How to find standard deviation - Algebra

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Question

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.)

Answer

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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Question

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

Mean: 72.1

Median: 70

Standard deviation: 4.6

Answer

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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Question

Find the standard deviation of the following set of numbers: Round your answer to the nearest tenth.

Answer

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:

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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Question

Find the standard deviation of the following set of numbers: Round your answer to the nearest hundredth.

Answer

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:

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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Question

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

Answer

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:

.

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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Question

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.

Answer

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:

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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Question

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.

Answer

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 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:

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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Question

The variance is . What is the standard deviation?

Answer

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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Question

Find the standard deviation of the data set:

Answer

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.

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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Question

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.

Answer

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. 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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Question

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.

Answer

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!

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

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

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

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

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

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

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

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

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

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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Question

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

Answer

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.

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

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

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

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Question

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

Answer

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.

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

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

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

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Question

Find the standard deviation of the following set of numbers:

Answer

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

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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Question

Find the standard deviation of the following set of numbers:

Answer

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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Question

Find the standard deviation of the following set of numbers.

Answer

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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Question

Find the standard deviation of the following data set:

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

Answer

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.

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.

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.

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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Question

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

Answer

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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Question

Approximate the standard deviation of the given set of numbers to three decimal places:

Answer

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:

$\mu$ 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:

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Question

If the variance of a data set is 60, what must be the standard deviation? Round to three decimal places.

Answer

The standard deviation is the square root of variance.

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

Substitute the variance.

$\sigma = \sqrt{\textup{60}} = 7.746$

The answer is:

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