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Algebra II : Deviation Concepts

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Example Question #13 : Standard Deviation

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

Possible Answers:

$\small 2.999999$

$\small 1$

$\small 4$

$\small 7.000001$

$\small 8$

Correct answer:

$\small 4$

Explanation:

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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Example Question #11 : Standard Deviation

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

Possible Answers:

$\small 9$

$\small 162$

$\small 81$

$\small 40.5$

$\small 3.214$

Correct answer:

$\small 9$

Explanation:

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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Example Question #15 : Standard Deviation

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

Possible Answers:

$\small 15$

$\small 47.5$

$\small 27$

$\small 34$

$\small 13.5$

Correct answer:

$\small 13.5$

Explanation:

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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Example Question #16 : Standard Deviation

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.

Possible Answers:

$\small 10$

$\small 4$

$\small 7$

$\small 3$

$\small 8$

Correct answer:

$\small 3$

Explanation:

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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Example Question #17 : Standard Deviation

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.

Possible Answers:

$\small 45$

$\small 60$

$\small 15$

$\small 30$

$\small 75$

Correct answer:

$\small 30$

Explanation:

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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Example Question #482 : Algebra Ii

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

Possible Answers:

Five

One

Four

Two

Three

Correct answer:

Two

Explanation:

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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Example Question #19 : Standard Deviation

Determine the population standard deviation of the following data set and round to three decimal places: [1,9,-7]

Possible Answers:

6.532

8.000

11.314

1.732

5.040

Correct answer:

6.532

Explanation:

Write the formula for population standard deviation.

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

represents the number of terms, x_i 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: 6.532

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Example Question #20 : Standard Deviation

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.

Possible Answers:

1.633

2.667

1.832

Correct answer:

Explanation:

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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Example Question #21 : Deviation Concepts

Determine the sample standard deviation if the sample variance is 225 .

Possible Answers:

Correct answer:

Explanation:

Write the formula for the standard deviation given the variance.

$\textup{Standard Deviation }(\sigma) = \sqrt{\textup{Variance}}$

Substitute the variance into the standard deviation.

$\sigma=\sqrt{225} = 15$

The answer is:

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Algebra II : Deviation Concepts

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #13 : Standard Deviation

Example Question #11 : Standard Deviation

Example Question #15 : Standard Deviation

Example Question #16 : Standard Deviation

Example Question #17 : Standard Deviation

Example Question #482 : Algebra Ii

Example Question #19 : Standard Deviation

Example Question #20 : Standard Deviation

Example Question #21 : Deviation Concepts

All Algebra II Resources

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