A set of n values X₁, X₂, …, Xₙ has a standard deviation of 5. The standard deviation of the n values X₁ − 7, X₂ − 7, …, Xₙ − 7 w…

Question

A set of n values X₁, X₂, …, Xₙ has a standard deviation of 5. The standard deviation of the n values X₁ − 7, X₂ − 7, …, Xₙ − 7 will be:

1. A. 0
2. B. 5 (Correct answer)
3. C. 10
4. D. 7

Correct Answer

Option B — 5

Detailed Solution & Explanation

The correct answer is 5.

Key Points

• Subtracting the same constant from every value shifts the whole distribution but does not spread it out or compress it.
• Formally, if every Xᵢ becomes Xᵢ − 7, the mean also falls by 7, so each deviation is unchanged:
  • (Xᵢ − 7) − (X̄ − 7) = Xᵢ − X̄
• Standard deviation is built entirely from those deviations, so it stays at 5.

Additional Information

• The general rules are worth holding together:
  • Adding or subtracting a constant leaves the standard deviation unchanged, but shifts the mean by that constant.
  • Multiplying by a constant k multiplies the standard deviation by |k| and the variance by k².
• So the SD of 3Xᵢ − 7 would be 15, not 5 — the shift is irrelevant but the scaling is not.
• Variance = (standard deviation)², here 25. Variance is in squared units, which is why SD is usually quoted instead.
• Measures of dispersion in order of sophistication: range, quartile deviation, mean deviation and standard deviation, the last being the most widely used because it responds to every observation.