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Algebra II : Statistical Concepts

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Example Question #11 : Deviation Concepts

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 #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 #5 : Standard Deviation

What is the standard deviation of 5, 7, 9 ?

Possible Answers:

$\frac{4}{3}$

$\frac{2\sqrt{3}}{3}$

Correct answer:

Explanation:

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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Example Question #1 : Standard Deviation

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

Possible Answers:

99.7

Correct answer:

Explanation:

By drawing a bell curve, the middle line is $50\%$ . One standard deviation left and right of the middle line is $34\%$ each. That means one standard deviation within is $68\%$ .

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Example Question #2 : Standard Deviation

If standard deviation is 3.5 and the mean is , what is the range of the number set if it's within one standard deviation?

Possible Answers:

4-7.5

0.5-4

3.5-4

0.5-7.5

2.25-5.75

Correct answer:

0.5-7.5

Explanation:

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.

(4-3.5,4+3.5) = (0.5,7.5)

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Example Question #11 : Standard Deviation

What is the standard deviation of this set?

2, 5, 8, 12, 13

Possible Answers:

21.5

{}4.15

17.2

4.64

Correct answer:

4.64

Explanation:

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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Example Question #11 : Standard Deviation

Standard Deviation can be calculated from what statistical term?

Possible Answers:

Variance

Mode

Median

Quartile

Range

Correct answer:

Variance

Explanation:

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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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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Algebra II : Statistical Concepts

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #11 : Deviation Concepts

Example Question #6 : How To Find Standard Deviation

Example Question #7 : How To Find Standard Deviation

Example Question #5 : Standard Deviation

Example Question #1 : Standard Deviation

Example Question #2 : Standard Deviation

Example Question #11 : Standard Deviation

Example Question #11 : Standard Deviation

Example Question #13 : Standard Deviation

Example Question #11 : Standard Deviation

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