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

A dataset of 2020 numerical measurements has a range of 3030 and a standard deviation of 6.46.4. If every measurement in the dataset is multiplied by −2-2 and then increased by 55 to construct a new dataset, what are the range and standard deviation of the new dataset, respectively?

  1. A
    Range is 6565 and standard deviation is 17.817.8
  2. B
    Range is −60-60 and standard deviation is −12.8-12.8
  3. Range is 6060 and standard deviation is 12.812.8Answer
  4. D
    Range is 6060 and standard deviation is 25.625.6
  5. E
    Range is 3030 and standard deviation is 6.46.4

Answer

The range of the new dataset is 6060 and the standard deviation is 12.812.8.
Under a linear transformation Y=aX+bY = aX + b, all measures of dispersion (range, interquartile range, standard deviation) are multiplied by ∣a∣|a| and are unaffected by the additive constant bb. Here, a=−2a = -2 and b=5b = 5. The absolute multiplier is ∣−2∣=2|-2| = 2. Therefore, the new range is 2×30=602 \times 30 = 60 and the new standard deviation is 2×6.4=12.82 \times 6.4 = 12.8.

Step-by-Step Solution

1
Analyze the general transformation formula for measures of dispersion under linear transformations of the form Y=aX+bY = aX + b.
For any constant multiplier aa and constant shift bb, the new range is ∣a∣×RangeX|a| \times \text{Range}_X and the new standard deviation is ∣a∣×σX|a| \times \sigma_X.
Measures of dispersion reflect spread; multiplying each data value by aa scales the spread by ∣a∣|a|, while adding a constant shift bb translates all values equally without expanding or contracting their relative distance.
2
Calculate the new range using a=−2a = -2 and b=5b = 5.
\text{New Range} = |-2| \times 30 = 2 \times 30 = 60.
The range scales by the absolute value of the multiplier ∣−2∣=2|-2| = 2, while the additive constant 55 has no impact on the spread.
3
Calculate the new standard deviation using a=−2a = -2 and b=5b = 5.
\text{New Standard Deviation} = |-2| \times 6.4 = 2 \times 6.4 = 12.8.
The standard deviation scales by ∣−2∣=2|-2| = 2, and the shift of +5+5 does not change the dispersion around the mean.

Key Concept

Effect of Linear Transformations on Measures of Dispersion
Estimated Time:1m 30s
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Topics

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

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