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Algebra 1 : How to find standard deviation

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Example Question #1 : How To Find Standard Deviation

Find the standard deviation of the following set of numbers: 10, 15, 19, 35, 27, 44. Round your answer to the nearest tenth.

Possible Answers:

11.7

137.7

Correct answer:

11.7

Explanation:

To find the standard deviation of a set of numbers, first find the mean (average) of the set of numbers:

$\frac{10+15+19+35+27+44}{6}=25$

Second, for each number in the set, subtract the mean and square the result:

(10-25)^2+(15-25)^2+(19-25)^2+(35-25)^2 +(27-25)^2+(44-25)^2=826

Then add all of the squares together and find the mean (average) of the squares, like this:

$\frac{826}{6}=137.67$

Finally, take the square root of the second mean:

$\sqrt{137.7}=11.7$ .

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Example Question #1 : How To Find Standard Deviation

Find the standard deviation of the following set of numbers: 10, 15, 7, 13. Round your answer to the nearest hundredth.

Possible Answers:

1.56

3.03

11.25

9.19

3.06

Correct answer:

3.03

Explanation:

To find the standard deviation of a set of numbers, first find the mean (average) of the set of numbers:

$\frac{110+15+7+13}{4}=11.25$

Second, for each number in the set, subtract the mean and square the result:

(10-11.25)^2+(15-11.25)^2+(7-11.25)^2+(13-11.25)^2=36.74

Then add all of the squares together and find the mean (average) of the squares, like this:

$\frac{36.74}{4}=9.19$

Finally, take the square root of the second mean: $\sqrt{9.19}=3.03$ .

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Example Question #3 : How To Find Standard Deviation

Kyle scored the following on his mathematics tests: 60, 75, 80, 75, 70 . What is the standard deviation of his test scores? Round your answer to the nearest tenth.

Possible Answers:

212

42.4

6.8

Correct answer:

6.8

Explanation:

To find the standard deviation of a set of numbers, first find the mean (average) of the set of numbers:

$\frac{60+75+80+75+70}{5}=72$

Second, for each number in the set, subtract the mean and square the result:

(60-72)^2+(75-72)^2+(80-72)^2+(75-72)^2 +(70-72)^2=230 .

Then add all of the squares together and find the mean (average) of the squares, like this:

$\frac{230}{5}=46$

Finally, take the square root of the second mean:

$\sqrt{46}=6.8$ .

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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

Give the interquartile range of a data set with the following characteristics.

Mean: 72.1

Median: 70

Standard deviation: 4.6

Possible Answers:

18.4

It cannot be determined from the information given.

4.2

8.4

9.2

Correct answer:

It cannot be determined from the information given.

Explanation:

The interquartile range is the difference between the first and third quartiles. The two pieces of information needed to determine interquartile range, the first and third quartiles, are missing; therefore, it is impossible to answer the question without more information.

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Example Question #6 : 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:

5.93

5.95

5.98

5.97

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 #7 : 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:

3.1

3.0

3.3

3.2

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 #1 : How To Find Standard Deviation

The variance is . What is the standard deviation?

Possible Answers:

7.682

3136

9.985

7.483

Correct answer:

7.483

Explanation:

Write the formula for standard deviation in terms of variance.

$\sigma=\sqrt{V}$

Substitute the variance.

$\sigma=\sqrt{56}= 7.483$

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Example Question #1 : How To Find Standard Deviation

In meteorology, the standard deviation of wind speed can be used to predict the likelihood of fog forming under given temperature conditions.

What is the standard deviation of the following wind speed measurements in kilometers per hour (kph), taken 1 hour apart at the same site for 10 hours? Round to the nearest tenth.

6, 4, 13, 11, 8, 6, 4, 5, 7, 7

Possible Answers:

50.41

85.3

3.1

23.2

2.9

Correct answer:

2.9

Explanation:

The first step in calculating standard deviation, or $\sigma$ , is to calculate the mean for your sample, or $\mu$ . Remember, to calculate mean, sum your data values and divide by the count, or number of values you have.

