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

Data set AA consists of 7 numbers that form an arithmetic progression with a non-zero common difference. Data set BB is formed by removing the smallest number and the largest number from Data set AA. If σA2\sigma_A^2 represents the population variance of Data set AA and σB2\sigma_B^2 represents the population variance of Data set BB, what is the value of σA2σB2\frac{\sigma_A^2}{\sigma_B^2}?

Answer: 2

Answer

2
By writing the 7 terms of Data set A symmetrically about their mean μ\mu with common difference dd, the sum of squared deviations is 28d228d^2, giving a population variance of σA2=28d27=4d2\sigma_A^2 = \frac{28d^2}{7} = 4d^2. Removing the minimum and maximum values leaves 5 terms centered at μ\mu with a sum of squared deviations of 10d210d^2, resulting in a population variance of σB2=10d25=2d2\sigma_B^2 = \frac{10d^2}{5} = 2d^2. The ratio of the variance of Data set A to the variance of Data set B is therefore 4d22d2=2\frac{4d^2}{2d^2} = 2.

Step-by-Step Solution

1
Represent the elements of Data set A using its mean μ\mu and common difference dd.
Let the 7 elements of Data set A be μ−3d,μ−2d,μ−d,μ,μ+d,μ+2d,μ+3d\mu - 3d, \mu - 2d, \mu - d, \mu, \mu + d, \mu + 2d, \mu + 3d, where d>0d > 0.
Symmetrical representation around the mean simplifies variance computation for arithmetic sequences.
2
Calculate the population variance σA2\sigma_A^2 of Data set A.
\sigma_A^2 = \frac{1}{7} \sum_{k=-3}^{3} (kd)^2 = \frac{d^2}{7}(9 + 4 + 1 + 0 + 1 + 4 + 9) = \frac{28d^2}{7} = 4d^2.
Variance is defined as the mean of the squared deviations from the dataset mean.
3
Determine the elements and population variance σB2\sigma_B^2 of Data set B.
Data set B contains the 5 remaining central elements, having mean μ\mu. Thus, $\sigma_B^2 = \frac{1}{5} \sum_{k=-2}^{2} (kd)^2 = \frac{d^2}{5}(4 + 1 + 0 + 1 + 4) = \frac{10d^2}{5} = 2d^2.
Removing the minimum and maximum terms leaves a symmetric 5-element dataset with unchanged mean μ\mu.
4
Compute the ratio σA2σB2\frac{\sigma_A^2}{\sigma_B^2}.
σA2σB2=4d22d2=2.\frac{\sigma_A^2}{\sigma_B^2} = \frac{4d^2}{2d^2} = 2.
The parameter d2d^2 cancels out completely, yielding a constant numerical ratio.

Key Concept

Effect of trimming extreme symmetric values on the population variance of an arithmetic sequence
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Topics

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

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