Q: Which formula correctly calculates the range of a quantitative data set?

$\text{Range} = (\text{Max. data entry}) - (\text{Min. data entry})$

Q: A sample of heights has mean $\bar{x} = 72.8$ inches and standard deviation $s = 3.3$ inches. What is the coefficient of variation?

$CV = \frac{3.3}{72.8} \times 100\% \approx 4.5\%$

Q: A data set has mean $\mu = 38.2$ and standard deviation $\sigma = 22.6$. Using Chebychev’s Theorem for $k = 2$, what is the minimum interval that contains at least 75% of the data?

$[38.2 - 2 \times 22.6, 38.2 + 2 \times 22.6] = [ -7.0, 83.4 ]$

Question 1

Which formula correctly calculates the range of a quantitative data set?

Accepted Answer

$	ext{Range} = (	ext{Max. data entry}) - (	ext{Min. data entry})$

Question 2

A sample of 12 football players has recovery times (in days): 8, 10, 4, 6, 7, 9, 7, 6, 5, 11, 6, 5. What is the sample mean recovery time?

Accepted Answer

$\bar{x} = \frac{90}{12} = 7.5$

Question 3

Which statement about standard deviation is TRUE?

Accepted Answer

Standard deviation measures the spread of data about the mean and has the same units as the data.

Question 4

Given a population data set $x_1, x_2, ..., x_N$ with mean $\mu$, which formula gives the population variance?

Accepted Answer

$\sigma^2 = \frac{\sum (x - \mu)^2}{N}$

Question 5

A sample has $n = 12$ and $\sum (x - \bar{x})^2 = 51$. What is the sample variance?

Accepted Answer

$s^2 = \frac{51}{11}$

Question 6

If a data set has all entries equal to 4, what is the population standard deviation?

Accepted Answer

$\sigma = 0$

Question 7

Using the Empirical Rule, approximately what percent of data in a bell-shaped distribution lies within two standard deviations of the mean?

Accepted Answer

95%

Question 8

Chebychev’s Theorem states that for any data set, at least what percent of the data lies within $k = 2$ standard deviations of the mean?

Accepted Answer

75%

Question 9

A sample of heights has mean $\bar{x} = 72.8$ inches and standard deviation $s = 3.3$ inches. What is the coefficient of variation?

Accepted Answer

$CV = \frac{3.3}{72.8} 	imes 100\% \approx 4.5\%$

Question 10

Which scenario best illustrates the use of the coefficient of variation?

Accepted Answer

Comparing the spread of heights and weights of basketball players, which have different units.

Question 11

For a frequency distribution with classes, which value is used to estimate the sample mean and standard deviation?

Accepted Answer

The midpoint of each class

Question 12

A population has mean $\mu = 51.5$ and standard deviation $\sigma = 3.0$. According to the Empirical Rule, what is the interval that contains approximately 68% of the data?

Accepted Answer

$[51.5 - 3.0, 51.5 + 3.0] = [48.5, 54.5]$

Question 13

If a sample has $n = 50$, sample mean $\bar{x} = 1.8$, and $\sum (x - \bar{x})^2 f = 144$, what is the sample standard deviation for grouped data?

Accepted Answer

$s = \sqrt{\frac{144}{49}} \approx 1.72$

Question 14

Which of the following is NOT a correct interpretation of standard deviation?

Accepted Answer

It is a measure of central tendency.

Question 15

A data set has mean $\mu = 38.2$ and standard deviation $\sigma = 22.6$. Using Chebychev’s Theorem for $k = 2$, what is the minimum interval that contains at least 75% of the data?

Accepted Answer

$[38.2 - 2 	imes 22.6, 38.2 + 2 	imes 22.6] = [ -7.0, 83.4 ]$

Question 16

Which formula is used to calculate the sample standard deviation?

Accepted Answer

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

Question 17

A sample of weights has mean $\bar{x} = 187.8$ pounds and standard deviation $s = 17.7$ pounds. What is the coefficient of variation?

Accepted Answer

$CV = \frac{17.7}{187.8} 	imes 100\% \approx 9.4\%$

Question 18

Why is the sample standard deviation divided by $n-1$ instead of $n$?

Accepted Answer

To correct for bias in estimating the population variance from a sample.

Question 19

In a bell-shaped distribution, what percent of data is expected to lie within three standard deviations of the mean according to the Empirical Rule?

Accepted Answer

99.7%

Measures of Variation in Descriptive Statistics

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