Introductory Statistical Concepts

7/28/2019 Introductory Statistical Concepts

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Introductory StatisticalConcepts

F. Michael Speed, Ph.D.

Department of Statistics

Texas A&M University


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Disclaimer I am not an expert SAS programmer.

Nothing that I say is confirmed or denied by TexasA&M University.


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Why Are We Here?Deming

To Learn

To Have Fun

Question: Who was Deming?


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Poll: What type of organization do you workfor?[PlaceWare Multiple Choice Poll. Use PlaceWare> Edit Slid e Propert ies.. . to edit.]

BusinessGovernment

Education

Nonprofit

Other


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Purpose of These LecturesA review of the statistical concepts used in most of the

SAS Analytics Lecture Series.We will look at questions such as the following:

What is the nature of statistical analyses?

Why are population parameters so important?

What is really being tested when you see ap-value?

Why does regression handle missing data so well?

What are residual analyses?


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Descriptive Statistics


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(Very impo rtant concepts)

Variable of Interest

The Distribution

Parameters

Mean Mode Range

Median Variance

Etc

The Population


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Learning OutcomesYou will learn

basic statistical concepts the definition of mean, median, mode and standard deviation

the difference between populations and samples

the difference between parameters and estimates

about confidence intervals

how to test a statistical hypothesis

how to run a regression analysis


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ParametersCharacteristics of the variable of interest

It is how we describe the variable of interest

Parameters are unknown


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Parameters

(Characteristics)

Central Tendency

Mode

Median

Mean

Measures of Variability

Range

Variance

Standard Deviation

Click Here for more information on Mode Mean Median

Click Here for an applet
http://dist.stat.tamu.edu/pub/speed/mean_mode.htmhttp://www.stat.tamu.edu/~west/ph/meanmedian.htmlhttp://www.stat.tamu.edu/~west/ph/meanmedian.htmlhttp://dist.stat.tamu.edu/pub/speed/mean_mode.htm


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Variability

Change in the Data


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What is an Index ?

How SUNNY is SUNNY?

THE UV Index

Click Here
http://www.epa.gov/sunwise/uviscale.htmlhttp://www.epa.gov/sunwise/uviscale.html


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Air Quality Index

What Does It Mean?


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DOW JONES INDUSTRIAL AVERAGE INDEX

What does 10,971.16 really mean?

What is better a DJIA of 10,000

Or a DJIA of 12,000?


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Variability IndexA Simple One

Find the Largest Value

Find the Smallest Value

Let Range = R = Largest Smallest


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A More Complex Variation Index

The Standard Deviation

Statisticians use this index to indicate variability

You will see it written as

Widely available from SAS, Excel, and other statistical packages

or S or s


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Details of the More Complex IndexExample Suppose that we observe the following three numbers

1 4 7

The mean of these number is:

( 1 +4+7)/3 = 4

We now subtract the mean from each number and square it

(1-4)*(1-4) + (4-4)*(4-4) +(7-4)*(7-4) = 18

The Standard Deviation = sqrt(18/2) = 3


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What does this Mean?By itself , it may be confusing to some.

Comparing populations, we can use it to say which

population varies the most.

Let us look at an appletClick Here
http://www.stat.tamu.edu/~west/ph/stddev.htmlhttp://www.stat.tamu.edu/~west/ph/stddev.html


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Using Graphs to Determine Variability

Box Plot

Click Here

3535N =

State

NEW_YORKCALIFORN

400000

300000

200000

100000

0
http://www.netmba.com/statistics/plot/box/http://www.netmba.com/statistics/plot/box/http://www.netmba.com/statistics/plot/box/


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Describe What Is Happening

You are giving the parameters of the picture


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Example Using SAS


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Distributions


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Known DistributionWith a known distribution, we know the following:

the shape

the mean

the variability (standard deviation)

and/or some other information


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Classical DistributionsNormal


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NormalOverlay


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Classical DistributionsUniform


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UniformOverlay


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Classical DistributionsChi-Square


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SurveyThe following are called parameters of the population:

mean, median, mode

variance, standard deviation, range, inter-quartile

range (IQR)

In general, are these known or unknown?

Known = yes (select using your seat indicator)

Unknown = no (select using your seat indicator)


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Generate a Sample from a Known Distribution

Why? This is a simulation.

It helps us to understand a process or analyses.

It helps to see if we are getting expected results.

It is fun.


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MPG ExampleSuppose we want to simulate mpg for a car that weighs

3000 lbs.

Let us assume that the mean mpg=24.

Let us assume that the standard deviation=1 mpg.

We will generate a number from the normal distributionwith mean 0 and standard deviation=1.

We will then add (subtract) that number from 24.


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MPGComposition

Let us generate 1000 mpg.

Observed +/-= Essential Part24 LeftoversN(0,1)


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Simulated MPG


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MPGHistogram

Compare withtrue values !


