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Algebra 1 : Algebra 1

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Example Questions

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:

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.

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

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!

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

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

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

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

(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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Example Question #11 : How To Find Standard Deviation

Find the standard deviation of the data set: (1,3,8)

Possible Answers:

5.635

8.667

1.872

2.483

2.944

Correct answer:

2.944

Explanation:

Write the formula of standard deviation.

$\sigma=\sqrt{\frac{1}{N}\sum_{a=1}^{N}(x_a-\mu)^2}= \sqrt{\textup{Variance}}$

Determine the mean, $\mu$ .

$\mu=\frac{1+3+8}{3}= \frac{12}{3}= 4$

Subtract each number in the dataset from the mean and square each result.

This is the $x_a-\mu$ term.

(1-4)^2= (-3)^2=9

(3-4)^2= (-1)^2=1

(8-4)^2= (4)^2=16

Find the mean of the squared differences, or the variance.

This is the $\frac{1}{N}\sum_{a=1}^{N}(x_a-\mu)^2$ term.

$\textup{Variance}= \frac{9+1+16}{3}=\frac{26}{3}$

Square root the variance for the standard deviation.

$\sigma=\sqrt{\frac{26}{3}}= 2.944$

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Example Question #1991 : Algebra 1

Find the standard deviation. Round your answer to the nearest tenth.

1, 3, 5, 7, 9

Possible Answers:

2.8

4.8

1.6

4.6

Correct answer:

2.8

Explanation:

To find standard deviation, we apply this formula: $\sqrt{\frac{1}{n}\Sigma (\bar{x}-x)^2}$ .

n represents how many data in the set.

$\bar{x}$ represents the average of the data in the set.

is any data in the set.

$\Sigma$ is summation of the difference between average and any data value squared.

So the mean is $\frac{1+3+5+7+9}{5}=5$ . Now, we apply the formula.

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

$=\sqrt{\frac{1}{5}[16+4+0+4+16])}$

$=\sqrt{\frac{1}{5}(40)}$

=2.8

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Example Question #13 : How To Find Standard Deviation

Find the standard deviation. Round your answer to the nearest tenth.

3, 6, 9, 12, 15

Possible Answers:

4.2

4.7

7.5

Correct answer:

4.2

Explanation:

To find standard deviation, we apply this formula: $\sqrt{\frac{1}{n}\Sigma (\bar{x}-x)^2}$ .

n represents how many data in the set.

$\bar{x}$ represents the average of the data in the set.

is any data in the set.

$\Sigma$ is summation of the difference between average and any data value squared.

So the mean is $\frac{3+6+9+12+15}{5}=9$ . Now, we apply the formula.

$\sqrt{\frac{1}{5}[(9-3)^2+(9-6)^2+(9-9)^2+(9-12)^2+(9-15)^2]}}$

$=\sqrt{\frac{1}{5}[36+9+0+9+36])}$

$=\sqrt{\frac{1}{5}(90)}$

=4.2

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Example Question #1992 : Algebra 1

Find the standard deviation of the following set of numbers:

7, 10, 13

Possible Answers:

$\sqrt{6}$

$\sqrt{7}$

$\sqrt{10}$

Correct answer:

$\sqrt{6}$

Explanation:

To find standard deviation, we will follow these steps:

Find the mean of the original set of numbers.
Subtract the mean from each of the original numbers.
Square each number.
Find the mean of the new set of numbers.
Take the square root of the mean.

So, we will take this step by step.

STEP 1

$\text{mean} = \frac{7 + 10 + 13}{3}$

$\text{mean} = \frac{30}{3}$

$\text{mean} = 10$

STEP 2

7, 10, 13

(7-10), (10-10), (13-10)

-3, 0, 3

STEP 3

(-3)^2, 0^2, 3^2

9, 0, 9

STEP 4

$\text{mean} = \frac{9+0+9}{3}$

$\text{mean} = \frac{18}{3}$

$\text{mean} = 6$

STEP 5

$\sqrt{6}$

Thefore, the standard deviation is $\sqrt{6}$ .

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Example Question #15 : How To Find Standard Deviation

Find the standard deviation of the following set of numbers:

1,3,4,7,10,13,16,17,19

Possible Answers:

$5\sqrt{14}$

$\sqrt{43.75}$

Correct answer:

$5\sqrt{14}$

Explanation:

To solve, you must use the following equation:

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

where,

$mean=x_{m}$

Thus,

$\sigma = \sqrt{\frac{(1-10)^2+(3-10)^2+(4-10)^2+(7-10)^2+(10-10)^2+(13-10)^2+(16-10)^2+(17-10)^2+(19-10)^2}{8}}$

$\\ \sigma = \sqrt{\frac{81+49+36+9+0+9+26+49+81}{8}}=\sqrt{\frac{350}{8}}=\sqrt{43.75} \approx 6.614$

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Example Question #16 : How To Find Standard Deviation

Find the standard deviation of the following set of numbers.

