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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