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Algebra II : Standard Deviation

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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:

$\frac{4}{3}$

$\frac{8}{3}$

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:

9.38

8.56

2.00

10.18

10.56

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 #3 : Standard Deviation

The formula for standard deviation is the following:

$\sigma = \sqrt{\frac{\sum (x - x_{avg})^2}{n-1}}$ .

Where,

$\sigma = \text{ standard deviation}$ ,

$x_{avg}=\text{ mean of the data set}$ ,

$n=\text{ number of data points}$ .

Five students took a test and recieved the following grades: , , , 100 , . What is the standard deviation of the test grades to the nearest decimal place?

Possible Answers:

2.4

8.3

6.6

7.9

None of the other answers.

Correct answer:

7.9

Explanation:

The first step in solving for standrd deviation is to find the mean of the data set.

For this problem:

$x_{avg} = \frac{95+90+80+100+85}{5} = \frac{450}{5} = 90$ .

Now we can evaluate the summation:

$\sum\left ( x - x_{avg}\right )^2 = (95-90)^2 + (90-90)^2 + (80-90)^2 + (100-90)^2 + (85-90)^2$

$\sum\left ( x - x_{avg}\right )^2 = 5^2 + 0^2 + (-10)^2 + 10^2 + (-5)^2$

$\sum\left ( x - x_{avg}\right )^2 = 25 + 0 + 100 + 100 + 25 = 250$

Now we can rewrite the standard deviation expression:

$\sigma = \sqrt{\frac{250}{n-1}}$

There are 5 data points, so n = 5

$\sigma = \sqrt{\frac{250}{4}} = \sqrt{62.5} = 7.9$

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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:

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

$4\ \textup{inches}$

$16\ \textup{inches}$

$8\ \textup{inches}$

$2\ \textup{inches}$

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 #5 : Standard Deviation

Find the standard deviation of the following set of numbers:

(7,4,11,13,10)

Possible Answers:

$3\sqrt{2}$

$\sqrt{14}$

$\sqrt{10}$

Correct answer:

$\sqrt{10}$

Explanation:

To begin, we must remember the formula for standard deviation:

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

where $\sigma$ is the standard deviation, N is the number of values in our set, x_i is the value we're currently evaluating in the summation, and $\mu$ is the mean of our set of numbers. All the summation part of the equation means is that we subtract our mean from each number in the set, square that value, and then add all of those values for each number together. So before we find the values that will be added together, we must first find our mean for the set of numbers:

$\mu =\frac{\sum_{i=1}^{N}x_i}{N}=\frac{7+4+11+13+10}{5}=9$

Now we can determine the value for our summation for each number in the set:

(7-9)^2=4

(4-9)^2=25

(11-9)^2=4

(13-9)^2=16

(10-9)^2=1

Looking at our equation for standard variation, now all we must do is sum all of the values above, divide by N, and take the square root:

$\sigma =\sqrt{\frac{1}{5}(4+25+4+16+1)}=\sqrt{10}$

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Example Question #1 : 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:

101.33

10.07

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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Example Question #1 : How To Find Standard Deviation

Mr. Bell gave out a science test last week to six honors students. The scores were 88, 94, 80, 79, 74, and 83. What is the standard deviation of the scores? (Round to the nearest tenth.)

Possible Answers:

81.5

6.5

Correct answer:

6.5

Explanation:

First, find the mean of the six numbers by adding them all together, and dividing them by six.

88 + 94 + 80 + 79 + 74 + 83 = 498

498/6 = 83

Next, find the variance by subtracting the mean from each of the given numbers and then squaring the answers.

88 – 83 = 5

5² = 25

94 – 83 = 11

11² = 121

80 – 83 = –3

–3² = 9

79 – 83 = –4

–4² = 16

74 – 83 = –9

–9² = 81

83 – 83 = 0

0² = 0

Find the average of the squared answers by adding up all of the squared answers and dividing by six.

25 + 121 + 9 +16 +81 + 0 = 252

252/6 = 42

42 is the variance.

To find the standard deviation, take the square root of the variance.

The square root of 42 is 6.481.

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Example Question #1 : How To Find Standard Deviation

On his five tests for the semester, Andrew earned the following scores: 83, 75, 90, 92, and 85. What is the standard deviation of Andrew's scores? Round your answer to the nearest hundredth.

Possible Answers:

5.97

5.95

5.93

5.98

Correct answer:

5.97

Explanation:

The following is the formula for standard deviation:

$s = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \overline{x})^2}$

Here is a breakdown of what that formula is telling you to do:

1. Solve for the mean (average) of the five test scores
2. Subtract that mean from each of the five original test scores. Square each of the differences.
3. Find the mean (average) of each of these differences you found in Step 2
4. Take the square root of this final mean from #3. This is the standard deviation

Here are those steps:

1. Find the mean of the test scores:

$(83+75+90+92+85) \div 5 = 425 \div 5 = 85$

2. Subtract the mean from each of the test scores, then square the differences:

83-85 = (-2)^2 = 4

75-85 = (-10)^2 = 100

90-85 = (5)^2 = 25

92-85 = (7)^2 = 49

85-85 = (0)^2 = 0

3. Find the mean of the squared values from Step 2:

$(4+100+25+49+0) \div5 = 178\div 5= 35.6$

4. Take the square root of your answer from Step 3:

$\sqrt{35.6} = 5.9667 = 5.97$

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Example Question #1 : How To Find Standard Deviation

In her last six basketball games, Jane scored 15, 17, 12, 15, 18, and 22 points per game. What is the standard deviation of these score totals? Round your answer to the nearest tenth.

Possible Answers:

3.2

3.1

3.3

3.0

Correct answer:

3.1

Explanation:

The following is the formula for standard deviation:

$s = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \overline{x})^2}$

Here is a breakdown of what that formula is telling you to do:

Here are those steps:

1. Find the mean of her score totals:

$(15+17+12+15+18+22) \div 6 = 99 \div 6 =16.5$

2. Subtract the mean from each of the test scores, then square the differences:

15-16.5 = (-1.5)^2 = 2.25

17-16.5 = (0.5)^2 = 0.25

12-16.5 = (-4.5)^2 = 20.25

15-16.5 = (-1.5)^2 = 2.25

18-16.5 = (1.5)^2 = 2.25

22-16.5 = (5.5)^2 = 30.25

3. Find the mean of the squared values from Step 2:

$(2.25+0.25+20.25+2.25+2.25+30.25) \div6 =$

$57.5\div 6= 9.58\bar{3}$

4. Take the square root of your answer from Step 3:

$\sqrt{9.58\bar{3}} = 3.096 = 3.1$

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Example Question #7 : Standard Deviation

What is the standard deviation of 5, 7, 9 ?

Possible Answers:

$\frac{2\sqrt{3}}{3}$

$\frac{4}{3}$

Correct answer:

Explanation:

Standard deviation is $\sqrt{\frac{\Sigma (x-\bar{x})^2}{n-1}}$ where represents the data point in the set, $\bar{x}$ is the mean of the data set and is number of points in the set.

The mean is the sum of the data set divided by the number of data points in the set.

$\frac{5+7+9}{3}=7$

Plugging in the values:

$\sqrt{\frac{(5-7)^2+(7-7)^2+(9-{7})^2}{3-1}}=2$

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Algebra II : Standard Deviation

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

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Example Question #1 : Standard Deviation

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