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Simulated SampleIn this example, we simulated taking a sample of size

1000 from one population of cars weighing 3000 poundswith a normal distribution with mean=24 and standard

deviation=1.

You can practice this after class.


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After Class PracticeSimulate 1000 data points for each of the following five

populations. Run and explore your data.


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SAS Code 4: Generate a Normal withMean 0 and Standard Deviation of s=1.5

data mpg1;s=1.5;mean = 24;do i=1 to 1000;

lo =s*normal(-1);

mpg = mean + lo;output;end;

run;


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This demonstration illustrates how to simulate

data for a given population.

Simulating DataSAS_Code4.sas


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View/Application Share: Demo: Simulation[PlaceWare View/Application Share. Use PlaceWare> Edit Slide Pro pert ies... to edit.]


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Summary


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Section 1.2

Populations and Samples


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Objectives Understand the relationships between

populations and samples parameters and estimates.

Look at an overview of hypotheses testing.


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Population

Mean, Variance, Median,

Mode, Distribution,

Parameters


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ExampleMpg of American-made cars that weigh between 2000

and 3500 pounds and were built in the 1970s.

Parameters mean, variance, and so on

In general, we do not know the parameters.


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Purpose of Statistical Analyses Estimate the parameters. (Make guesses.)

Example: What is the population mean?

Test hypothesis about the parameters. (Ask questions.)

Example: Is the population mean=30mpg?


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Role of Samples Taking a sample of the population enables you to

make estimates of the population parameters answer the questions about the population

parameters.


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Population and Sample

Mean, Variance, Median,

Mode, Distribution,

Parameters

Sample mean

Sample variance

Sample

S

Inference:

Estimates

Test of hypotheses


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Results of Summary Statistics


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Results of Histogram

continued...


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Results of Histogram


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This demonstration illustrates how to estimate

and plot the sampling distribution of variousstatistics.

Sampling Distribution Appletsampling_dist


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View/Application Share: Demo: SamplingDistributions Applet[PlaceWare View/Application Share. Use PlaceWare> Edit Slide Pro pert ies... to edit.]


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http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.h...[PlaceWare Web Page. Use PlaceWare> Edit Slid e Propert ies.. . to edit.]


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Confidence Intervals on the Population MeanLevel of Comfort

50% {21.57 to 22.21}

95% {20.96 to 22.82}

99.9% {20.30 to 23.48}

What does this mean?


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Test That the Population Mean = 30 mpgUse t-test One Sample t-test

Requirements for running this test:

Large n > 35

Or leftovers are normal

What is thep-value or sig value?


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Testing Mean = 30

: 30

: 30

o mpg

A mpg

H

H


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Conclusions of the TestChoose an alpha level, usually alpha=.05.

If sig


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Sig andp-valuesWhen you see a sig value orp-value:

You know that some hypothesis is being tested. You know whether or not the hypothesis is being

rejected.

You probably do not know what the hypothesis really

is.

Ask yourself these questions:

What are the population parameters being tested?

How is what is being tested related to those

parameters?


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Requirements for Doing This TestLarge n n > 35

Or leftovers are normally distributed.

Use Histogram to test for normality.


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This demonstration illustrates the testing of

hypotheses using the data setcars_american.

Testing Hypotheses

Vi /A li i Sh D T i


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View/Application Share: Demo: TestingHypotheses[PlaceWare View/Application Share. Use PlaceWare> Edit Slide Pro pert ies... to edit.]


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P l ti Whi h O Si il ?


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PopulationsWhich Ones are Similar?

P l ti Whi h O Si il ?


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PopulationsWhich Ones are Similar?Take samples.

T k S l


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Take SamplesUse the samples to answer this question:

Which populations are similar?

Statistical translations:

Which populations are similar? is the same as asking

Are the following the same:

distribution?

mean?

variance?

B k d/R i t


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Background/RequirementsBefore we jump into the analysis, we must ask the

following questions: How many populations are there?

How many population parameters are we interested in

and what are they?

What tests do we want to do, and what are therequirements for doing those?

Are we using everything we know?

E l


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ExampleSuppose that we are interested in the mpg of American

and European cars. How many populations are there?

American Cars

Mpg

DistributionMean

Variance

European Cars

Mpg

DistributionMean

Variance

P ll H l ti th ?


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Poll: How many populations are there?[PlaceWare Multiple Choice Poll. Use PlaceWare> Edit Slid e Propert ies.. . to edit.]

One - MPG

Two - American and European

Depends on the sample size

P t


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Parameters

Population 1 Population 2

American Cars European Cars

Variable of interest: mpg Variable of interest: mpg

Distribution: Normal? Distribution: Normal?

Mean: Mean:

Variance: Variance:

A

E

2

A

2

E

A l


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Analyses1. We want to look at the distributions.

2. We want to estimate the parameters.3. We want to answer these questions:

Are the populations means the same?

Are the population variances the same?

