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Algebra II : Algebra II

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Example Questions

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Example Question #5 : Mean, Standard Deviation, And Normal Distribution

Find the standard deviation of the following set of numbers:

(7,4,11,13,10)

Possible Answers:

$3\sqrt{2}$

$\sqrt{14}$

$\sqrt{10}$

Correct answer:

$\sqrt{10}$

Explanation:

To begin, we must remember the formula for standard deviation:

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

where $\sigma$ is the standard deviation, N is the number of values in our set, x_i is the value we're currently evaluating in the summation, and $\mu$ is the mean of our set of numbers. All the summation part of the equation means is that we subtract our mean from each number in the set, square that value, and then add all of those values for each number together. So before we find the values that will be added together, we must first find our mean for the set of numbers:

$\mu =\frac{\sum_{i=1}^{N}x_i}{N}=\frac{7+4+11+13+10}{5}=9$

Now we can determine the value for our summation for each number in the set:

(7-9)^2=4

(4-9)^2=25

(11-9)^2=4

(13-9)^2=16

(10-9)^2=1

Looking at our equation for standard variation, now all we must do is sum all of the values above, divide by N, and take the square root:

$\sigma =\sqrt{\frac{1}{5}(4+25+4+16+1)}=\sqrt{10}$

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Example Question #5 : Mean, Standard Deviation, And Normal Distribution

The formula for standard deviation is the following:

$\sigma = \sqrt{\frac{\sum (x - x_{avg})^2}{n-1}}$ .

Where,

$\sigma = \text{ standard deviation}$ ,

$x_{avg}=\text{ mean of the data set}$ ,

$n=\text{ number of data points}$ .

Five students took a test and recieved the following grades: , , , 100 , . What is the standard deviation of the test grades to the nearest decimal place?

Possible Answers:

None of the other answers.

2.4

6.6

8.3

7.9

Correct answer:

7.9

Explanation:

The first step in solving for standrd deviation is to find the mean of the data set.

For this problem:

$x_{avg} = \frac{95+90+80+100+85}{5} = \frac{450}{5} = 90$ .

Now we can evaluate the summation:

$\sum\left ( x - x_{avg}\right )^2 = (95-90)^2 + (90-90)^2 + (80-90)^2 + (100-90)^2 + (85-90)^2$

$\sum\left ( x - x_{avg}\right )^2 = 5^2 + 0^2 + (-10)^2 + 10^2 + (-5)^2$

$\sum\left ( x - x_{avg}\right )^2 = 25 + 0 + 100 + 100 + 25 = 250$

Now we can rewrite the standard deviation expression:

$\sigma = \sqrt{\frac{250}{n-1}}$

There are 5 data points, so n = 5

$\sigma = \sqrt{\frac{250}{4}} = \sqrt{62.5} = 7.9$

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Example Question #6 : Mean, Standard Deviation, And Normal Distribution

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

Possible Answers:

50%

68%

95%

34%

99.7%

Correct answer:

68%

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 #7 : Mean, Standard Deviation, And Normal Distribution

If the mean is $\small 0$ and the standard deviation is $\small 3.4$ , which of the following is NOT within two standard deviations?

Possible Answers:

$\small -6.8$

$\small \small -6.81$

$\small 6.80$

$\small 6.79$

$\small -6.79$

Correct answer:

$\small \small -6.81$

Explanation:

Two standard deviations means to double the standard deviation value.

$2\cdot 3.4 =6.8$

This means the range is $\small -6.8$ to $\small 6.8$ .

$(0-6.8, 0+6.8)\rightarrow (-6.8,6.8)$

Only $\small -6.81$ is not in the range.

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Example Question #8 : Mean, Standard Deviation, And Normal Distribution

Ibram was paid $5000 for editing a 2200-page encyclopedia. What was his rate earned per page?

Possible Answers:

$0.44

$1.27

$2.27

$0.56

Correct answer:

$2.27

Explanation:

This question is really just asking for the average dollar per page. This could be easily calculated:

$\frac{5000}{2200}=2.27$

However, you might also think of this as a rate problem:

D=RT

Using the information from the question, we can see that D=5000 and T=2200.

This comes out to what we solved above:

$R=\frac{D}{T}=\frac{5000}{2200}=2.27$

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Example Question #2 : Mean, Standard Deviation, And Normal Distribution

Standard Deviation can be calculated from what statistical term?

Possible Answers:

Range

Median

Quartile

Variance

Mode

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 deviation can be calculated.

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

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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #5 : Mean, Standard Deviation, And Normal Distribution

Example Question #5 : Mean, Standard Deviation, And Normal Distribution

Example Question #6 : Mean, Standard Deviation, And Normal Distribution

Example Question #7 : Mean, Standard Deviation, And Normal Distribution

Example Question #8 : Mean, Standard Deviation, And Normal Distribution

Example Question #2 : Mean, Standard Deviation, And Normal Distribution

All Algebra II Resources

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