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Algebra II : Statistical Concepts

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Example Question #1 : Deviation Concepts

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

(37,42,39,33,x)

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

Possible Answers:

28.8

31.6

35.4

24.7

19.9

Correct answer:

28.8

Explanation:

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, x_i 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:

(37-40)^2=9

(42-40)^2=4

(39-40)^2=1

(33-40)^2=49

(49-40)^2=81

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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Example Question #4 : Variance

What is the variance for the values , , and ?

Possible Answers:

$\frac{2}{9}$

$\frac{2}{3}$

$\frac{4}{5}$

$\frac{3}{2}$

Correct answer:

$\frac{2}{3}$

Explanation:

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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Example Question #5 : Variance

Determine the variance if the standard deviation is 8.6 .

Possible Answers:

2.93

73.70

2.15

73.96

25.22

Correct answer:

73.96

Explanation:

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: 73.96

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Example Question #6 : Variance

Evaluate the variance if the standard deviation is 19.5 .

Possible Answers:

7414.875

380.25

1.29

4.416

2.692

Correct answer:

380.25

Explanation:

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: 380.25

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Example Question #7 : Variance

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

Possible Answers:

Correct answer:

Explanation:

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

9^2=81

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Example Question #8 : Variance

Determine the population variance given the following data set: [5,7,27]

Possible Answers:

$\frac{296}{3}$

148

280

Correct answer:

$\frac{296}{3}$

Explanation:

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.

(5-13)^2+(7-13)^2+(27-13)^2 = 64+36+196 = 296

Divide this value by three.

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

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Example Question #1 : Standard Deviation

Calculate the standard deviation of the following set of values.

$\left \{ 2,3,3,4,5,7\right \}$

Possible Answers:

$\frac{8}{3}$

$\frac{4}{3}$

Correct answer:

Explanation:

To get the standard deviation, we need to calculate the variance, which is the average of the squared differences from the mean, so we will start by getting the mean.

$\frac{(2+3+3+4+5+7)}{6}=4$

Then, subtract the mean from each value...

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

$(3-4)^{2}=(-1)^{2}=1$

$(4-4)^{2}=(0)^{2}=0$

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

$(7-4)^{2}=(3)^{2}=9$

...and take the mean of these resulting values, which is equal to the variance.

$\frac{(4+1+1+0+1+9)}{6}=2\frac{2}{3}=\frac{8}{3}$

The square root of this value is the standard deviation. The answer is presented as $\sqrt{\frac{8}{3}}$ ,

but you may also calculate it and find it equal to about 1.63

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Example Question #2 : Standard Deviation

Determine the standard deviation for the following data set:

12, 15, 30, 5, 27, 19

Possible Answers:

2.00

10.56

8.56

9.38

10.18

Correct answer:

8.56

Explanation:

Formula for the standard deviation:

$\sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}$

1. Find the mean

$\frac{12+15+30+5+27+19}{6}=\frac{108}{6}=18$

2. Subtract the mean from each number in the data set

$(12-18)^{2}=(-6)^{2}=36$

$(15-18)^{2}=(-3)^{2}=9$

$(30-18)^{2}=(12)^{2}=144$

$(5-18)^{2}=(-13)^{2}=169$

$(27-18)^{2}=(9)^{2}=81$

$(19-18)^{2}=(1)^{2}=1$

3. Sum up the square of the differences and divide by n

$\frac{36+9+144+169+81+1}{6}=\frac{440}{6}=\frac{220}{3}$

4. Take the square root of the variance

$\sqrt{\frac{220}{3}}= 8.56$

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Example Question #1 : Standard Deviation

In the population of high school boys, the variance in height, measured in inches, was found to be 16. Assuming that the height data is normally distributed, 95% of high school boys should have a height within how many inches of the mean?

Possible Answers:

$8\ \textup{inches}$

$16\ \textup{inches}$

$\textup{Cannot be determined from the information given}$

$4\ \textup{inches}$

$2\ \textup{inches}$

Correct answer:

$8\ \textup{inches}$

Explanation:

The 68-95-99.7 rule states that nearly all values lie within 3 standard deviations of the mean in a normal distribution. In this case the question asks for 95% so we want to know what 2 standard deviations from the mean is.

We are given the variance, so to find the standard deviation, take the square root.

$\sqrt{16}=4$

So two standard devations is 8 inches. 95% of heights should be within 8 inches of the mean.

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Example Question #4 : Standard Deviation

At the end of the fall semester, a math class of ninth graders had the following grades: 85, 75, 97, 83, 62, 75, 88, 84, 92, and 89.

What is the standard deviation of this class?

Possible Answers:

10.07

101.33

Correct answer:

10.07

Explanation:

The standard deviation of a set of numbers is how much the numbers deviate from the mean. More formally, the standard deviation is

$\sigma =\sqrt{\frac{\sum{(x-M)^2} }{N-1}}$

where is a number in the series, is the mean, and is the number of data points. So, to calculate the standard deviation, we must first calculate the mean. The mean of this data set is

$M = \frac{85+75+97+83+62+75+88+84+92+89}{10}=83$

Now that we know the mean, we can start calculating the standard deviation. We first need to find the sum of each data point minus the average squared.

(85-83)^2+(75-83)^2+(97-83)^2+(62-83)^2+(75-83)^2+(88-83)^2+(84-83)^2+(92-83)^2+(89-83)^2

Calculating that, we get that the variance from the mean is 912 . Plugging that into our equation for standard deviation, with being ten data points, we get

$\sigma = \sqrt{\frac{912}{9}}=10.07$

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Algebra II : Statistical Concepts

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Deviation Concepts

Example Question #4 : Variance

Example Question #5 : Variance

Example Question #6 : Variance

Example Question #7 : Variance

Example Question #8 : Variance

Example Question #1 : Standard Deviation

Example Question #2 : Standard Deviation

Example Question #1 : Standard Deviation

Example Question #4 : Standard Deviation

All Algebra II Resources

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