Example: Our Data Set car am eu


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Example: Our Data Set car_am_euSuppose that we are interested in the mpg of American

and European cars.

Sample

American Cars

Mpg

DistributionMean

Variance

European Cars

Mpg

DistributionMean

Variance

Sample

Results from the Sample


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Results from the Sample

continued...


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Box Plots


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Box Plots

American European

Histograms


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Histograms

American

European

Poll: Are the populations the same?


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Poll: Are the populations the same?[PlaceWare Yes/No Poll. Use PlaceWare> Edit Slid e Propert ies.. . to edit.]

Yes

No

Conclusion Based on Sample Numbers and


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Conclusion Based on Sample Numbers andGraphs

Easy -- Based on the samples, the populations are

differentno statistical jargon

But I must have ap-value for my boss, for my paper, and

so on.

Formal Tests


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Formal TestsThe classical approach in determining whether two

populations are the same is to test to see whether the twopopulation means are equal.

But first we check to see whether the two population

variances are equal:

2 2:o A EH :

o A E

H

continued...

Formal Tests


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Formal TestsWe use t-test Two Sample.

Test 2

Test 1


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This demonstration shows how to compare two

populations using the data set car_am_eu.

Comparing Two Populations

View/Application Share: Demo: Comparing


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View/Application Share: Demo: ComparingTwo populations[PlaceWare View/Application Share. Use PlaceWare> Edit Slide Pro pert ies... to edit.]

Example


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Example

1. Run summary statistics.

2. Ask for histogram and box plot.

What do you get?

data temp1;x = 1;output;

run;


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Section 1.3

Simple Linear Regression

Objectives


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Objectives Identify the following:

the population parameters the appropriate model

number of populations sampled

the correct hypotheses

what should be tested for normality

what equal variances means.

MPG Example


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MPG Example

Weight = 3000

1

2

1

3

2

3

2

2

2

4

2

4

Weight = 2600

Weight = 2900Weight = 2300

Take a sample of

size 1 from each

population!

Data


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DataWe should be in deep trouble with one sample from each

population.We have eight unknown population parameters.

Can you name them?

But what do we know?

Survey


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SurveyName the population parameters.

Essential Part and Leftovers


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Essential Part and LeftoversWe want to model the data as follows:

MPG = Essential Part + Leftover

or

MPG = Mean + Leftover

Know or Assumptions


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Know or AssumptionsFirst, we know that

Second, each population mean is related to weight by the

following:

The population means fall on a straight line!!

How many unknowns are there now?

2 2 2 2 21 2 3 4

i

*

ia b weight

Poll: How many unknowns are there?


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Poll: How many unknowns are there?[PlaceWare Multiple Choice Poll. Use PlaceWare> Edit Slid e Propert ies.. . to edit.]

1

2

3

4

5n

Graph


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Graph

Observed, Essential Part, Leftover


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Observed, Essential Part, Leftover

The Official Regression Model


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or

or

or

mpg = a + b*weight+leftover

The Official Regression Model

The errors are known

to be normal with mean0 and variance .2

mpg = a + b*weight+error

mpg = a + b*weight+

o 1

mpg = + *weight+


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This demonstration illustrates the fundamental

concepts of simple linear regression.

Assumptions for Simple

Linear RegressionAppendix A

View/Application Share: Demo: Linear.doc


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View/Application Share: Demo: Linear.doc[PlaceWare View/Application Share. Use PlaceWare> Edit Slide Pro pert ies... to edit.]

How Can We Estimate the Unknown


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Parameters?

The Principle of Least Squares:

or

or

Now, choose a andb so that is as small as

possible.

or

Minimize .

i i i

leftover mpg (a+b*weight )

i i i

Let leftover mpg (essential part)

i i i

r mpg (a+b*weight )

2 2 2 2

1 2 3 4r r r r

1(r2 2 2 2

2 3 4r r r )


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This demonstration references David Lanes

applet at Rice University.

Regression AppletReg_by_eye

View/Application Share: Demo: David Lane's


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ppApplet[PlaceWare View/Application Share. Use PlaceWare> Edit Slide Pro pert ies... to edit.]

http://www.ruf.rice.edu/~lane/stat_sim/reg_by_


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p _ g_ y_eye[PlaceWare Web Page. Use PlaceWare> Edit Slid e Propert ies.. . to edit.]


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View/Application Share: Demo: Output of SAS


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pp pRegression[PlaceWare View/Application Share. Use PlaceWare> Edit Slide Pro pert ies... to edit.]

OUTPUT_0


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OUTPUT


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OUTPUT_2


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OUTPUT_3


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OUTPUT_4


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Missing Values


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Suppose that we want to estimate the mean mpg when

weight=2500.

Predicted (Estimated) Mean MPG = 44.05 - .0078*weight

Why does this work?

Survey


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Can anyone explain why this works?


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Introductory Statistical Concepts