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Show how a linear transformation of a random variable alters its median, interquartile range, mean, standard deviation, variance, skewness and kurtosis.

FRM® Part I · Quantitative Analysis · Random Variables · FRM1.QA.2.f

The idea

Location and scale measures transform in a simple way under a linear transformation, where a is the shift and b is the multiplier: Y = a + bX. The mean becomes a + b × E(X). The median becomes a + b × median(X) for either sign of b, but other quantiles change places when b is negative, so the order of quantiles reverses. The standard deviation becomes |b| × σ and the variance becomes b² × Var(X), so the shift a has no effect on dispersion. The interquartile range becomes |b| times the original. Skewness and kurtosis are standardised moments, so they ignore both a and the size of b. Kurtosis is unchanged. Skewness is unchanged when b is positive but changes sign when b is negative, because the distribution is mirrored. The usual errors are carrying the sign of b into a standard deviation or IQR, and adding a to the variance.

Formulas

Variance after a linear transformation

Var(a + bX) = b² × Var(X)

b
the multiplier
Var(X)
the variance of X
a
the shift

The shift a does not appear on the right because it has no effect on dispersion. The standard deviation becomes |b| × σ, never a negative number.

A practice question

From the question bank, with the reason each wrong option is wrong.

Y = a + bX where b is a non-zero constant of either sign. Which statistic of Y always equals the corresponding statistic of X?

  1. Kurtosis - correct
  2. SkewnessWhy not: Skewness is unchanged only for b > 0; with b < 0 the distribution is mirrored and the sign flips.
  3. Standard deviationWhy not: The standard deviation is scaled by |b|, so it changes unless |b| = 1.
  4. MedianWhy not: The median is shifted by a and scaled by b, so it changes in general.

Kurtosis is a standardised even-order moment, so it does not respond to shifts, rescaling or reflection. Skewness flips sign for negative b.

A harder one

X has a mean of 4 and a standard deviation of 3. What is the standard deviation of Y = 10 - 2X?

  1. -6Why not: Carries the sign of -2 into the standard deviation, which cannot be negative: it scales by |b|.
  2. 4Why not: Applies the whole transformation to the standard deviation: 10 - 2 x 3.
  3. 6 - correct
  4. 36Why not: Reports the variance, b² x 9, instead of the standard deviation.

SD(a + bX) = |b| SD(X) = 2 x 3 = 6. The intercept 10 shifts the mean (to 2) but not the dispersion.

The hardest tier

X has a median of 1, a first quartile of -0.5 and a third quartile of 2.5. What are the median and interquartile range of Y = 4 - 2X?

  1. Median 2, IQR 6 - correct
  2. Median 2, IQR -6Why not: Carries the negative sign of the multiplier into the IQR; a range is scaled by |b| and stays positive.
  3. Median 2, IQR 3Why not: Leaves the IQR unscaled, forgetting to multiply by |b| = 2.
  4. Median -2, IQR 6Why not: Drops the intercept 4 when transforming the median.

Median(Y) = 4 - 2(1) = 2. With b < 0 the quartiles swap: Q1(Y) = 4 - 2(2.5) = -1 and Q3(Y) = 4 - 2(-0.5) = 5, so IQR = 6 = |b| x 3.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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