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# All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given ? (1) median (2) mode (3) variance (4) mean

Latika Leekha
6 years ago
All the three statistical measures, mean, median and mode depend on absolaute values while variance measures spread of data points. Hence, if all the points are shifted by the same amount, the spread will not change and hence the variance remains the same. We show how the variance remains unchanged if the teacher gives grace marks of 10 to each student:
Let the original marks be xi, i = 1 to N.
Then, new marks are yi = xi + 10, i = 1 to N.
Let $\overline{x} and \overline{y}$ be the original and new mean values.
Then, by definition, mean, N . $\overline{x}$ = $\sum_{i=1}^{n}{X_i}$
Similarly, N. $\overline{y}$ = $\sum_{i=1}^{n}{Y_i}$
N. $\overline{y}$ = xi + 10
Hence, $\overline{y} = \overline{x} + 10$
Now we try to find the original and new variance
Original variance, i.e. σ2org = [Σ (xi – mean)2] / N
New variance, i.e. σ2new = [Σ (yi – mean)2] / N
= Σ[(xi + 10) – ($\overline{x} + 10$)]2 / N
= Σ (xi – mean)2 / N
= σ2org
Hence, the two variance are the same.
Thanks & Regards
Latika Leekha