Deck 5: Measures of Variability

Question 1

The vertical line in the center of a box plot

A) represents the sample mean.
B) represents the sample median.
C) serves to anchor the box.
D) can represent anything you want it to.

Answer 1

represents the sample median.

Question 2

The difference between s and σ is that σ is

A) the value of the standard deviation in a sample.
B) the long range average of the variance over repeated sampling.
C) the biased estimate of s.
D) the value of the standard deviation in a population.

Answer 2

the value of the standard deviation in a population.

Question 3

If the whiskers on a boxplot are much longer on the right than on the left, we would suspect that the distribution is

A) positively skewed.
B) negatively skewed.
C) symmetric.
D) distorted.

Answer 3

positively skewed.

Question 4

People in the stock market refer to a measure called the "standard deviation," although it is calculated somewhat differently from the one discussed here. It is a good guess that this measure refers to

A) the riskiness of the stock.
B) the value of the stock.
C) how much the stock price is likely to fluctuate.
D) how much money you are likely to earn from buying that stock.

Question 5

If we multiply a set of data by a constant, such as converting feet to inches, we will

A) leave the mean and variance unaffected.
B) multiply the mean and the standard deviation by the constant.
C) multiply the mean by the constant but leave the standard deviation unchanged.
D) leave the mean unchanged but alter the standard deviation.

Question 6

What do we mean by an unbiased statistic?

A) a statistic that equals the sample mean
B) a statistic whose average is very stable from sample to sample
C) a statistic used to measure racial diversity
D) a statistic whose long range average is equal to the parameter it estimates

Question 7

When calculating the standard deviation we divide by N -1 rather than N because the result is

A) smaller.
B) less biased.
C) easier to interpret.
D) equal to the population mean.

Question 8

We generally like the standard deviation when we are trying to describe a sample of data because

A) it is larger than the variance.
B) it allows for more intuitive interpretation with respect to the data than does the variance.
C) it is less biased than the variance.
D) all of the above

Question 9

The variance can best be thought of as the

A) average of the squared deviations from the mean.
B) average of the absolute deviations from the mean.
C) average of the deviations from the median.
D) square of the mean.

Question 10

A trimmed sample is one that

A) has been distorted by lopping off the highest scores.
B) is strongly influenced by outliers.
C) is unrepresentative of what it purports to measure.
D) has been truncated equally at the two extremes.

Question 11

Data points that lie outside the whiskers in a boxplot are often referred to as

A) incorrect values.
B) outliers.
C) representative values.
D) deviates.

Question 12

Which of the following is NOT a measure of variability?

A) the density
B) the range
C) the standard deviation
D) the interquartile range

Question 13

Data points at the extremes of the distribution have

A) little effect on the variance.
B) more effect on the variance than scores at the center of the distribution.
C) are undoubtedly incorrect.
D) distort the usefulness of the median.

Question 14

Dispersion refers to

A) the degree to which data cluster toward one end of the scale.
B) the centrality of the distribution.
C) the degree to which individual data points are distributed around the mean.
D) all of the above

Question 15

Which of the following sets of data is likely to have the smallest standard deviation?

A) the distribution of SAT scores for students from your high school
B) the distribution of heights of students in an elementary school
C) the grade point averages of students from your high school's honors biology class
D) the amount that you and your friends pay for college tuition

Question 16

The whiskers in a boxplot

A) always enclose all of the data points.
B) always run from the smaller inner fence to the larger inner fence.
C) encompass the H-spread only.
D) contain all data points outside the box except the outliers.

Question 17

In a boxplot the width of the box encompasses

A) all of the observed values.
B) all but the most extreme values.
C) approximately 50% of the observed values.
D) the center-most 10% of the values.

Question 18

The population variance is

A) an estimate of the sample variance.
B) calculated exactly like the sample variance.
C) a biased estimate.
D) usually an unknown that we try to estimate.

Question 19

If we eliminate the top and bottom 25% of the data and take the range of what remains we have the

A) range.
B) adjusted range.
C) interquartile range.
D) quartile variance.

Question 20

An outlier

A) can be an extreme score.
B) can be an error that snuck into the data.
C) will never have a large influence on many measures of variability.
D) both a and b

Question 21

The problem with measuring dispersion by merely averaging all the deviations between each score and the overall mean is that

A) positive and negative deviations will balance out.
B) squared values make intuitive interpretation difficult.
C) dividing by (N-1) gives a biased statistic.
D) There are no problems with measuring dispersion this way.

Question 22

The equation
is used to calculate the

A) median.
B) hinge location.
C) outer fence.
D) inner fences.

Question 23

The range is

A) the difference between the inner fences.
B) the H-spread.
C) not influenced very much by outliers.
D) the difference between the highest and lowest score.

Question 24

The university counseling center has treated a large number of students for depression. They find that the standard deviation of depression scores for their pool of students is substantially higher after treatment than before treatment. The most likely explanation is

A) some students improved more than others.
B) some students improved substantially while others actually got worse.
C) depression therapy at the counseling center affects different students differently.
D) all of the above

Question 25

A "hinge" is another word for

A) the median.
B) a quartile.
C) the range.
D) boundary.

Question 26

A boxplot is better than a statistic such as the mean when your purpose is

A) to describe the central tendency of a population.
B) to describe the variability of a population.
C) to understand what a distribution of data looks like.
D) It is only worthwhile if you care only about medians.

