Practice Variance in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The variance is the average of the squared deviations from the mean: Οƒ2=1nβˆ‘(xiβˆ’xΛ‰)2\sigma^2 = \frac{1}{n}\sum (x_i - \bar{x})^2. It is the square of the standard deviation.

Another spread measureβ€”variance =SD2= \text{SD}^2. Same idea, different scale.

Showing a random 20 of 50 problems.

Example 1

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Use the computational formula Οƒ2=βˆ‘xi2nβˆ’ΞΌ2\sigma^2 = \frac{\sum x_i^2}{n} - \mu^2 for {1,2,3,4,5}\{1, 2, 3, 4, 5\}.

Example 2

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A set has variance 7. Add 5 to every value. New variance?

Example 3

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Find the sample variance of {2,4,6,8}\{2, 4, 6, 8\}.

Example 4

hard
The variance of {a,a+d,a+2d}\{a, a+d, a+2d\} (population) is 2d23\frac{2d^2}{3}. Verify for a=1a=1, d=2d=2.

Example 5

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A data set has population variance Οƒ2=16\sigma^2 = 16 measured in seconds2^2. What is the standard deviation, in seconds?

Example 6

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Find the population variance of {βˆ’2,0,2}\{-2, 0, 2\}.

Example 7

hard
Fair die: let XX be the value rolled. Compute Var⁑(X)\operatorname{Var}(X).

Example 8

challenge
A binomial XX has n=12n=12, p=0.25p=0.25. Find Var(X)\text{Var}(X) and explain via additivity.

Example 9

challenge
Show the variance of {a,βˆ’a}\{a, -a\} (population) equals a2a^2.

Example 10

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Compute the sample variance s2=1nβˆ’1βˆ‘(xiβˆ’xΛ‰)2s^2 = \frac{1}{n-1}\sum(x_i - \bar{x})^2 for {2,4,6,8}\{2, 4, 6, 8\}.

Example 11

hard
A sample variance is s2=12s^2 = 12 for n=9n = 9. What is βˆ‘(xiβˆ’xΛ‰)2\sum (x_i - \bar{x})^2?

Example 12

challenge
A data set of nn values has mean ΞΌ\mu and variance Οƒ2\sigma^2. A new value equal to ΞΌ\mu is added. What is the new population variance?

Example 13

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Find the population variance of {3,6,9}\{3, 6, 9\}.

Example 14

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If variance is 36, what is the standard deviation?

Example 15

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Three numbers have mean 55 and population variance 66. Two of them are 22 and 55. Find the third.

Example 16

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The mean of {1,3}\{1, 3\} is 2. Find the population variance.

Example 17

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For XX taking values 1,2,31, 2, 3 each with probability 1/31/3, compute Var⁑(X)=E[X2]βˆ’(E[X])2\operatorname{Var}(X) = E[X^2] - (E[X])^2.

Example 18

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Population vs. sample variance: which divides by nβˆ’1n-1?

Example 19

hard
A data set of n=10n=10 has βˆ‘xi=50\sum x_i = 50 and βˆ‘xi2=300\sum x_i^2 = 300. Find the population variance.

Example 20

hard
XX and YY are independent with Var⁑(X)=4\operatorname{Var}(X) = 4 and Var⁑(Y)=9\operatorname{Var}(Y) = 9. Find Var⁑(Xβˆ’Y)\operatorname{Var}(X - Y).
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