Descriptive Statistics
When every observation in a dataset is first multiplied by three and then increased by five, how does the variance of the resulting transformed dataset relate to the original variance?
Select the correct option:
Solution
It becomes nine times the original variance
Variance measures spread about the mean and obeys the transformation rule Var(aX + b) = a² Var(X), because adding a constant shifts all data equally without changing dispersion while scaling stretches deviations. This linear-transformation property is frequently tested at JEE Advanced. Here the transformation is Y = 3X + 5, so a = 3 and b = 5. The additive constant 5 merely relocates the dataset and leaves spread untouched, while the multiplier 3 scales every deviation by 3 and therefore scales squared deviations by 3² = 9. Thus Var(Y) = 9 Var(X). The option fourteen times wrongly adds the constant's effect as if 3² + 5 mattered. The option three times forgets to square the scaling factor. The option that variance is unchanged ignores the multiplicative scaling entirely and treats the whole map as a shift. The governing identity is Var(aX + b) = a² Var(X), derived from the definition of variance as expected squared deviation. Plausibility check: since variance is non-negative and scaling by 3 must enlarge spread, a factor of 9 > 1 is sensible, and the additive 5 rightly drops out as it cannot alter relative scatter.
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About This Question
- Subject
- mathematics
- Chapter
- statistics and probability
- Topic
- descriptive statistics
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
It becomes nine times the original variance
Variance measures spread about the mean and obeys the transformation rule Var(aX + b) = a² Var(X), because adding a constant shifts all data equally without changing dispersion while scaling stretches deviations. This linear-transformation property is frequently tested at JEE Advanced. Here the transformation is Y = 3X + 5, so a = 3 and b = 5. The additive constant 5 merely relocates the dataset and leaves spread untouched, while the multiplier 3 scales every deviation by 3 and therefore scales squared deviations by 3² = 9. Thus Var(Y) = 9 Var(X). The option fourteen times wrongly adds the constant's effect as if 3² + 5 mattered. The option three times forgets to square the scaling factor. The option that variance is unchanged ignores the multiplicative scaling entirely and treats the whole map as a shift. The governing identity is Var(aX + b) = a² Var(X), derived from the definition of variance as expected squared deviation. Plausibility check: since variance is non-negative and scaling by 3 must enlarge spread, a factor of 9 > 1 is sensible, and the additive 5 rightly drops out as it cannot alter relative scatter.
This medium difficulty mathematics question is from the chapter statistics and probability, covering the topic of descriptive statistics. It appeared in the 2025 exam.
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