Q: If a data set has a mean of $$85$$ and a standard deviation of $$5$$, what is the coefficient of variation (CV) as a percentage?

$$CV = \frac{5}{85} \times 100\% = 5.88\%$$

Q: If the mean height of men in a group is $$69.4$$ inches with a standard deviation of $$2.9$$ inches, what interval contains approximately 95\% of the heights?

$$[69.4 - 2 \times 2.9, 69.4 + 2 \times 2.9] = [63.6, 75.2]$$

Question 1

Which of the following best describes the purpose of the standard deviation in a data set?

Accepted Answer

It measures the spread of data values around the mean.

Question 2

Given the following data set: $$\{4, 8, 6, 10, 12\}$$, what is the range?

Accepted Answer

$$12 - 4 = 8$$

Question 3

A sample of 5 students has scores: $$60, 70, 80, 90, 95. $$What is the sample mean $$\bar{x}$$?

Accepted Answer

$$\bar{x} = \frac{60 + 70 + 80 + 90 + 95}{5} = 79$$

Question 4

Which formula correctly calculates the sample variance $$s^2$$ for a data set with $$n$$ values $$x_i$$ and mean $$\bar{x}$$?

Accepted Answer

$$s^2$$ = \frac{\sum (x_i - \bar{x}$$)^2$$}{n-1}$$

Question 5

If a data set has a mean of $$85$$ and a standard deviation of $$5$$, what is the coefficient of variation (CV) as a percentage?

Accepted Answer

$$CV = \frac{5}{85} 	imes 100\% = 5.88\%$$

Question 6

According to the Empirical Rule, approximately what percentage of data lies within 2 standard deviations of the mean in a normal distribution?

Accepted Answer

95\%

Question 7

A history class has an average score of $$100$$ with a standard deviation of $$7.27. An $$English class has an average of $$85$$ with a standard deviation of $$5. $$Which class is more variable according to the coefficient of variation?

Accepted Answer

History class, because $$CV = 7.27\%$$

Question 8

For a data set with $$n = 35$$ students and grades $$0, 1, 2, 3, 4$$ with frequencies $$6, 9, 12, 5, 3$$ respectively, what is the sample mean?

Accepted Answer

$$\bar{x} = \frac{80}{35} = 2.29$$

Question 9

Using Chebyshev's Theorem, what is the minimum percentage of data that lies within $$k = 2$$ standard deviations of the mean for any distribution?

Accepted Answer

75\%

Question 10

A group of 5 students has scores: $$85, 75, 50, 100, 85. $$What is the sample variance $$s^2$$? (Use $$\sum x = 395$$, $$\sum x^2$ = 32575)$$

Accepted Answer

$$s^2$$ = \frac{32575 - \frac{$$395^2$$}{5}}{4} = 342.5$$

Question 11

If the mean height of men in a group is $$69.4$$ inches with a standard deviation of $$2.9$$ inches, what interval contains approximately 95\% of the heights?

Accepted Answer

$$[69.4 - 2 	imes 2.9, 69.4 + 2 	imes 2.9] = [63.6, 75.2]$$

Question 12

Which of the following statements about the range is TRUE?

Accepted Answer

The range is the difference between the maximum and minimum values.

Question 13

A data set has a mean of $$100$$ and a standard deviation of $$15. $$According to the Empirical Rule, what percentage of data falls between $$85$$ and $$115$$?

Accepted Answer

68\%

Question 14

For a frequency table, which formula is used to calculate the sample variance $$s^2$$?

Accepted Answer

$$s^2$$ = \frac{\sum f $$x^2$$ - \frac{(\sum f $$x)^2$$}{n}}{n-1}$$

Question 15

If all values in a data set are the same, what is the value of the standard deviation?

Accepted Answer

It is zero.

Question 16

A survey of 1000 adults about travel spending gives a sample mean of $$192$$ and a sample variance of $$160.3. $$What is the sample standard deviation?

Accepted Answer

$$s = \sqrt{160.3} = 12.66$$

Question 17

Which of the following is a disadvantage of using the range as a measure of variability?

Accepted Answer

It only uses the two extreme values and ignores the rest.

Question 18

According to Chebyshev's Theorem, what is the minimum percentage of data within 3 standard deviations of the mean for any distribution?

Accepted Answer

88.9\%

Question 19

If a data set is more 'spread out', what happens to its standard deviation?

Accepted Answer

The standard deviation increases.

Question 20

A data set has a mean of $$70$$ and a standard deviation of $$8. $$What interval contains approximately 68\% of the data according to the Empirical Rule?

Accepted Answer

$$[70 - 8, 70 + 8] = [62, 78]$$

Measures of Variability and Dispersion in Statistics

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