2. Learning Objectives
LO1 Explain the concept of central tendency.
LO2 Identify and compute the arithmetic mean.
LO3 Compute and interpret the weighted mean.
LO4 Determine the median.
LO5 Identify the mode.
LO6 Calculate the geometric mean.
LO7 Explain and apply measures of dispersion.
LO8 Compute and interpret the standard deviation.
LO9 Explain Chebyshev’s Theorem and the Empirical
Rule.
LO10 Compute the mean and standard deviation of
grouped data.
3-2
4. Population Mean
For ungrouped data, the population mean is the sum of all the population values divided by the total number of
population values. The sample mean is the sum of all the sample values divided by the total number of sample
values.
EXAMPLE:
LO2 Identify and compute
the arithmetic mean.
LO1 Explain the concept
of central tendency
3-4
5. Properties of the Arithmetic Mean
Every set of interval- or ratio-level data has a mean.
Recall from Chapter 1 that ratio-level data include such
data as ages, incomes, and weights, with the distance
between numbers being constant.
All the values are included in computing the mean.
The mean is unique. That is, there is only one mean in
a set of data. Later in the chapter, we will discover an
average that might appear twice, or more than twice, in a
set of data.
The sum of the deviations of each value from the
mean is zero.
7. The Median
PROPERTIES OF THE MEDIAN
1. There is a unique median for each data set.
2. It is not affected by extremely large or small values and is therefore a valuable measure of central tendency
when such values occur.
3. It can be computed for ratio-level, interval-level, and ordinal-level data.
4. It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended
class.
EXAMPLES:
MEDIAN The midpoint of the values after they have been ordered from the smallest to the largest, or the largest to
the smallest.
The ages for a sample of five college students are:
21, 25, 19, 20, 22
Arranging the data in ascending order gives:
19, 20, 21, 22, 25.
Thus the median is 21.
The heights of four basketball players, in inches, are:
76, 73, 80, 75
Arranging the data in ascending order gives:
73, 75, 76, 80.
Thus the median is 75.5
LO4 Determine the median.
3-7
8. The Mode
MODE The value of the observation that appears most frequently.
LO5 Identify the mode.
3-8
10. The Geometric Mean
Useful in finding the average change of percentages, ratios, indexes, or growth rates over time.
It has a wide application in business and economics because we are often interested in finding the
percentage changes in sales, salaries, or economic figures, such as the GDP, which compound or
build on each other.
The geometric mean will always be less than or equal to the arithmetic mean.
The formula for the geometric mean is written:
LO6 Calculate the geometric mean.
EXAMPLE:
The return on investment earned by Atkins Construction Company for four successive years was: 30
percent, 20 percent, -40 percent, and 200 percent. What is the geometric mean rate of return on
investment?
3-10
11. The Geometric Mean
A second application of the geometric
mean is to find an average percentage
change over a period of time.
For example, if you earned $30,000 in
2000 and $50,000 in 2010, what is your
annual rate of increase over the period?
12. The Geometric Mean
Answer: It is 5.24 percent.
The rate of increase is determined from
the following formula.
13. EXAMPLE
During the decade of the 1990s, and into the 2000s, Las
Vegas, Nevada, was the fastest-growing city in the
United States. The population increased from 258,295 in
1990 to 607,876 in 2009. This is an increase of 349,581
people, or a 135.3 percent increase over the period. The
population has more than doubled. What is the average
annual percent increase?
14. SOLUTION
There are 19 years between 1990 and 2009, so n = 19.
Then formula (3–5) for the geometric mean as applied to
this problem is:
The value of .0461 indicates that the average annual
growth over the period was 4.61 percent.
To put it another way, the population of Las Vegas
increased at a rate of 4.61 percent per year from 1990 to
2009.
15. The Weighted Mean
The weighted mean is a special case of
the arithmetic mean.
It occurs when there are several
observations of the same value.
16. Harmonic Mean
The harmonic mean is a type of numerical
average.
It is calculated by dividing the number of
observations by the reciprocal of each
number in the series.
