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Difficulty: HardMeasures of Dispersion

The mean of five numbers 3,5,7,x,3, 5, 7, x, and yy is 66. Given that the variance of the numbers is 88 and x<yx < y, calculate the value of yy.

Answer: 11

Answer

11
The total sum of the five numbers is 5×6=305 \times 6 = 30, giving x+y=15x + y = 15. The sum of squared deviations from the mean 66 is 5×8=405 \times 8 = 40. The known numbers 3,5,73, 5, 7 contribute (−3)2+(−1)2+12=11(-3)^2 + (-1)^2 + 1^2 = 11 to this sum, leaving (x−6)2+(y−6)2=29(x-6)^2 + (y-6)^2 = 29. Substituting y=15−xy = 15 - x yields 2x2−30x+88=0  ⟹  x2−15x+44=02x^2 - 30x + 88 = 0 \implies x^2 - 15x + 44 = 0. Factoring gives roots 44 and 1111. Since x<yx < y, we find y=11y = 11.

Step-by-Step Solution

1
Use the definition of the arithmetic mean to write a linear relationship between xx and yy.
x+y=15x + y = 15, or y=15−xy = 15 - x.
The total sum of 5 numbers with a mean of 6 is 5×6=305 \times 6 = 30. Subtracting the known numbers 3+5+7=153 + 5 + 7 = 15 leaves x+y=15x + y = 15.
2
Apply the variance formula for population data.
∑(xi−6)2=40\sum (x_i - 6)^2 = 40.
Variance is the mean of squared deviations from the mean: ∑(xi−xˉ)25=8  ⟹  ∑(xi−6)2=40\frac{\sum (x_i - \bar{x})^2}{5} = 8 \implies \sum (x_i - 6)^2 = 40.
3
Compute the sum of squared deviations for the known elements and simplify the variance equation.
(x−6)2+(y−6)2=29(x - 6)^2 + (y - 6)^2 = 29.
The squared deviations for 3,5,73, 5, 7 are (−3)2=9(-3)^2 = 9, (−1)2=1(-1)^2 = 1, and 12=11^2 = 1. Subtracting 9+1+1=119 + 1 + 1 = 11 from 4040 gives 2929.
4
Substitute y=15−xy = 15 - x into the simplified equation and solve the resulting quadratic equation.
x=4x = 4 or x=11x = 11.
Substituting y=15−xy = 15 - x gives (x−6)2+(9−x)2=29  ⟹  2x2−30x+88=0  ⟹  x2−15x+44=0  ⟹  (x−4)(x−11)=0(x - 6)^2 + (9 - x)^2 = 29 \implies 2x^2 - 30x + 88 = 0 \implies x^2 - 15x + 44 = 0 \implies (x - 4)(x - 11) = 0.
5
Select the correct pair (x,y)(x, y) using the condition x<yx < y.
x=4x = 4 and y=11y = 11.
Since x<yx < y, xx must be the smaller value (44) and yy must be the larger value (1111).

Key Concept

Calculation of variance and mean for ungrouped data containing unknown elements
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Topics

Measures of Dispersion

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