§3.4–Descriptive Measures for Populations
Use of Samples
Tom Lewis
Fall Term 2009
Tom Lewis ()
§3.4–Descriptive Measures for Populations Use of Samples
Fall Term 2009
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Outline
1
Population mean
2
Population Standard Deviation
3
Parameters versus Statistics
4
Standardizing
Tom Lewis ()
§3.4–Descriptive Measures for Populations Use of Samples
Fall Term 2009
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Population mean
Definition (Population mean)
For a variable x, the mean of all possible observations for the entire
population is called the population mean or mean of the variable x. It is
denoted by µx or, when no confusion will arise, simply µ. For a finite
population,
P
xi
µ=
N
where N is the populations size.
Tom Lewis ()
§3.4–Descriptive Measures for Populations Use of Samples
Fall Term 2009
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Population mean
Definition (Population mean)
For a variable x, the mean of all possible observations for the entire
population is called the population mean or mean of the variable x. It is
denoted by µx or, when no confusion will arise, simply µ. For a finite
population,
P
xi
µ=
N
where N is the populations size.
Example
Consider the results for the 2009 SunRise Run. For this “population,” the
mean finish time is 45.93 minutes.
Tom Lewis ()
§3.4–Descriptive Measures for Populations Use of Samples
Fall Term 2009
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Population mean
Use of sample means
In an inferential study, we will use sample data to describe the entire
population. Thus, for example, we will use the mean of a sample to
describe or predict the mean of a population.
Tom Lewis ()
§3.4–Descriptive Measures for Populations Use of Samples
Fall Term 2009
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Population mean
Use of sample means
In an inferential study, we will use sample data to describe the entire
population. Thus, for example, we will use the mean of a sample to
describe or predict the mean of a population.
Example
50 runners were selected from the participants in the 2009 SunRise Run
(sunsample.xls). What is the mean of this sample? How does it compare
with the mean of the population?
Tom Lewis ()
§3.4–Descriptive Measures for Populations Use of Samples
Fall Term 2009
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Population Standard Deviation
Definition (Population Standard Deviation)
For a variable x, the standard deviation of all possible observations for the
entire population is called the population standard deviation or standard
deviation of the variable x. It is denoted by σx or σ. For a finite
population, the defining formula
rP
(xi − µ)2
,
σ=
N
where N is the population size.
The population standard deviation can also be found from the computing
formula
s
P 2
xi
σ=
− µ2 .
N
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§3.4–Descriptive Measures for Populations Use of Samples
Fall Term 2009
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Parameters versus Statistics
Definition
Tom Lewis ()
§3.4–Descriptive Measures for Populations Use of Samples
Fall Term 2009
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Parameters versus Statistics
Definition
Parameter: A descriptive measure for a population.
Tom Lewis ()
§3.4–Descriptive Measures for Populations Use of Samples
Fall Term 2009
6/7
Parameters versus Statistics
Definition
Parameter: A descriptive measure for a population.
Statistic: A descriptive measure for a sample.
Tom Lewis ()
§3.4–Descriptive Measures for Populations Use of Samples
Fall Term 2009
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Parameters versus Statistics
Definition
Parameter: A descriptive measure for a population.
Statistic: A descriptive measure for a sample.
Example (Parameters and Statistics)
Population
µ
σ
Tom Lewis ()
Sample
x
s
§3.4–Descriptive Measures for Populations Use of Samples
Fall Term 2009
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Standardizing
Definition (Standardized Variable)
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§3.4–Descriptive Measures for Populations Use of Samples
Fall Term 2009
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Standardizing
Definition (Standardized Variable)
For a variable x, the variable
z=
x −µ
σ
is called the standardized version of x or the standardized variable
corresponding to the variable x.
Tom Lewis ()
§3.4–Descriptive Measures for Populations Use of Samples
Fall Term 2009
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Standardizing
Definition (Standardized Variable)
For a variable x, the variable
z=
x −µ
σ
is called the standardized version of x or the standardized variable
corresponding to the variable x.
For an observed value x, the standardized variable z is called the
z-score of the observation.
Tom Lewis ()
§3.4–Descriptive Measures for Populations Use of Samples
Fall Term 2009
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