Variance - Algebra II

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Question

Calculate the variance of the following set of values.

$\left { 3,3,4,5,8,7\right }$

Answer

Variance is the average of the squared differences from the mean. So start by working out the mean of the set of values.

$\frac{(3+3+4+5+8+7)}{6}=5$

Then, for each number, subtract the mean and square the result.

$(3-5)^{2}=(-2)^{2}=4$

$(4-5)^{2}=(-1)^{2}=1$

$(5-5)^{2}=(0)^{2}=0$

$(8-5)^{2}=(3)^{2}=9$

$(7-5)^{2}=(2)^{2}=4$

The average of the results is the variance.

$\frac{(4+4+1+0+9+4)}{6}=3.67$

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Question

What is the variance of the following set of numbers?

Answer

Remember that the variance of a set of numbers is equal to its standard deviation squared. So if the formula for standard deviation is known to be:

$\sigma =\sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu )^2}$

Then the variance is simply $\sigma^2$ , given by the following equation:

$\sigma^2 =\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu )^2$

Where $\sigma^2$ is the variance, N is the number of values in the set, is the number currently being evaluated in the summation, and $\mu$ is the mean of the set. So all we must do to find the variance is determine our mean, and then sum the values given by subtracting the mean from each number in the set and squaring it, dividing the final sum by N to give us our variance. We start by determining the mean of the set:

$\mu=\frac{\sum_{i=1}^{N}x_i}{N}=\frac{17+13+21+14+10}{5}=15$

We then find each value that will be added together by subtracting our mean from each of the numbers in our set and then squaring the result:

Looking at our formula for variance, all we must do now is add these values together and divide by N:

$\sigma^2 =\frac{1}{5}(4+4+36+1+25)=14$

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Question

A set of five numbers is given as follows, where one of the numbers, , is unknown:

If the mean of the set is , what is the variance?

Answer

Before we can determine the variance of the set, we must first determine the value of x, our unknown. The problem tells us the mean of the set, and we know each number in the set except for x, so we can solve for x as follows:

$\mu =\frac{\sum_{i=1}^{N}x_i}{N}=\frac{37+42+39+33+x}{5}=40$

$151+x=200\rightarrow x=49$

Now that we know all the values of the set, the mean, and the number of values, we can solve for the variance using the following equation:

$\sigma ^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu )^2$

Where $\sigma ^2$ is the variance, N is the number of values in the set, is the value currently being evaluated in the set, and $\mu$ is the mean of the set. We start by determing the value of each term in the summation part of the formula:

Now that we now each value of the summation, all we have left to do is add them all together and divide the final result by N:

$\sigma ^2=\frac{1}{5}(9+4+1+49+81)=\frac{144}{5}=28.8$

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Question

What is the variance for the values , , and ?

Answer

Write the formula for variance.

$\sigma^2 = \frac{\sum (X-\mu)^2 }{N}$

The value of represents each number in the data set.

The value of is 3, since there are 3 numbers in the data set.

Find the mean, $\mu$ .

$\mu= \frac{2+3+4}{3} = \frac{9}{3}=3$

Rewrite the variance formula:

$\sigma^2 = \frac{(2-3)^2 +(3-3)^2+(4-3)^2}{3} = \frac{1+0+1}{3}=\frac{2}{3}$

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Question

The standard deviation of a distribution is 9. What is the variance?

Answer

The variance of a distribution is equal to the standard deviation squared.

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Question

Determine the variance if the standard deviation is .

Answer

Write the formula relationship between standard deviation and variance.

The standard deviation is the square root of variance.

$\sigma = \sqrt{\textup{Variance}}$

Substitute the standard deviation.

$8.6 = \sqrt{\textup{Variance}}$

Square both sides.

$\textup{Variance} =( 8.6)^2 = 73.96$

The answer is:

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Question

Evaluate the variance if the standard deviation is .

Answer

Write the formula for the variance.

$\textup{Variance} =\sigma^2$

$\textup{Standard Deviation}= \sigma$

The standard deviation is the square root of variance.

$\sigma = \sqrt{\textup{Variance}}$

Substitute the value of standard deviation in the formula.

$19.5 = \sqrt{\textup{Variance}}$

Square both sides.

$\textup{Variance }= 380.25$

The answer is:

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Question

Determine the population variance given the following data set:

Answer

To determine the variance, we will need to find the squared differences of the mean, and take the average.

Find the mean.

$\frac{5+7+27}{3} = \frac{39}{3}=13$

Subtract each number from the mean and square the quantities.

Divide this value by three.

The answer is: $\frac{296}{3}$

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