Q: A data set has a mean of 50 and a standard deviation of 5. Using the Empirical Rule, what is the approximate range of values that contains about 99.7% of the data?

$$50 \pm 3 \times 5 = 35 to$$ $$65$$

Question 1

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

Accepted Answer

The difference between the maximum and minimum data entries in the set.

Question 2

Given the data set: 37, 39, 41, 41, 42, 44, 44, 45, 47, 52, what is the range?

Accepted Answer

$$52 - 37 = 15$$

Question 3

Which formula correctly represents the population variance?

Accepted Answer

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

Question 4

A sample of 10 students has test scores with a mean of 80. The sum of squared deviations from the mean is 90. What is the sample variance?

Accepted Answer

$$s^2$$ = \frac{90}{9} = 10$$

Question 5

Which of the following statements about standard deviation is TRUE?

Accepted Answer

The standard deviation is always greater than or equal to zero.

Question 6

For a normally distributed data set, approximately what percentage of data falls within two standard deviations of the mean according to the Empirical Rule?

Accepted Answer

About 95%

Question 7

A population has the following data: 2, 4, 6, 8, 10. What is the population standard deviation?

Accepted Answer

$$\sigma = \sqrt{\frac{(2-6)^2$ + (4-6)^2$ + (6-6)^2$ + (8-6)^2$ + (10-6)^2$}{5}} = \sqrt{8} \approx 2.83$$

Question 8

Which formula is used to calculate the sample standard deviation for grouped data (frequency distribution)?

Accepted Answer

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

Question 9

A data set has a mean of 50 and a standard deviation of 5. Using the Empirical Rule, what is the approximate range of values that contains about 99.7% of the data?

Accepted Answer

$$50 \pm 3 	imes 5 = 35 to$$ $$65$$

Question 10

According to Chebyshev's Theorem, what is the minimum proportion of data that must lie within $$k = 2$$ standard deviations of the mean for any data set?

Accepted Answer

$$1 - \frac{1}{2^2$} = 0.75 or 75$$%

Question 11

Which of the following best describes the coefficient of variation (CV)?

Accepted Answer

It is the ratio of the standard deviation to the mean, often expressed as a percentage.

Question 12

A researcher wants to compare the variability of two different data sets with different units. Which measure should they use?

Accepted Answer

Coefficient of variation

Question 13

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

Accepted Answer

0

Question 14

A sample of 8 values has a mean of 10. Each value is either 9 or 11. What is the sample standard deviation?

Accepted Answer

$$s = 1$$

Question 15

Which of the following is NOT a step in calculating the sample variance?

Accepted Answer

Divide the sum of squared deviations by $$n ($$the sample size).

Question 16

A data set is highly skewed to the right. Which measure of variation is most affected by this skewness?

Accepted Answer

Range

Question 17

If the standard deviation of a data set is increased, what happens to the spread of the data?

Accepted Answer

The data become more spread out from the mean.

Question 18

A frequency distribution is given for grouped data. Which formula should be used to estimate the sample standard deviation?

Accepted Answer

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

Question 19

A hospital administrator wants to compare the variation in patient wait times between two departments. Department A has a mean wait time of 30 minutes and a standard deviation of 5 minutes. Department B has a mean wait time of 60 minutes and a standard deviation of 10 minutes. Which department has greater relative variation?

Accepted Answer

Both departments have the same relative variation.

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

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