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Describing Data:
Numerical Measures
Chapter 3
©The McGraw-Hill Companies, Inc. 2008
• Calculate the arithmetic mean, weighted mean, median,
mode, and geometric mean.
• Explain the characteristics, uses, advantages, and
disadvantages of each measure of location.
• Identify the position of the mean, median, and mode for
both symmetric and skewed distributions.
• Compute and interpret the range, mean deviation,
variance, and standard deviation.
• Understand the characteristics, uses, advantages, and
disadvantages of each measure of dispersion.
• Understand Chebyshev’s theorem and the Empirical
Rule as they relate to a set of observations.
Parameters v. Statistics
Definition: A parameter is a numerical characteristic
of a population.
Example: The fraction of U. S. voters who support
Sen. McCain for President is a parameter.
 Definition: A statistic is a numerical characteristic of
a sample.
Example: If we select a simple random sample of
n = 1067 voters from the population of all U. S.
voters, the fraction of people in the sample who
support Sen. McCain is a statistic.
Characteristics of the Mean
The arithmetic mean is the most widely used
measure of location. It requires the interval
scale. Its major characteristics are:
All values are used.
It is unique.
The sum of the deviations from the mean is 0.
It is calculated by summing the values and dividing by the
number of values.
Population Mean
For ungrouped data, the population mean is the
sum of all the population values divided by the
total number of population values:
EXAMPLE – Population Mean
Sample Mean
For ungrouped data, the sample mean
is the sum of all the sample values
divided by the number of sample
EXAMPLE – Sample Mean
Properties of the Arithmetic Mean
Every set of interval-level and ratio-level data has a mean.
All the values are included in computing the mean.
A set of data has a unique mean.
The mean is affected by unusually large or small data values.
The arithmetic mean is the only measure of central tendency
where the sum of the deviations of each value from the mean is
The Mean is Affected by Extreme
Suppose that, in the SunCom example, the data
value 119 is replaced by an extreme value, 229.
The mean of the original data set was 97.5.
The mean of the data set with this new extreme
value is 106.6667.
There are many situations in which a data set
naturally has some rather extreme values; e.g., data
on personal income – there are relatively few people
with extremely large values of personal income. The
few extreme values would affect the calculated value
of the mean.
Weighted Mean
The weighted mean of a set of numbers X1,
X2, ..., Xn, with corresponding weights w1,
w2, ...,wn, is computed from the following
Often each weight represents the number of
items in the data set having a particular
EXAMPLE – Weighted Mean
The Carter Construction Company pays its hourly
employees $16.50, $19.00, or $25.00 per hour.
There are 26 hourly employees, 14 of which are paid
at the $16.50 rate, 10 at the $19.00 rate, and 2 at the
$25.00 rate. What is the mean hourly rate paid the
26 employees?
The Median
The Median is the midpoint of the values
after they have been ordered from the
smallest to the largest.
– There are as many values above the median as below it in
the data array.
For an even set of values, the median will be the arithmetic
average of the two middle numbers.
Properties of the Median
There is a unique median for each data set.
It is not affected by extremely large or small
values and is therefore a valuable measure
of central tendency when such values occur.
It can be computed for ratio-level, intervallevel, and ordinal-level data.
It can be computed for an open-ended
frequency distribution if the median does not
lie in an open-ended class.
The ages for a sample of
five college students
21, 25, 19, 20, 22
Arranging the data in
ascending order gives:
The heights of four
basketball players, in
inches, are:
76, 73, 80, 75
Arranging the data in
ascending order gives:
73, 75, 76, 80.
19, 20, 21, 22, 25.
Thus the median is 75.5
Thus the median is 21.
Example: Using the Median When
There are Extreme Values
The U.S. Department of Commerce Bureau of Labor
Statistics regularly publishes information about the
distribution of personal incomes in the U.S. This
distribution, of course, has a floor value of $0.00,
and a relatively few number of extremely large
values (Think Bill Gates). Hence the Bureau uses
the median income, rather than the mean, as the
appropriate measure of central tendency.
The Mode
The mode is the value of the observation
that appears most frequently.
Example - Mode
Mode of Categorical Data
There is one situation in which the mode is the only
measure of central tendency that can be used –
when we have categorical, or non-numeric data. In
this situation, we cannot calculate a mean or a
median. The mode is the most typical value of the
categorical data.
Example: Suppose I have collected data on religious
affiliation of citizens of the U.S. The modal, or most
Typical value, is Roman Catholic, since The Roman
Catholic Church is the largest religious organization in
the U.S.
