Descriptive Statistics

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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?

RankGuru · Accepted Answer

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.

Descriptive Statistics

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