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Subject 4. Measures of center tendency.

This subject is for the following CFA Learning Outcome Statement(s):

e. calculate and interpret measures of central tendency, including the population mean, sample mean, arithmetic mean, weighted average or mean, geometric mean, harmonic mean, median, and mode;
m. compare the use of arithmetic mean or geometric mean when analyzing investment returns.

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Practice Question 1

Mileage tests were conducted on a randomly selected sample of 100 newly developed automobile tires. The average tread wear was found to be 50,000 miles with a standard deviation of 3,500 miles. What is the best estimate of the average tread life in miles for the entire population of these tires?

A. 50,000
B. (3,500/100)
C. (50,000/100).

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Practice Question 2

Suppose you are taking a class in which your final grade is figured as follows: 50% from your test mean, 25% from your final exam score, 15% from your quiz mean, and 10% from your homework grade. If your grades are 80 (test mean), 88 (final exam), 75 (quiz mean), and 98 (homework), what is the weighted mean of your scores?

A. 84.72
B. 80.33
C. 83.05

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Practice Question 3

Find the measures of central tendencies (mean, median, mode) for the following lengths of operating times in months of 15 randomly selected car batteries: 18, 21, 22, 22, 23, 23, 23, 25, 25, 26, 27, 30, 30, 31, 32.

A. 23, 25, 32
B. 25.2, 23, 25
C. 25.2, 25, 23

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Practice Question 4

From March 1981 to February 1983 the number of burglaries committed each month in a Georgia town was recorded. They are given in the chart below. Between months 12 and 13 a law enacted requiring citizens to own a gun. Town officials felt this law might decrease the number of burglaries, by acting to deter criminals.

The mean number of burglaries for months 13 to 24, i.e., those months after the law was enacted, is

A. 2.83
B. 3.38
C. 2.5

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Practice Question 5

For a physics course containing 11 students, the maximum point total for the quarter was 200. The point totals for the 11 student are given in the stem plot below.

Which of the following statements is/are true about the stem plot?

I. Because there are 7 stems, the median must be contained in the 4th smallest stem and be between 140 and 150.
II. We can compute the mean number of points for the 11 students from the information in the stem plot.
III. Because of the small sample size, it would be impossible to split the stems in this stem plot.

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Practice Question 6

You receive a fax with 6 bids (in millions of dollars): 2.2, 1.3, 1.9, 1.2, 2.4 and x, where x is some number that is too blurry to read. Without knowing what x is, the median

A. is 1.9.
B. must be between 1.3 and 2.2.
C. could be any number between 1.2 and 2.4.

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Practice Question 7

For the grouped frequency distribution shown below, which of the following is false?

A. the median is in class #3
B. the median is in class #4
C. the exact value of the minimum cannot be determined

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Practice Question 8

For a symmetric distribution, the mean and median are

A. the same
B. always different
C. possibly the same, possibly different.

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Practice Question 9

Which of the following measures of central tendency tends to be most influenced by an extreme score?

A. median
B. mode
C. mean

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Practice Question 10

In a group of 12 scores, the largest score is increased by 36 points. What effect will this have on the mean of the scores?

A. it will be increased by 12 points
B. it will be increased by 3 points
C. there is no way of knowing exactly how many points the mean will be increased.

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Practice Question 11

The mean is a measure of:

I. variability.
II. position.
III. skewness.
IV. central tendency.
V. symmetry.

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Practice Question 12

If a teacher computes the mean for a set of test scores and then subtracts this mean from each score, the SUM of the resulting set of difference scores will equal

A. zero.
B. the mean.
C. n times the mean.

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Practice Question 13

A stockbroker placed the following order:

50 shares of Kaiser Aluminum preferred at $104 a share
100 shares of GTE preferred at $25 1/4 a share
20 shares of Boston Edison preferred at $9 1/8 a share

What is the weighted mean price per share?

A. $25.25
B. $103.5
C. None of these answers

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Practice Question 14

If one or more numbers in a dataset are negative, which of the following is not always defined?

A. median
B. mode
C. geometric mean

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Practice Question 15

An employee had the following percentage increases in salary over the last 5 years: 4%, 7%, 10%, 15%, 12%. The geometric mean of his salary increases equals ________.

A. 9.22%
B. 8.72%
C. 9.53%

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Practice Question 16

The following data represents the annualized total returns in percentages for twelve bond funds.

14.2, 12.7, 11.7, 11.5, 11.2, 10.8, 10.0, 9.2, 9.1, 8.4, 7.9, -2.1

The median of the distribution is closest to which of the following?

A. 9.47
B. 9.55
C. 10.4

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Practice Question 17

You are examining the market prices of precious metals. The current prices per ounce are 433, 565, 20, 0.005, and 50. What is the range of precious metal prices?

A. 19.995.
B. 432.995.
C. 564.995.

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Practice Question 18

For the following frequency distribution:

The median of the distribution is:

A. 24.5
B. 25.5
C. 26.5

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Practice Question 19

In what sense is the mean of any distribution the "best guess" of the score of any single individual selected at random from the group?

I. In a series of such guesses, the sum of the errors in one direction will balance the sum of the errors in the other direction.
II. The mean score will occur more often than any other single score.
III. The chances are 50-50 that any individual will be above or below the mean.

A. I only.
B. II and III.
C. I and II.

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Practice Question 20

Consider the following data: 1, 7, 3, 3, 6, 4

The mean and median for this data are:

A. 4.8 and 3
B. 4 and 3 1/2
C. 4 and 3 1/3

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Practice Question 21

A distribution of 6 scores has a median of 21. If the highest score increases 3 points, the median will become _______.

A. 21
B. 24
C. Cannot be determined without additional information.

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Practice Question 22

The measure of central tendency which is sensitive to extreme scores on the higher or lower end of a distribution is the:

A. median.
B. mean.
C. mode.

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Practice Question 23

Assume a student received the following grades for the semester: History, B; Statistics, A; Spanish, C; and English, C. History and English are 5 credit hour courses, Statistics a 4 credit hour course and Spanish a 3 credit hour course. If 4 grade points are assigned for an A, 3 for a B and 2 for a C, what is the weighted mean for the semester grade?

A. 2.88
B. 2.76
C. 3.01

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Practice Question 24

What is the median of 26, 30, 24, 32, 32, 31, 27 and 29?

A. 29.5
B. 29
C. 30

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Practice Question 25

Which measures of central tendency are not affected by extremely low or extremely high values?

A. Geometric mean and mean
B. Mode and median
C. Mean and median

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Practice Question 26

The difference between an arithmetic average and a geometric average of returns:

A. increases as the variability of the returns increases.
B. increases as the variability of the returns decreases.
C. depends on the specific returns being averaged, but is not necessarily sensitive to their variability.

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Practice Question 27

Suppose that four different portfolios have produced the following returns over the past year: 10%, 30%, 5% and 15%.

I. The mean return is 15%.
II. The range is 25%.
III. The mean absolute deviation is 7.5%.
IV. The variance is 87.5%2.
V. The standard deviation is 9.354%.

Which statement(s) is/are FALSE?

A. II and IV only.
B. III and V only.
C. None of the above.

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Practice Question 28

What is the median of the following observations: 34,21,89,45,66,11,38,57?

A. 41.5
B. 55.5
C. 23.0

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Practice Question 29

Which of the following is most likely correct?

A. If the geometric mean is negative, it means that the arithmetic mean must also be negative.
B. The geometric mean return is a more appropriate measure of return compared to the arithmetic mean return.
C. If the arithmetic mean returns is positive, it implies that the return over the entire period is positive.

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