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Algebra 1 : Statistics and Probability

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Example Questions

Example Question #15 : How To Find Standard Deviation

Find the standard deviation of the following set of numbers:

1,3,4,7,10,13,16,17,19

Possible Answers:

$5\sqrt{14}$

$\sqrt{43.75}$

Correct answer:

$5\sqrt{14}$

Explanation:

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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Example Question #16 : How To Find Standard Deviation

Find the standard deviation of the following set of numbers.

1,4,5,5,6,15,15,16,23

Possible Answers:

$\frac{\sqrt{438}}{3}$

$\sqrt{438}$

Correct answer:

$\frac{\sqrt{438}}{3}$

Explanation:

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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Example Question #11 : How To Find Standard Deviation

Find the standard deviation of the following data set:

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

Possible Answers:

Correct answer:

Explanation:

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.

4 - 6 = -2

2 - 6 = -4

6 - 6 = 0

8 - 6 = 2

2 - 6 = -4

8 - 6 = 2

10 - 6 = 4

6 - 6 = 0

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.

(-2)^2 = 4

(-4)^2 = 16

0^2 = 0

2^2 = 4

(-4)^2= 16

2^2 = 4

4^2 = 16

0^2 = 0

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.

4 + 16 + 0 + 4 + 4 + 16 + 4 + 16 + 0 = 64

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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Example Question #11 : How To Find Standard Deviation

If the variance of a set of sample data is 100 , what is the standard deviation?

Possible Answers:

3.5

5.5

12.5

Correct answer:

Explanation:

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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Example Question #11 : How To Find Standard Deviation

Approximate the standard deviation of the given set of numbers to three decimal places: [1, 2, 9]

Possible Answers:

3.426

4.136

3.874

3.559

4.742

Correct answer:

3.559

Explanation:

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: N=3

$\mu$ 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: 3.559

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Example Question #11 : How To Find Standard Deviation

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

Possible Answers:

7.328

7.546

7.746

6.372

8.532

Correct answer:

7.746

Explanation:

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: 7.746

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Example Question #1 : How To Find Range

We want to create a three digit code for a combination lock. Only digits from 0-9 are allowed and digits can be repeated. How many such codes can be generated?

Possible Answers:

$10^{3}$

$3^{10}$

$10^{4}$

$10^{2}$

$10\times 9\times 8$

Correct answer:

$10^{3}$

Explanation:

The first position can be filled 10 ways and since repetition is allowed, the 2nd position can be filled in 10 ways and similarly the third position can be filled in 10 ways as well, giving us the correct answer of

$10^{3}$

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Example Question #1 : How To Find Range

You are given the following number set:

2, 8, 3, 6, 9, 10, 5

Find the range.

Possible Answers:

Correct answer:

Explanation:

The range is the difference between the lowest and the highest number in the number set.

The lowest number in the number set is 2 while the highest number is 10. Therefore, the range is calculated as

10-2=8

So 8 is the range.

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Example Question #2 : How To Find Range

Evaluate:

$C\binom{5}{3}$

Possible Answers:

120

Correct answer:

Explanation:

This is a combination problem. All we need to solve this problem is the combination formula: $C\binom{n}{k}=\frac{n!}{k!(n-k)!}$

$C\binom{5}{3} = \frac{5!}{2!\times 3!}$

Now $5! = 5\times 4\times 3\times 2\times\times 1$

$3!= 3\times 2\times 1$

and $2! = 2\times 1$

Instead of finding answers for all of these factorials, notice that they have many of the same terms and can therefore be cancelled.

$\frac{5\times 4\times3\times2\times1}{3\times2\times1\times2\times1}=\frac{5\times4}{2}=5\times2=10$

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Example Question #1 : How To Find Range

A teacher is grading quizzes and recording the scores in her records. The scores are recorded as follows: 81, 91, 83, 88, 74, 98, 81, 94, 68, 92, 77, 79, 83, 91, 81, 84, 85, 81, 85, 79, 83, 81, 83.

What is the range?

Possible Answers:

Correct answer:

Explanation:

The range of the statistics is simply the difference between the largest value and the smallest value.

The largest number in the set is 98, and the smallest number is 68.

Subtract to find the range:

98-68=30

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Algebra 1 : Statistics and Probability

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #15 : How To Find Standard Deviation

Example Question #16 : How To Find Standard Deviation

Example Question #11 : How To Find Standard Deviation

Step 1: Find the mean.

Step 2: Subtract the mean from each number in the data set.

Step 3: Square each number in the data set.

Step 4: Add each number in the data set together.

Step 5: Take the square root of the answer.

Example Question #11 : How To Find Standard Deviation

Example Question #11 : How To Find Standard Deviation

Example Question #11 : How To Find Standard Deviation

Example Question #1 : How To Find Range

Example Question #1 : How To Find Range

Example Question #2 : How To Find Range

Example Question #1 : How To Find Range

All Algebra 1 Resources

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