Mean, Median, Mode Using Excel
Table 2–4 in Chapter 2 shows the prices of the 80 vehicles sold last month at Whitner Autoplex in
Raytown, Missouri. Determine the mean and the median selling price. The mean and the median
selling prices are reported in the following Excel output. There are 80 vehicles in the study. So the
calculations with a calculator would be tedious and prone to error.
Mean, Median, Mode Using Excel
Example: Test Score Data
We have 25 scores on a final exam, as follows:
86, 83, 56, 98, 82, 52, 71, 88, 75, 91, 69, 88, 64, 78,
81, 74, 77, 83, 90, 85, 64, 79, 71, 83, 64.
We want to calculate the mean, median, and mode for
this data, and to look at the relationship among them.
We enter the data in an Excel spreadsheet, click on the
MegaStat pull-down menu, choose Descriptive
Statistics, enter the appropriate range of the input list,
and choose which descriptive statistics we want.
The Relative Positions of the Mean,
Median and the Mode
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 geometric mean of a set of n positive numbers is defined as the
nth root of the product of n values.
The formula for the geometric mean is written:
EXAMPLE – Geometric Mean
Suppose you receive a 5 percent increase in
salary this year and a 15 percent increase
next year. The average annual percent
increase is 9.886, not 10.0. Why is this so?
We begin by calculating the geometric mean.
GM  ( 1.05 )( 1.15 )  1.09886
EXAMPLE – Geometric Mean (2)
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?
GM  4 ( 1.3 )( 1.2 )( 0.6 )( 3.0 )  4 2.808  1.294
Example: Geometric Mean (3)
The 2006 population size of Duval County was
837,964. The population grew by 7.6% between 2000
and 2006. We want to project the size of the
population in 2030, assuming that the growth rate
remains the same; i.e., 7.6% every 6 years. The
Projected population size in 2030 is (1.0764 X 837,964)
= 1123245. The average growth rate over the 24 years
is found by calculating the geometric mean:
GM  4 1.0761.0761.0761.076  1.076
The average growth rate is just what we expect.
Why Study 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.
Samples of Dispersions
Measures of Dispersion
Mean Deviation
Variance and Standard
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.
Range = Largest – Smallest value
= 80 – 20 = 60
EXAMPLE – Mean Deviation
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.
EXAMPLE – Variance and Standard
The number of traffic citations issued during the last five months in
Beaufort County, South Carolina, is 38, 26, 13, 41, and 22. What
is the population variance?
EXAMPLE – Sample Variance
The hourly wages for
a sample of parttime employees at
Home Depot are:
$12, $20, $16, $18,
and $19. What is
the sample
Why Variance and Standard Deviation?
Both the variance and the standard deviation give
the same information about the dispersion of the data
values. Why have both? They are used for different
purposes. There is a branch of statistics called
Analysis of Variance, having to do with analyzing
cause-and-effect relationships for experimental data.
The standard deviation is often a more useful measure
of variability, because its value “looks reasonable” as a
measure of variability, and because it has the same
unit of measurement as the data values themselves.
Example: Variance and Standard
The household grocery data set (p. 46, Exercise 30).
 The range of values is $570 – $41 = $529.
 The variance is 11,735.98 square dollars (?).
 The standard deviation is $108.33.
The standard deviation looks like a more “reasonable”
measure of spread of the data, being about one-fifth of
the range.
Example: Sample Range, Variance,
and Standard Deviation
For the test score data, the range is 98 – 52 = 46.
The variance is 127.79, and the standard deviation is
11.30. All of these values may be found using
MegaStat. Note that the numeric value of the variance
is not readily interpretable in this case, while the
standard deviation makes somewhat more sense
as a measure of the spread of the data values. The
calculated value is about one-fifth of the range of the
data values.
Chebyshev’s Theorem
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?
The Empirical Rule
Example: Stanford-Binet IQ Test
The Stanford-Binet IQ test is constructed and scored
so that the score histogram for the entire adult
population is bell-shaped, the population mean is 100,
and the population standard deviation is 16.
What fraction of the adult population have IQ scores
between 84 and 116? Between 52 and 148?
The Arithmetic Mean of Grouped Data
We can obtain an approximate sample mean for
grouped data if we have the grouped frequency table
for the data.
The Arithmetic Mean of Grouped Data Example
Recall in Chapter 2, we
constructed a frequency
distribution for the vehicle
selling prices. The
information is repeated
below. Determine the
arithmetic mean vehicle
selling price.
The Arithmetic Mean of Grouped Data Example
Standard Deviation of Grouped Data
Standard Deviation of Grouped Data Example
Refer to the frequency distribution for the Whitner Autoplex data
used earlier. Compute the standard deviation of the vehicle
selling prices
End of Chapter 3
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