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LOS m. compare the use of arithmetic mean or geometric mean when analyzing investment returns.
Measures of center tendency.
Measures of central tendency specify where the data are centered. They attempt to use a typical value to represent all the observations in the data set.

Population Mean

The average for a finite population. It is unique; that is, a given population has only one mean.

where

  • N = the number of observations in the entire population.
  • Xi = the ith observation
  • ΣXi = add up Xi, where i is from 0 to N.

Sample Mean

The average for a sample. It is a statistic and used to estimate the population mean.

where n = the number of observations in the sample.

Arithmetic Mean

It is what is commonly called the average. The population mean and sample mean are both examples of the arithmetic mean:

  • If the data set encompasses an entire population, the arithmetic mean is called a population mean.
  • If the data set includes a sample of values taken from a population, the arithmetic mean is called a sample mean.

It is the most widely used measure of central tendency. When the word "mean" is used without a modifier, it can be assumed that it refers to the arithmetic mean. The mean is the sum of all the scores divided by the number of scores. It is used to measure prospective (expected future) performance (return) of an investment over a number of periods.

  • All interval and ratio data sets (e.g. incomes, ages, rates of return) have an arithmetic mean.
  • All data values are considered and included in the arithmetic mean computation.
  • A data set has only one arithmetic mean. This says that the mean is unique.
  • The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is always zero. Deviation from the arithmetic mean is the distance between the mean and an observation in the data set.

The arithmetic mean has the following disadvantages:

  • The mean can be affected by extremes, that is, unusually large or small values.
  • The mean cannot be determined for an open-ended data set (i.e., n is unknown).

Geometric Mean

It has three important properties:

  • It exists only if all the observations are greater than or equal to zero. In another word, it cannot be determined if any value of the data set is zero or negative.
  • If values in the data set are all equal, both the arithmetic and geometric means will be equal to that value.
  • It is always less than the arithmetic mean if values in the data set are not equal.

It is typically used when calculating returns over multiple periods. It is a better measure of the compound growth rate of an investment. When returns are variable by period, the geometric mean will always be less than the arithmetic mean. The more disperse the rates of returns, the greater the difference between the two. It is not so highly influenced by extreme values as is the arithmetic mean.

Weighted Mean

It is computed by weighting each observed value according to its importance. In contrast, the arithmetic mean assigns equal weight to each value. Notice that the return of a portfolio is the weighted mean of the returns of the individual assets in the portfolio. The assets are weighted on their market values relative to the market value of the portfolio. When we take a weighted average of forward-looking data, the weighted mean is called expected value.

Example

A year ago, a certain share had a price of $6. Six months ago, the same share had a price of $6.20. The share is now trading at $7.50. Because the recent price is the most reliable, we decide to attach more relevance to this value. So, suppose we decide to "weight" the prices in the ratio 1:2:4, so that the current share price is twice as important as the price six months back, which in turn is twice as important as the price last year.

The weighted mean would then be: (1 x 6 + 2 x 6.2 + 4 x 7.5) / (1 + 2 + 4) = $6.91. If we just calculated the mean without weights, we'd get: (6 + 6.2 + 7.5) / 3 = $6.57. So the fact that we've given more importance to the most recent (higher) share price inflates the weighted mean relative to the un-weighted mean.

Median

In English, the word "mediate" means to go between or to stand in the middle of two groups, in order to act as a referee, so to speak. The median does the same thing; it is the value that stands in the middle of the data set, and divides it into two equal halves, with an equal number of data values in each half.

To determine the median, arrange the data from the highest to the lowest (or lowest to highest) and find the middle observation. If there are an odd number of observations in the data set, the median is the middle observation (n + 1)/2 of the data set. If the number of observations is even, there is no single middle observation (there are two actually). To find the median, take the arithmetic mean of the two middle observations.

Unlike the mean, the median is less sensitive to extreme scores than the mean. This makes it a better measure than the mean for highly skewed distributions. The median income is usually more informative than the mean income, for example. The sum of the absolute deviations of each number from the median is lower than is the sum of absolute deviations from any other number.

Note that whenever you calculate a median, it is imperative that you place the data in order first. It does not matter whether you order the data from smallest to biggest or from biggest to smallest, but it does matter that you do order the data.

Mode

Mode means fashion. So the mode is the "most fashionable" number in the data set: it is the most frequently occurring score in a distribution and is used as a measure of central tendency. A set of data can have more than one mode, or even no mode. When all values are different, the data set has no mode. When a distribution has one value that appears most frequently, it is said to be unimodal. A data set that has two modes is said to be bimodal.

The advantage of the mode as a measure of central tendency is that its meaning is obvious. Like the median, the mode is not affected by extreme values. Further, it is the only measure of central tendency that can be used with nominal data. The mode is greatly subject to sample fluctuations and, therefore, is not recommended to be used as the only measure of central tendency. A further disadvantage of the mode is that many distributions have more than one mode. These distributions are called "multimodal".

Harmonic Mean

The harmonic mean of n numbers xi (where i = 1, 2, ..., n) is:

The special cases of n = 2 and n = 3 are given by:

and so on.

For n = 2, the harmonic mean is related to the arithmetic mean A and geometric mean G by:

The mean, median, and mode are equal in symmetric distributions. The mean is higher than the median in positively skewed distributions and lower than the median in negatively skewed distributions. Extreme values affect value of mean, while median is less affected by outliers. Mode helps to identify shape and skewness of distribution.
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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Practice Question 30

When using stock return data, a geometric mean return calculation is most likely preferred over a geometric mean calculation because:

A. return data can be less than one.
B. return data can be negative.
C. the geometric mean return is closer in value to the arithmetic mean.

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Practice Question 31

An analyst collects the following set of past stock returns: -2.3%, -5.1%, 7.6%, 8.2%, 9.1%, and 9.8%. Which of the following measures of return is most likely the highest?

A. Arithmetic mean return
B. Geometric mean return
C. Median return

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