Question 1

Which of the following statements best explains why the range is not always a reliable measure of variation?

Accepted Answer

The range only considers the minimum and maximum values, ignoring all other data points.

Question 2

Given the sample data set $$x = \{8, 9, 10, 13\}$$, what is the sample variance $$s^2$$?

Accepted Answer

$$s^2$$ = \frac{14}{3} = 4.666\ldots$$

Question 3

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

Accepted Answer

Dividing by $$n-1$$ corrects for the bias in estimating the population variance from a sample.

Question 4

If a sample has values $$y = \{2, 5, 11, 22\}$$, what is the sample standard deviation $$s$$?

Accepted Answer

$$s = \sqrt{78} \approx 8.8$$

Question 5

Which formula represents the sample variance for a data set $$x_1$$, $$x_2$$, ..., x_n$$?

Accepted Answer

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

Question 6

A medical researcher collects the following sample of blood pressure readings: $$120, 122, 121, 119. $$What is the range of this sample?

Accepted Answer

Range $$= 122 - 119 = 3$$

Question 7

Which of the following best describes the purpose of squaring the deviations from the mean when calculating variance?

Accepted Answer

Squaring removes negative signs so that all deviations contribute positively to the measure of variation.

Question 8

Suppose a sample has a variance of $$s^2 = 25$. What is the standard deviation?

Accepted Answer

$$s = \sqrt{25} = 5$$

Question 9

A data set has values $$x = \{4, 6, 8, 10\}. If $$the mean is $$7$$, what is the sum of the squared deviations from the mean?

Accepted Answer

$$(4-7)^2$$ + $$(6-7)^2$$ + $$(8-7)^2$$ + $$(10-7)^2 = 9 + 1 + 1 + 9 = 20$

Question 10

Which of the following is NOT a reason why the standard deviation is preferred over the range as a measure of variation?

Accepted Answer

Standard deviation is less affected by extreme values than range.

Question 11

If the sample size is $$n = 5$$, which denominator should be used when calculating the sample variance?

Accepted Answer

Divide by $$n-1 = 4$$

Question 12

A chemistry lab records the following sample temperatures: $$15^	ext{o}C, 18^	ext{o}C, 21^	ext{o}C, 24^	ext{o}C. $$What is the sample mean?

Accepted Answer

$$\bar{x} = \frac{15 + 18 + 21 + 24}{4} = 19.5$$

Question 13

Which of the following is the correct formula for sample standard deviation?

Accepted Answer

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

Question 14

A sample of exam scores is $$\{70, 75, 80, 85\}. If $$the mean is $$77.5$$, what is the sample variance?

Accepted Answer

$$s^2$$ = \frac{$$(70-77.5)^2$$ + $$(75-77.5)^2$$ + $$(80-77.5)^2$$ + $$(85-77.5)^2$$}{3} = \frac{56.25 + 6.25 + 6.25 + 56.25}{3} = \frac{125}{3} \approx 41.67$$

Question 15

In a real-world scenario, why might a company prefer to use standard deviation rather than range to assess the consistency of product weights?

Accepted Answer

Standard deviation accounts for all product weights, providing a more comprehensive measure of consistency.

Question 16

If a sample has values $$x = \{5, 7, 9, 11\}$$, what is the sample mean and sample variance?

Accepted Answer

Mean $$= 8$$, Variance $$= \frac{(5-8)^2$ + (7-8)^2$ + (9-8)^2$ + (11-8)^2$}{3} = \frac{9 + 1 + 1 + 9}{3} = \frac{20}{3} \approx 6.67$$

Question 17

Which of the following statements is TRUE regarding the calculation of sample variance?

Accepted Answer

Sample variance is calculated by dividing the sum of squared deviations by $$n-1.$$

Question 18

A sample of four values has a mean of $$10$$ and squared deviations of $$4, 1, 0, 9. $$What is the sample standard deviation?

Accepted Answer

$$s = \sqrt{\frac{4 + 1 + 0 + 9}{3}} = \sqrt{\frac{14}{3}} \approx 2.2$$

Measures of Variation: Range, Variance, and Standard Deviation

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