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Chapter 2: Q56E (page 103)

Answer the following questions about the variability of data sets:

a.What is the primary disadvantage of using the range to compare the variability of data sets?

b.Describe the sample variance using words rather than a formula. Do the same with the population variance.

c.Can the variance of a data set ever be negative? Explain. Can the variance ever be smaller than the standard deviation? Explain.

Short Answer

Expert verified

a) Same range but different data variation

b) Sample variance is the squared summation of the differences in observations from the mean divided by the number of observations minus one. Population variance is the summation of squared differences from the mean divided by the total number of observations.

c) No, the variance can never be negative. Yes, the variance can be smaller than the standard deviation.

Step by step solution

01

Disadvantage of using range to compare data variation

It is not advisable to use range to compare data variation because the range might be the same for two data sets. Still, there might be a very significant difference in the variation of the data.

For example, the range of heights in two groups of students may be the same, but there are chances that the height of students in group 1 is spread out symmetrically, and the height of students in group 2 is concentrated to the right.

02

Describing sample variance and population variance in words.

Sample variance is the squared summation of the differences of observations from the mean divided by the number of observations minus one.

Population variance is the summation of squared differences from the mean divided by the total number of observations.

03

Variance of a data set ever be negative or not

No, the variance cannot be negative because it is the square of deviations from the mean. And based on the basic mathematic rules, a square can never be negative.

The variance can be smaller than the standard deviation when the observations range between 0 and 1.

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Most popular questions from this chapter

Performance of stock screeners.Investment companies provide their clients with automated tools—called stock screeners—to help them select a portfolio of stocks to invest in. The American Association of Individual Investors (AAII) provides statistics on stock screeners at its Website, www.aaii.com. The next table lists the annualized percentage return on investment (as compared to the Standard & Poor’s 500 Index) for 13 randomly selected stock screeners. (Note:A negative annualized return reflects a stock portfolio that performed worse than the S&P 500.)

(9.0, -.1, -1.6, 14.6, 16.0, 7.7, 19.9, 9.8, 3.2, 24.8, 17.6, 10.7, 9.1)

a.Compute the mean for the data set. Interpret its value.

b.Compute the median for the data set. Interpret its value.

Give the percentage of measurements in a data set that are above and below each of the following percentiles:

a. 75th percentile

b. 50th percentile

c. 20th percentile

d. 84th percentile

Describe how the mean compares to the median for distribution as follows:

a.Skewed to the left

b.Skewed to the right

c.Symmetric

Monitoring weights of flour bags.When it is working properly, a machine that fills 25-pound bags of flour dispenses an average of 25 pounds per fill; the standard deviation of the amount of fill is .1 pound. To monitor the performance of the machine, an inspector weighs the contents of a bag coming off the machine’s conveyor belt every half hour during the day. If the contents of two consecutive bags fall more than 2 standard deviations from the mean (using the mean and standard deviation given above), the filling process is said to be out of control, and the machine is shut down briefly for adjustments. The data

given in the following table are the weights measured by the inspector yesterday. Assume the machine is never shut down for more than 15 minutes at a time. At what times yesterday was the process shut down for adjustment? Justify your answer.

Time

Weight (pounds)

8:00 a.m

25.10

8:30

25.15

9:00

24.81

9:30

24.75

10:00

25.00

10:30

25.05

11:00

25.23

11:30

25.25

12:00

25.01

12:30 p.m

25.06

1:00

24.95

1:30

24.80

2:00

24.95

2:30

25.21

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24.90

3:30

24.71

4:00

25.31

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25.15

5:00

25.20

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