However,variance is a measure of dispersion. The variance of a set of observations $x_i$ is given by $\sigma^2 = \frac{1}{n} \sum (x_i - \bar{x})^2$.For the new observations,the new mean is $\bar{x}' = \bar{x} + 10$.The new variance is $\sigma'^2 = \frac{1}{n} \sum ((x_i + 10) - (\bar{x} + 10))^2 = \frac{1}{n} \sum (x_i - \bar{x})^2 = \sigma^2$.Thus,the variance remains unchanged.

All students in a classroom performed poorly in mathematics. The teacher decided to increase the marks of each student by $10$. Which of the following statistical measures will not change even after adding the extra marks?

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