Let x¯, M and σ2 be respectively the mean, mode and variance of n observations x1, x2,...., xn and di=−xi−a, i=1, 2,...., n, where a is any number.Statement I: Variance of d1, d2,..., dn is σ2.Statement II: Mean and mode of d1, d2,...., dn are −x¯−a and −M−a, respectively.Statement I and Statement II are both true Statement I and Statement II are both false Statement I is true and Statement II is falseStatement I is false and Statement II is true
Question
Let x¯, M and σ2 be respectively the mean, mode and variance of n observations x1, x2,...., xn and di=−xi−a, i=1, 2,...., n, where a is any number.Statement I: Variance of d1, d2,..., dn is σ2.Statement II: Mean and mode of d1, d2,...., dn are −x¯−a and −M−a, respectively.Statement I and Statement II are both true Statement I and Statement II are both false Statement I is true and Statement II is falseStatement I is false and Statement II is true
Solution
Statement I is true and Statement II is true.
Here's why:
Statement I: Variance is a measure of how spread out the numbers in a data set are. It is calculated as the average of the squared differences from the mean. In this case, the variance of d1, d2,..., dn is indeed σ2 because variance is not affected by changes in location (i.e., adding or subtracting a constant from each data point).
Statement II: The mean and mode of d1, d2,..., dn are -x¯-a and -M-a, respectively. This is because when you add or subtract a constant from each data point, the mean and mode shift by that constant. In this case, subtracting 'a' from each xi results in a shift of the mean and mode by '-a'. Therefore, the mean and mode of the new data set are -x¯-a and -M-a, respectively.
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