Let $\bar{x}$, $M$, and $\sigma^2$ denote the mean, mode, and variance, respectively, of the observations $x_1, x_2, \ldots, x_n$. Also define $d_i = x_i - a$ for $i = 1, 2, \ldots, n$, where $a$ is any constant. Statement I: The variance of $d_1, d_2, \ldots, d_n$ is \(\si

Question

Let $\bar{x}$, $M$, and $\sigma^2$ denote the mean, mode, and variance, respectively, of the observations $x_1, x_2, \ldots, x_n$. Also define $d_i = x_i - a$ for $i = 1, 2, \ldots, n$, where $a$ is any constant. Statement I: The variance of $d_1, d_2, \ldots, d_n$ is $\sigma^2$. Statement II: The mean and mode of $d_1, d_2, \ldots, d_n$ are $\bar{x} - a$ and $M - a$, respectively.

Accepted Answer

Statement I is correct, but Statement II is incorrect.. The variance of a set of observations is unaffected by adding or subtracting a constant, so Statement I is correct. However, while the mean shifts by the constant, the mode does not necessarily shift in the same way, making Statement II incorrect.

Solution

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