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

Dataset XX consists of 2020 distinct real numbers with standard deviation sX>0s_X > 0 and interquartile range IQRX>0IQR_X > 0. Dataset YY is created by multiplying each number in Dataset XX by −3-3 and then adding 77. Dataset ZZ is created by adding a single 21st value, equal to the arithmetic mean of Dataset XX, to Dataset XX. Which of the following statements must be true regarding the measures of dispersion of these datasets?

  1. The standard deviation of Dataset YY is 3sX3s_X, and the standard deviation of Dataset ZZ is strictly less than sXs_X.Answer
  2. B
    The standard deviation of Dataset YY is 3sX3s_X, and the standard deviation of Dataset ZZ is equal to sXs_X.
  3. C
    The standard deviation of Dataset YY is −3sX+7-3s_X + 7, and the standard deviation of Dataset ZZ is strictly less than sXs_X.
  4. D
    The standard deviation of Dataset YY is 3sX+73s_X + 7, and the standard deviation of Dataset ZZ is strictly greater than sXs_X.
  5. E
    The standard deviation of Dataset YY is −3sX-3s_X, and the standard deviation of Dataset ZZ is equal to sXs_X.

Answer

The standard deviation of Dataset YY is 3sX3s_X, and the standard deviation of Dataset ZZ is strictly less than sXs_X.
Linear transformation scales standard deviation by the absolute value of the multiplier, making sY=∣−3∣sX=3sXs_Y = |-3|s_X = 3s_X. Constant additions do not affect spread. Adding a data point equal to the mean does not change the total sum of squared deviations, but increases the sample size nn, thereby reducing the average squared deviation and yielding sZ<sXs_Z < s_X.

Step-by-Step Solution

1
Analyze the linear transformation on Dataset X to form Dataset Y.
For any linear transformation of data yi=axi+by_i = a x_i + b, the standard deviation transforms according to sY=∣a∣sXs_Y = |a| s_X. Here a=−3a = -3 and b=7b = 7, so sY=∣−3∣sX=3sXs_Y = |-3| s_X = 3s_X.
Adding a constant shift bb shifts all points equally without altering their spread relative to the mean, whereas multiplying by aa scales distance by ∣a∣|a|.
2
Analyze the effect of inserting the mean into Dataset X to form Dataset Z.
Adding μX\mu_X to Dataset XX yields a new dataset with mean μZ=μX\mu_Z = \mu_X. The sum of squared deviations ∑(zi−μZ)2=∑(xi−μX)2+(μX−μX)2=∑(xi−μX)2\sum (z_i - \mu_Z)^2 = \sum (x_i - \mu_X)^2 + (\mu_X - \mu_X)^2 = \sum (x_i - \mu_X)^2 remains identical.
The extra point contributes 00 squared distance from the mean.
3
Compare the standard deviation sZs_Z to sXs_X.
Because the sum of squared deviations remains unchanged while the total count of elements increases from 2020 to 2121, the average squared deviation (variance) decreases. Hence, sZ<sXs_Z < s_X.
Dividing the same numerator by a larger denominator (2121 or 2020 depending on sample/population formula) results in a smaller variance and smaller standard deviation.

Key Concept

Effect of linear transformations and mean-value insertion on standard deviation

Alternative Method

Consider extreme simple cases: if Dataset X has two values {−1,1}\{ -1, 1 \}, mean is 0, sX=1s_X = 1. Dataset Y becomes {10,4}\{ 10, 4 \}, mean is 7, sY=3=3sXs_Y = 3 = 3s_X. Adding 0 to X gives {−1,0,1}\{ -1, 0, 1 \}, mean remains 0, variance becomes 2/32/3, so sZ=2/3<1s_Z = \sqrt{2/3} < 1.
Estimated Time:2m 0s
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Topics

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

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