Q: What is the formula for the coefficient of variation (CV) for a sample?

$CV = \frac{s}{\overline{x}} \times 100$

Question 1

Which of the following best describes the range of a data set?

Accepted Answer

The difference between the maximum and minimum data values.

Question 2

Given the following wait times (in minutes): 50, 25, 75, 35, 50, 25, 30, 50, 45, 25, 20, what is the range?

Accepted Answer

$75 - 20 = 55$

Question 3

Which property of the range makes it a less reliable measure of variation?

Accepted Answer

It uses only the maximum and minimum values, making it sensitive to outliers.

Question 4

What is the formula for the sample standard deviation $s$?

Accepted Answer

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

Question 5

Why do we divide by $n-1$ instead of $n$ when calculating the sample variance?

Accepted Answer

To make the sample variance an unbiased estimator of the population variance.

Question 6

Which of the following is NOT a property of the standard deviation?

Accepted Answer

It is resistant to outliers.

Question 7

A sample has a mean of 100 and a standard deviation of 15. According to the empirical rule, approximately what percentage of values fall between 70 and 130?

Accepted Answer

About 95%

Question 8

Using Chebyshev’s theorem, what is the minimum proportion of data that must lie within 3 standard deviations of the mean for any data set?

Accepted Answer

At least $\frac{8}{9}$ (or about 89%)

Question 9

What is the formula for the coefficient of variation (CV) for a sample?

Accepted Answer

$CV = \frac{s}{\overline{x}} 	imes 100$

Question 10

If a data set has a range of 40, what is the estimated standard deviation using the range rule of thumb?

Accepted Answer

$s = \frac{40}{4} = 10$

Question 11

Which of the following statements about variance is TRUE?

Accepted Answer

Variance is the square of the standard deviation.

Question 12

A researcher collects a sample of 12 measurements. When calculating the sample variance, what denominator should be used?

Accepted Answer

11

Question 13

Which of the following best describes a biased estimator?

Accepted Answer

An estimator that systematically overestimates or underestimates the parameter.

Question 14

Suppose the mean wait time for a ride is 30 minutes with a standard deviation of 8 minutes. According to the range rule of thumb, what is the approximate range of most wait times?

Accepted Answer

$30 - 2(8)$ to $30 + 2(8)$, or 14 to 46 minutes

Question 15

Which formula is used to calculate the population standard deviation $\sigma$?

Accepted Answer

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

Question 16

A data set has a mean of 60 and a standard deviation of 5. According to the empirical rule, what interval contains about 99.7% of the data?

Accepted Answer

From 40 to 80

Question 17

Which of the following is a correct interpretation of Chebyshev’s theorem for $K=2$?

Accepted Answer

At least 75% of data values are within 2 standard deviations of the mean, regardless of the data's distribution.

Question 18

If the sample standard deviation $s$ is 12 and the sample mean $\overline{x}$ is 48, what is the coefficient of variation (CV)?

Accepted Answer

$25\%$

Question 19

Which of the following is TRUE about the sample variance $s^2$?

Accepted Answer

It is an unbiased estimator of the population variance $\sigma^2$.

Measures of Variation in Statistics: Range, Standard Deviation, and Variance

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