1,4,5,5,6,15,15,16,23

Possible Answers:

$\frac{\sqrt{438}}{3}$

$\sqrt{438}$

Correct answer:

$\frac{\sqrt{438}}{3}$

Explanation:

To solve, simply use the formula for standard deviation. Thus,

$\sigma =\sqrt{\frac{\sum_{i=1}^{n}(x_i-x_m)^2}{n}}$ where xm is the mean. Thus,

$\sigma =\sqrt{\frac{(1-10)^2+(4-10)^2+(5-10)^2+(5-10)^2+(6-10)^2+(15-10)^2+(15-10)^2+(16-10)^2+(23-10)^2}{9}}$

$\sigma =\sqrt{\frac{9^2+6^2+5^2+5^2+4^2+5^2+5^2+6^2+13^2}{9}}=\sqrt{\frac{438}{9}}=\frac{\sqrt{438}}{3}$

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Example Question #11 : How To Find Standard Deviation

Find the standard deviation of the following data set:

4, 2, 6, 8, 8, 2, 8, 10, 6.

Possible Answers:

Correct answer:

Explanation:

To find the standard deviation, we will go through a few steps.

Step 1: Find the mean.

We will find the mean of the data set.

$\text{mean} = \frac{4+2+6+8+8+2+8+10+6}{9}$

$\text{mean} = \frac{54}{9}$

$\text{mean} = 6$

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

We will now take each number in the data set and subtract the mean from it.

4 - 6 = -2

2 - 6 = -4

6 - 6 = 0

8 - 6 = 2

2 - 6 = -4

8 - 6 = 2

10 - 6 = 4

6 - 6 = 0

Giving us a new data set of -2, -4, 0, 2, 2, -4, 2, 4, 0.

Step 3: Square each number in the data set.

(-2)^2 = 4

(-4)^2 = 16

0^2 = 0

2^2 = 4

(-4)^2= 16

2^2 = 4

4^2 = 16

0^2 = 0

Giving us a new data set of 4, 16, 0, 4, 4, 16, 4, 16, 0.

Step 4: Add each number in the data set together.

4 + 16 + 0 + 4 + 4 + 16 + 4 + 16 + 0 = 64

Giving us an answer of 64.

Step 5: Take the square root of the answer.

$\sqrt{64} = 8$

That is the standard deviation! So, the standard deviation of the data set

4, 2, 6, 8, 8, 2, 8, 10, 6

is 8.

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Example Question #11 : How To Find Standard Deviation

If the variance of a set of sample data is 100 , what is the standard deviation?

Possible Answers:

3.5

5.5

12.5

Correct answer:

Explanation:

The formula for finding the standard deviation given the variance is:

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

Substitute the variance.

$SD = \sqrt{100}=10$

The standard deviation is .

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Example Question #11 : How To Find Standard Deviation

Approximate the standard deviation of the given set of numbers to three decimal places: [1, 2, 9]

Possible Answers:

3.426

4.136

3.874

3.559

4.742

Correct answer:

3.559

Explanation:

Write the formula for standard deviation.

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

represents the total numbers in the data set: N=3

$\mu$ is the mean: $\mu = \frac{1+2+9}{3} =\frac{12}{3} = 4$

Find the mean of the squared differences. Subtract each number in the data set with the mean, square the quantities, and take the average.

This will be the variance:

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

$\frac{1}{3} \cdot [ (9-4)^2+(2-4)^2+(1-4)^2]$

$\frac{1}{3} \cdot [ (5)^2+(-2)^2+(-3)^2] = \frac{1}{3}[25+4+9]= \frac{38}{3}$

The standard deviation is the square root of the variance, which is the square root of this fraction.

$\sqrt{\frac{38}{3}} = 3.559$

The answer is: 3.559

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Algebra 1 : Algebra 1

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #1 : How To Find Standard Deviation

Example Question #11 : How To Find Standard Deviation

Example Question #1991 : Algebra 1

Example Question #13 : How To Find Standard Deviation

Example Question #1992 : Algebra 1

STEP 1

STEP 2

STEP 3

STEP 4

STEP 5

Example Question #15 : How To Find Standard Deviation

Example Question #16 : How To Find Standard Deviation

Example Question #11 : How To Find Standard Deviation

Step 1: Find the mean.

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

Step 3: Square each number in the data set.

Step 4: Add each number in the data set together.

Step 5: Take the square root of the answer.

Example Question #11 : How To Find Standard Deviation

Example Question #11 : How To Find Standard Deviation

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