Questions: Variance and Standard Deviation

Question 1

A dataset has mean 70 and standard deviation 10. Every value is increased by 5 points. What happens to the variance?

• A. Variance increases by 25, because 5² = 25
• B. Variance increases by 5, matching the shift
• C. Variance is unchanged
• D. Variance decreases because scores are now closer to the new mean

Answer: (C)

Adding a constant b to every value shifts the entire distribution without changing its spread. Formally, Var(X + b) = Var(X) — the b appears in every term of E[(X + b − (μ + b))²] = E[(X − μ)²] = Var(X), and cancels exactly. The mean shifts to 75, but every deviation from the new mean is identical to the old deviation from the old mean. Variance measures spread around the mean, so a pure location shift leaves it untouched. Option D contains a subtle logic error — the scores are not 'closer together'; each score moved by the same amount.

Question 2

X and Y are independent random variables with Var(X) = 9 and Var(Y) = 16. What is Var(X + Y)?

• A. 5, by analogy with the rule for standard deviations
• B. 7, averaging the two variances
• C. 25, summing the variances
• D. 12, taking the geometric mean

Answer: (C)

For independent random variables, Var(X + Y) = Var(X) + Var(Y) = 9 + 16 = 25. Note that the standard deviation of X + Y is √25 = 5, which happens to equal σ_X + σ_Y = 3 + 4 in this case — but that's a coincidence of this specific example. In general, σ(X+Y) = √(σ_X² + σ_Y²) ≠ σ_X + σ_Y. This is precisely why variance (not standard deviation) is the fundamental object: it has the clean additivity property for independent variables.

Question 3

Squaring the deviations before averaging them — rather than taking the average of absolute deviations — means that large deviations from the mean are penalized proportionally more than small ones.

• T. True
• F. False

Answer: True

Squaring is superlinear: a deviation of 10 contributes 100, while a deviation of 2 contributes only 4 — a 5-fold difference in raw deviation becomes a 25-fold difference after squaring. This means variance is especially sensitive to outliers and extreme values. This property, combined with the mathematical convenience that differentiating squared terms is clean (leading to the normal equations, least squares, etc.), is why squaring is the canonical choice for measuring spread.

Question 4

If Var(X) = 4 and Var(Y) = 9 and X, Y are independent, then the standard deviation of X + Y equals σ_X + σ_Y = 2 + 3 = 5.

• T. True
• F. False

Answer: False

Standard deviations do not add. σ(X+Y) = √(Var(X) + Var(Y)) = √(4 + 9) = √13 ≈ 3.61, not 5. The variance additivity property (Var(X+Y) = Var(X) + Var(Y)) holds for independent variables, but taking the square root of a sum is not the same as the sum of square roots: √(a² + b²) ≠ a + b except in degenerate cases. This is a crucial reason why variance, not standard deviation, is the object that appears in most statistical formulas.

Question 5

Why is variance defined using squared deviations from the mean rather than simply averaging the absolute deviations |X − μ|? What mathematical advantages does squaring provide?

Model answer: Squaring has two key advantages: (1) it eliminates the sign problem — without squaring, positive and negative deviations would cancel, and the average deviation would be zero for any symmetric distribution. (2) It produces mathematically tractable formulas — squared expressions are differentiable and lead cleanly to the additivity property Var(X+Y) = Var(X)+Var(Y) for independent variables, which absolute values do not satisfy. The mean absolute deviation is a valid measure of spread but loses these properties.

A third advantage of squaring is that it naturally emphasizes outliers more than absolute value does, which is often desirable. The tradeoff is that variance is in squared units (dollars² if X is in dollars), requiring the square root to restore interpretability as standard deviation. Mean absolute deviation (MAD) is more robust to outliers and interpretable directly, but it lacks the additivity and differentiability that make variance so useful in probability theory, estimation, and linear algebra.