Thus, the harmonic mean is the
reciprocal of the arithmetic mean of the
reciprocals.
17. Harmonic Mean
The harmonic mean of 1, 4, and 4 is:
Solution:
3/(1/1+1/4+1/4) = 3/1.5 = 2
The harmonic mean helps to find multiplicative or
divisor relationships between fractions without
worrying about common denominators.
Harmonic means are often used in averaging things
like rates (e.g., the average travel speed given a
duration of several trips).
18. Harmonic Mean
The harmonic mean is the weighted harmonic mean,
where the weights are equal to 1. The weighted
harmonic mean of x1, x2, x3 with the corresponding
weights w1, w2, w3 is given as:
19. Harmonic Mean
KEY TAKEAWAYS:
The harmonic mean is the reciprocal of the
arithmetic mean of the reciprocals.
Harmonic means are used in finance to
average data like price multiples.
Harmonic means can also be used by
market technicians to identify patterns
such as Fibonacci sequences.
20. Measures of Dispersion
A measure of location, such as the mean or the median, only describes the center of the data. It is valuable from
that standpoint, but it does not tell us anything about the spread of the data.
For example, if your nature guide told you that the river ahead averaged 3 feet in depth, would you want to wade
across on foot without additional information? Probably not. You would want to know something about the variation
in the depth.
A second reason for studying the dispersion in a set of data is to compare the spread in two or more distributions.
RANGE
MEAN DEVIATION
VARIANCE AND STANDARD DEVIATION
LO7 Explain and apply
measures of dispersion.
3-20
21. EXAMPLE – Mean Deviation
EXAMPLE:
The number of cappuccinos sold at the Starbucks location in the Orange Country
Airport between 4 and 7 p.m. for a sample of 5 days last year were 20, 40, 50, 60,
and 80. Determine the mean deviation for the number of cappuccinos sold.
Step 1: Compute the mean
Step 2: Subtract the mean (50) from each of the observations, convert to positive if difference
is negative
Step 3: Sum the absolute differences found in step 2 then divide by the number of
observations
50
5
80
60
50
40
20
n
x
x
LO7
3-21
22. Variance and Standard Deviation
The variance and standard deviations are nonnegative and are zero only
if all observations are the same.
For populations whose values are near the mean, the variance and
standard deviation will be small.
For populations whose values are dispersed from the mean, the
population variance and standard deviation will be large.
The variance overcomes the weakness of the range by using all the
values in the population
VARIANCE The arithmetic mean of the squared deviations from the mean.
STANDARD DEVIATION The square root of the variance.
LO7
3-22
23. EXAMPLE – Population Variance and Population
Standard Deviation
The number of traffic citations issued during the last five months in Beaufort County, South Carolina,
is reported below:
What is the population variance?
Step 1: Find the mean.
Step 2: Find the difference between each observation and the mean, and square that difference.
Step 3: Sum all the squared differences found in step 3
Step 4: Divide the sum of the squared differences by the number of items in the population.
29
12
348
12
10
34
...
17
19
N
x
124
12
488
,
1
)
( 2
2
N
X
LO7
3-23
24. Sample Variance and
Standard Deviation
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EXAMPLE
The hourly wages for a sample of part-time
employees at Home Depot are: $12, $20,
$16, $18, and $19.
What is the sample variance?
LO7
3-24
25. Chebyshev’s Theorem and Empirical Rule
The arithmetic mean biweekly amount
contributed by the Dupree Paint
employees to the company’s profit-
sharing plan is $51.54, and the standard
deviation is $7.51. At least what percent
of the contributions lie within plus 3.5
standard deviations and minus 3.5
standard deviations of the mean?
LO9 Explain Chebyshev’s
Theorem and the Empirical Rule.
3-25
26. The Arithmetic Mean and Standard Deviation of Grouped Data
EXAMPLE:
Determine the arithmetic mean vehicle
profit given in the frequency table
below.
EXAMPLE
Compute the standard deviation of the
vehicle profit in the frequency table below.
LO10 Compute the mean and
standard deviation of grouped data.
3-26
