1 mark, −0.333StatisticsDescriptive statistics and dispersionThe arithmetic mean and its propertiesModerateabout 1 min

IIT JAM 2021 Question 3: Descriptive statistics and dispersion

Let $\{x_1, x_2, \cdots, x_n\}$ be a sample with mean $\bar x$ , and let $k$ be a real number other than $\bar x$ . Define $S_1 = \sum_{i=1}^n (x_i - \bar x)^2$ and $S_2 = \sum_{i=1}^n (x_i - k)^2$ . Then,

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Answer and worked solution

Answer: (C)

The sum of squared deviations is smallest when taken about the mean. Algebraically $S_2 = S_1 + n(\bar x - k)^2$ .

Since $k \neq \bar x$ , the added term $n(\bar x - k)^2$ is strictly positive, so $S_2 > S_1$ , i.e. $S_1 < S_2$ . The answer is (C), regardless of whether $k$ is above or below the mean.

The trap: (A)

S_1 > S_2

The direction does not depend on whether $k$ exceeds the mean; the squared term is positive either way, so the ordering is unconditional.

Plain-text version

Let {x₁, x₂, ⋯, xₙ} be a sample with mean x̄, and let k be a real number other than x̄. Define S₁ = Σᵢ₌₁ⁿ (xᵢ - x̄)² and S₂ = Σᵢ₌₁ⁿ (xᵢ - k)². Then, (A) S₁ > S₂ (B) S₁ > S₂ only if x̄ < k (C) S₁ < S₂ (D) S₁ > S₂ only if x̄ > k

Also written as: Let {x_1, x_2,..., x_n} be a sample with mean x_bar, and let k be a real number other than x_bar. Define S_1 = sum_(i=1)^n (x_i - x_bar)^2 and S_2 = sum_(i=1)^n (x_i - k)^2. Then,

Descriptive statistics and dispersion has come up 117 times in 6 exams since 2004.

What this question tests

This question is on The arithmetic mean and its properties, part of Descriptive statistics and dispersion in Statistics.

Computing the mean from raw and grouped data, and what changes when data are shifted, scaled or corrected.

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