$\mu= \frac{x_{1} +x_{2} + ...x_{n}}{n} = \frac{6+4+13+11+8+6+4+5+7+7}{10} = 7.1$

Next, we must find the difference between each recorded value and the mean. At the same time, we will square these differences, so it does not matter whether you subtract the mean from the value or vice versa.

We use $\sum_{i=1}^{N} (x_{i}-\mu)^{2}$ to represent this, but all it really means is that you square the difference between each value $x_{i}$ , where is the position of the value you're working with, and the mean, $\mu$ . Then we sum all those differences up (the part that goes $\sum_{i=1}^{N}$ , where is your count. i=1 just refers to the fact that you start at the first value, so you include them all.)

It's probably easier to do than to think about at first, so let's dive in!

$(6-7.1)^{2} = 1.21$

$(4-7.1)^{2} = 9.61$

$(13-7.1)^{2} = 34.81$

$(11-7.1)^{2} = 15.21$

$(8-7.1)^{2} = .81$

$(6-7.1)^{2} = 1.21$

$(4-7.1)^{2} = 9.61$

$(5-7.1)^{2} = 4.41$

$(7-7.1)^{2} = .01$

$(10-7.1)^{2} = 8.41$

Now, add the deviations, and we're nearly there!

$\sum_{i=1}^{N} (x_{i}-\mu)^{2} = 1.21 + 6.91 + 34.81 + ... + 8.41 = 85.29$

Next, we must divide this number by our :

$\frac{85.29}{N} = 8.529$

This number, 8.529, is our variance, or $\sigma^{2}$ . Since standard variation is $\sigma$ , you may have guessed what we must do next. We must take the square root of the summed squares of deviations.

$\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N} (x_{i}-\mu)^{2}} = \sqrt{8.529} = 2.92$

So, our standard deviation is 2.9 kph (remembering the problem told us to round to 1 decimal point.)

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Example Question #1 : How To Find Standard Deviation

Actuaries (people who determine insurance premiums for things like life and car insurance) often have to look at the average insurance costs in an area. One way to do this without letting outliers affect their data is to take the standard deviation of insurance costs in an area over a given period of time.

Calculate the standard deviation from the data set of insurance claims for a region over one-year periods (units in millions of dollars). Round your final answer to the nearest million dollars.

32, 38, 16, 21, 25, 28, 23, 31, 16, 25, 32

Possible Answers:

Correct answer:

Explanation:

$\mu= \frac{x_{1} +x_{2} + ...x_{n}}{n} = \frac{32+38+16+21+25+28+23+31+16+25+32}{11} \approx 26.1$

We use $\sum_{i=1}^{N} (x_{i}-\mu)^{2}$ to represent this, but all it really means is that you square the difference between each value $x_{i}$ , where is the position of the value you're working with, and the mean, $\mu$ . Then we sum all those differences up (the part that goes $\sum_{i=1}^{N}$ , where is your count. $_{i=1}$ just refers to the fact that you start at the first value, so you include them all.)

It's probably easier to do than to think about at first, so let's dive in!

$(32-26.1)^{2} = 34.81$

$(38-26.1)^{2} = 141.61$

$(16-26.1)^{2} = 102.01$

$(21-26.1)^{2} = 26.01$

(25-26.1)^2 = 1.21

$(28-26.1)^{2} = 3.61$

$(23-26.1)^{2} = 9.61$

$(31-26.1)^{2} = 24.01$

$(16-26.1)^{2} = 102.01$

(25-26.1)^2 = 1.21

$(32-26.1)^{2} = 34.81$

Now, add the deviations, and we're nearly there!

$\sum_{i=1}^{N} (x_{i}-\mu)^{2} = 34.81 + 141.61 + 102.01 + ... + 34.81 = 480.9$

Next, we must divide this number by our :

$\frac{480.9}{11} = 43.72$

This number, 43.35, is our variance, or $\sigma^{2}$ . Since standard variation is $\sigma$ , you may have guessed what we must do next. We must take the square root of the summed squares of deviations.

$\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N} (x_{i}-\mu)^{2}} = \sqrt{43.72} = 6.612$

So, our standard deviation is 7 million dollars (remembering to round to the nearest million, per our instructions.)

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Algebra 1 : How to find standard deviation

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Example Question #1 : How To Find Standard Deviation

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