Let x̄, M, and σ² denote the mean, mode, and variance, respectively, of the observations x₁, x₂, …, xₙ. Also define d_i = x_i - a for i = 1, 2, …, n, where a is any constant.

Statement I: The variance of d₁, d₂, …, dₙ is σ².
Statement II: The mean and mode of d₁, d₂, …, dₙ are x̄ - a and

Question

Let x̄, M, and σ² denote the mean, mode, and variance, respectively, of the observations x₁, x₂, …, xₙ. Also define d_i = x_i - a for i = 1, 2, …, n, where a is any constant.

Statement I: The variance of d₁, d₂, …, dₙ is σ².
Statement II: The mean and mode of d₁, d₂, …, dₙ are x̄ - a and M - a, respectively.

Accepted Answer

Both Statement I and Statement II are correct.. Since d_i = x_i - a is a shift by a constant, the variance is unchanged: Var(d) = sigma^2, so Statement I is correct. The mean and mode also shift by a, giving x-bar - a and M - a, so Statement II is also correct. Both statements are correct.

Solution

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