Deviation Concepts - Algebra II

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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

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

Calculate the standard deviation of the following set of values.

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

Answer

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

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Question

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?

Answer

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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Question

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?

Answer

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.

Calculating that, we get that the variance from the mean is . 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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Question

Determine the standard deviation for the following data set:

12, 15, 30, 5, 27, 19

Answer

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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Question

What is the standard deviation of ?

Answer

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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Question

In a normal distribution, what percentage is covered within one standard deviation?

Answer

By drawing a bell curve, the middle line is . One standard deviation left and right of the middle line is each. That means one standard deviation within is .

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Question

If standard deviation is and the mean is , what is the range of the number set if it's within one standard deviation?

Answer

Standard deviation is the dispersion of the data set. Since it's asking for within one standard deviation, we need to take the mean and add the standard deviation to find the upper bound of the range. Then, we will need to subtract the standard deviation from the mean to identify the lower bound of the range.

=

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Question

What is the standard deviation of this set?

Answer

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.

$\frac{2+5+8+12+13}{5}=8$

Plugging in the values:

$\small \sqrt{\frac{(2-8)^2+(5-8)^2+(8-8)^2+(12-8)^2+(13-8)^2}{5-1}}=4.64$

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Question

Standard Deviation can be calculated from what statistical term?

Answer

Another way to calculate standard deviation is the square root of variance.

Variance is,

$\small {\frac{\Sigma (x-\bar{x})^2}{n-1}}$ .

Taking the square root of this is how standard devation can be calculated.

$\sigma=\sqrt{\small {\frac{\Sigma (x-\bar{x})^2}{n-1}}}$

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Question

If the mean is $\small 5$ with a standard deviation of $\small 2$ , then which of the following values is within one standard deviation?

Answer

If mean is $\small 5$ with standard deviation of $\small 2$ , then one standard deviation within has a range of $\small \small 3$ to $\small 7$ .

Remember, we find the range by adding the standard deviation to the mean and subtracting the standard deviation from the mean.

$(5-2,5+2)\rightarrow (3,7)$

Only $\small 4$ is in the range.

The rest of the numbers are more than one standard deviation.

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Question

If variance is $\small 81$ , what is the standard deviation?

Answer

Variance is,

$\small {\frac{\Sigma (x-\bar{x})^2}{n-1}}$ .

To find standard deviation is to take the sqaure root of the variance.

$\small \sqrt{81}=9$ .

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Question

What is the difference between two standard deviations on the right tail with one standard deviation on the right tail? Assume a normal distribution.

Answer

Two standard deviations represents $\small 95%$ . One standard deviation represents $\small 68%$ . The difference is $\small 27%$ . However, the question is focusing on the right side of the tail. Since it's normal distribution, both tails of the graph are equal. Divide $\small 27$ by $\small 2$ and we get $\small 13.5$ .

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Question

If the number set is within a standard deviation and the range is $\small 4$ to $\small 10$ , what is the standard deviation? Assume a normal distribution.

Answer

Since it's normally distributed, that means we can find the mean. The middle number in the number set is the mean. In this case that value is $\small 7$ . Since it's within one standard deviation, we can take the difference of either $\small 7$ and $\small 4$ or $\small 10$ and $\small 7$ which the answer is $\small 3$ either way.

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Question

If the number set is within $\small 2.5$ standard deviations and the range is $\small \small 20$ to $\small 170$ , what is the standard deviation? Assume a normal distribution.

Answer

Since it's normally distributed, that means we can find the mean. The middle number in the number set is the mean. In this case that value is $\small \small 95$ .

$\mu=\frac{20+170}{2}=\frac{190}{2}=95$

Since it's $\small 2.5$ standard deviations, we need to set up an equation.

$\small 2.5x+20=95$ This is written like this because $\small 20$ is the lowest number in the set, $\small 2.5x$ represents the $\small 2.5$ standard deviations and $\small x$ is one standard deviation. By subtracting $\small 20$ both sides and dividing both sides by $\small 2.5$ , we get $\small 30$ .

$\small 2.5x=75$

$\small x=30$

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Question

How many standard deviations represents $\small 95%$ of a set in a normal distribution?

Answer

One standard deviation represents $\small 68%$ . Two standard deviations represents $\small 95%$ . Three standard deviations represents $\small 99.7%$ .

As you add more standard deviations, the percent coverages get a bit bigger.

The answer is just two.

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Question

Determine the population standard deviation of the following data set and round to three decimal places:

Answer

Write the formula for population standard deviation.

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

represents the number of terms, represents the terms in the data set, and $\mu$ is the mean.

Calculate the mean, $\mu$ .

$\frac{1+9-7}{3} = \frac{3}{3}=1$

$\mu =1$

Evaluate the variance, $\frac{1}{N}\sum_{i}^{N}(x_i-\mu)^2$ .

$\frac{1}{3}[(1-1)^2+(9-1)^2+(-7-1)^2] = \frac{1}{3}[64+64]=\frac{128}{3}$

The standard deviation is the square root of the variance.

$\sigma= \sqrt{\frac{1}{N}\sum_{i}^{N}(x_i-\mu)^2} = \sqrt{\frac{128}{3}}= 6.532$

The answer is:

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Question

A jellybean machine dispenses 3 jellybeans on the first trial, 5 jellybeans on the second trial, and 7 jellybeans on the third trial. Determine the sample standard deviation.

Answer

The standard deviation measures the spread of the results.

Write the formula for the sample standard deviation.

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

Determine the mean $\overline{x}$ .

$\overline{x} = \frac{3+5+7}{3} = \frac{15}{3} =5$

The term $\sum_{i=1}^{N}(x_i-\overline{x})^2$ means that we are summing the squared differences from the mean.

$\sum_{i=1}^{N}(x_i-\overline{x})^2 = (3-5)^2+(5-5)^2+(7-5)^2$

Simplify the terms.

$\sum_{i=1}^{N}(x_i-\overline{x})^2= (-2)^2+0+(2)^2 = 4+4 = 8$

represents the sample size. There are three numbers in the data.

$s=\sqrt{\frac{1}{N-1}\sum_{i=1}^{N}(x_i-\overline{x})^2} = \sqrt{\frac{1}{3-1}(8)}=\sqrt{4}=2$

The answer is:

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