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Difficulty: HardMeasures of Dispersion and Position (Range, IQR, Standard Deviation, Percentiles)

Dataset XX consists of 5050 distinct real numbers with range RR, interquartile range QQ, and standard deviation ss, where R>Q>s>0R > Q > s > 0. A new dataset, Dataset YY, is formed by applying the transformation y=−3x−7y = -3x - 7 to each data point xx in Dataset XX. If RYR_Y, QYQ_Y, and sYs_Y represent the range, interquartile range, and standard deviation of Dataset YY, respectively, which of the following expressions represents the sum RY+QY+sYR_Y + Q_Y + s_Y?

  1. 3(R+Q+s)3(R + Q + s)Answer
  2. B
    3(R+Q+s)−213(R + Q + s) - 21
  3. C
    3(R+Q+s)−73(R + Q + s) - 7
  4. D
    −3(R+Q+s)−7-3(R + Q + s) - 7
  5. E
    −3(R+Q+s)-3(R + Q + s)

Answer

The sum of the dispersion metrics for Dataset YY is 3(R+Q+s)3(R + Q + s).
Under a linear transformation y=ax+by = ax + b, all measures of dispersion (range, interquartile range, standard deviation) scale by ∣a∣|a| and are unaffected by bb. With a=−3a = -3 and b=−7b = -7, each measure of dispersion is multiplied by ∣−3∣=3|-3| = 3. Therefore, RY=3RR_Y = 3R, QY=3QQ_Y = 3Q, and sY=3ss_Y = 3s, making their sum 3(R+Q+s)3(R + Q + s).

Step-by-Step Solution

1
Identify the effect of linear transformations on measures of dispersion.
For any linear transformation of data y=ax+by = ax + b, measures of dispersion (Range, Interquartile Range, Standard Deviation) scale by the absolute value of the multiplicative constant, ∣a∣|a|, and are completely unaffected by the constant addition or subtraction bb.
Measures of dispersion quantify spread and distances between data points, which shift uniformly when a constant is added but stretch by ∣a∣|a| when scaled.
2
Calculate individual dispersion measures for Dataset YY.
RY=∣−3∣R=3RR_Y = |-3| R = 3R, QY=∣−3∣Q=3QQ_Y = |-3| Q = 3Q, and sY=∣−3∣s=3ss_Y = |-3| s = 3s.
The multiplicative factor is a=−3a = -3, so ∣a∣=∣−3∣=3|a| = |-3| = 3. The constant shift b=−7b = -7 has zero effect on spread.
3
Sum the three dispersion measures for Dataset YY.
RY+QY+sY=3R+3Q+3s=3(R+Q+s)R_Y + Q_Y + s_Y = 3R + 3Q + 3s = 3(R + Q + s).
Factoring out 33 yields the simplified combined expression.

Key Concept

Linear Transformations on Dispersion Metrics
Estimated Time:2m 0s
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Topics

Measures of Dispersion and Position (Range, IQR, Standard Deviation, Percentiles)

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