Question 27

The disadvantage of using an interquartile range is that

A) it discards too much of the data.
B) it removes outliers only extremely high in value.
C) the positive and negative deviations balance out.
D) it is disproportionately influenced by outliers.

Question 28

Given the numbers 1, 2, and 3, the standard deviation is

A) 0
B) 1
C) 0.667
D) the square of the variance

Question 29

If the average adult male in the United States is 5' 9" tall, and the standard deviation for height is 2", approximately how many adult males would you expect to be between 5' 7" and 5'11" tall?

A) 50% of them
B) 66.7% of them
C) 75% of them
D) 90% of them

Question 30

If I continue to draw observations from a population and recalculate the mean each time I add an observation, the mean will approach _______ as the sample size increases.

A) its expected value
B) the true population mean
C) the median of the population if the population is symmetric
D) all of the above

Question 31

As you increase the number of observations in a sample from 50 to 500, you are most likely to

A) leave the mean and standard deviation approximately unchanged.
B) increase the variability as the sample size increases.
C) decrease the variability as the sample size increases.
D) make the shape of the distribution more skewed.

Question 32

Which of the following is NOT a method of describing data that reduces the role of outliers on the measurement of a data set's variability?

A) interquartile range
B) boxplot
C) range
D) trimmed statistics

Question 33

The US Census Bureau collected data on family composition and found that samples from different parts of the country gave very different results for the mean number of family members living in households. If all of the data were combined to one data set,

A) the standard deviation of number of family members would probably be very small.
B) the standard deviation of number of family members would probably be relatively high.
C) the interquartile range would be small.
D) the median would equal the mean.

Question 34

Errors that can lead to outliers can occur in

A) measurement.
B) data recording.
C) data entry.
D) all of the above

Question 35

You would obtain a negative value for the variance if

A) all observations were at the mean.
B) the distribution is very negatively skewed.
C) the distribution if positively skewed.
D) you would never obtain a negative variance.

Question 36

If we know that a set of test scores has a mean of 75 and a standard deviation of 8, we would conclude that

A) the average deviation from the mean is about 8 points.
B) the average person will have a score of 75 + 8 = 83.
C) more people are above 75 than below it.
D) You can't tell anything about how scores lie relative to the mean.

Question 37

We normally compute the variance using N - 1 in the denominator because

A) it is easier that way.
B) it leads to an unbiased estimate of the sample variance.
C) it leads to an unbiased estimate of the population variance.
D) it overestimates that population variance.

Question 38

The standard deviation for the numbers 8, 9, and 10 is

A) -3.0
B) 0.0
C) .67
D) 1.0

Question 39

A data set of intelligence scores was collected from high school seniors. The IQ scores ranged from 82 to 113. Which of the following is probably NOT a reasonable estimate of the standard deviation?

A) 6.2
B) 4.7
C) 35.4
D) All of the above are reasonable estimates.

Question 40

The interquartile range

A) is the 50^th percentile score in a data set.
B) contains as few as 25% of scores or as many as 75% of scores in a data set
C) contains the middle 50% of scores in a data set.
D) is the same as the range.

Question 41

The variance of a sample is typically a larger value than the standard deviation.

Question 42

Answer the next two questions based on the following box-plot.

The median of this distribution is 16.

Question 43

Trimmed statistics are calculated based on the entire sample.

Question 44

Measures of variability refer to the dispersion of data around the mean or the center.

Question 45

Of all of the measures of variability, the standard deviation is most susceptible to distortion due to outliers.

Question 46

A sample of 20 families reported how many children they have. Answer the following questions based on the summary table below.

a) What is the range?
b. What is the variance?
c. What is the standard deviation?

Question 47

Answer the following questions based on this set of numbers:
1               2               2               3               3               3               4               5
 
a. What is the range?
b. What is the variance?
c. What is the standard deviation?

Question 48

The sample variance is a biased statistic.

Question 49

Create a box plot for the above data.

Question 50

Answer the following questions based on this distribution of exam scores.

a. What is the median?
b. Are there outliers?
c. Does the distribution seem skewed?   If so, is it positively, or negatively skewed?

Question 51

The population variance is

A) an estimate of the sample variance.
B) usually an unknown that we try to estimate.
C) calculated exactly like the sample variance.
D) a biased estimate.

Question 52

The interquartile range is the range of the middle 25% of values.

Question 53

What happens to the standard deviation when a constant is added to each score? Use the following set of data, and a constant of 2 to illustrate your answer.
1               2               3               4

Question 54

The median is a measure of variability.

Question 55

The difference between the lowest to the highest score in a distribution is the range.

Question 56

Answer the next two questions based on the following box-plot.

There no outliers.

Question 57

Data points at the extremes of the distribution have

A) little effect on the variance.
B) distort the usefulness of the median.
C) more effect on the variance than scores at the center of the distribution.
D) are undoubtedly incorrect.

Question 58

Compare the distribution of exam scores for students who did and did not read the textbook prior to taking the exam. Discuss measures of variability and of central tendency.

Question 59

Based on the same data, calculate:
a. The median location
b. The median
c. The hinge location
d. The upper hinge
e. The lower hinge
f. H spread
g. Lower fence
h. Upper fence
i. Lower adjacent value
j. Upper adjacent value

Question 60

Create two sets of scores with equal ranges, but different variances.

Question 61

Given the following distribution, which would be the least useful measure of central tendency? Explain your answer.

Question 62

Construct two small sets of data that have the same mean, but a different standard deviation.