Introductory Statistical Concepts

Disclaimer

– I am not an expert SAS programmer.– Nothing that I say is confirmed or denied by Texas

A&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 Poll: What type of organization do you work for?work for?

• [PlaceWare Multiple Choice Poll. Use PlaceWare > Edit Slide Properties... to edit.]

• Business• Government• Education• Nonprofit• Other

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Purpose of These Lectures

• A 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 a p-value?– Why does regression handle missing data so well?– What are residual analyses?

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Descriptive Statistics

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The Population

(Very important concepts)

Variable of Interest

The DistributionParameters

Mean Mode RangeMedian Variance

Etc

Learning Outcomes• You 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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Parameters

• Characteristics 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

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Click Here for more information on Mode Mean MedianClick Here for an applet

Variability

Change in the Data

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What is an Index ?

How SUNNY is SUNNY?

THE UV Index

Click Here

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Air Quality IndexWhat Does It Mean?

DOW JONES INDUSTRIAL AVERAGE INDEX

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What does 10,971.16 really mean?

What is “better” a DJIA of 10,000 Or a DJIA of 12,000?

Variability Index

• A 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

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or S or s

Details of the More Complex Index

• Example – 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 applet – Click Here

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Using Graphs to Determine Variability• Box Plot• Click Here

3535N =

State

NEW_YORKCALIFORN

To

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iole

nt

Cri

me

400000

300000

200000

100000

0

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Distributions

Known Distribution

• With 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 Distributions─Normal

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Normal─Overlay

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Classical Distributions─Uniform

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Survey

• The 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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MPG─Histogram

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Compare with“true” values !

Simulated Sample

• In this example, we simulated taking a sample of size 1000 from one population of cars weighing 3000 pounds with a normal distribution with mean=24 and standard deviation=1.

• You can practice this after class.

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Section 1.2

Populations and Samples

Objectives

– Understand the relationships between• populations and samples• parameters and estimates.

– Look at an overview of hypotheses testing.

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Population

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Mean, Variance, Median, Mode, Distribution, …

Parameters

Example

• Mpg 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

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Mean, Variance, Median, Mode, Distribution, …

Parameters

Sample meanSample variance

Sample

S

Inference:EstimatesTest of hypotheses

Example: cars_american

• This is a sample of American-made cars that weigh between 2000 and 3500 pounds and that were built in the 1970s.

• We are interested in the mpg.

• Use summary statistics to analyze the data.

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Results of Summary Statistics

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Results of Histogram

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continued...

Results of Histogram

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Sampling Distribution Applet sampling_dist

• This demonstration illustrates how to estimate and plot the sampling distribution of various statistics.

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View/Application Share: Demo: View/Application Share: Demo: Sampling Distributions AppletSampling Distributions Applet

• [PlaceWare View/Application Share. Use PlaceWare > Edit Slide Properties... to edit.]

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http://www.ruf.rice.edu/~lane/http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.h...stat_sim/sampling_dist/index.h...

• [PlaceWare Web Page. Use PlaceWare > Edit Slide Properties... to edit.]

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Confidence Intervals on the Population Mean

• Level of Comfort

• 50% {21.57 to 22.21}

• 95% {20.96 to 22.82}

• 99.9% {20.30 to 23.48}

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What does this mean?

Test That the Population Mean = 30 mpg

• Use t-test One Sample t-test

• Requirements for running this test:– Large n > 35– Or leftovers are normal

• What is the p-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 Test

• Choose an alpha level, usually alpha=.05.

• If sig<alpha, then reject.

• Otherwise, fail to reject.

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Sig and p-values• When you see a sig value or p-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?46

Requirements for Doing This Test

• Large n n > 35

• Or leftovers are normally distributed.

• Use Histogram to test for normality.

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Populations─Which Ones are Similar?

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Populations─Which Ones are Similar?

• Take samples.

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Take Samples• Use 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?

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Background/Requirements

• Before 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 the

requirements for doing those?– Are we using everything we “know?”

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Example

• Suppose that we are interested in the mpg of American and European cars. How many populations are there?

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American CarsMpg

DistributionMean

Variance

European CarsMpg

DistributionMean

Variance

Poll: How many populations are there?Poll: How many populations are there?• [PlaceWare Multiple Choice Poll. Use PlaceWare > Edit Slide Properties... to edit.]

• One - MPG• Two - American and European• Depends on the sample size

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ParametersPopulation 1 Population 2

American Cars European Cars

Variable of interest: mpg Variable of interest: mpg

Distribution: Normal? Distribution: Normal?

Mean: Mean:

Variance: Variance:

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A E2A 2

E

Analyses

1. 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?

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Example: Our Data Set car_am_eu

• Suppose that we are interested in the mpg of American and European cars.

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Sample

American CarsMpg

DistributionMean

Variance

European CarsMpg

DistributionMean

Variance

Sample

Results from the Sample

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continued...

Results

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Tests of Normality

.110 248 .000

.111 70 .033

Country of OriginAmerican

European

Miles per GallonStatistic df Sig.

Kolmogorov-Smirnova

Lilliefors Significance Correctiona.

Box Plots

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American European

Histograms

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American

European

Poll: Are the populations the same?Poll: Are the populations the same?• [PlaceWare Yes/No Poll. Use PlaceWare > Edit Slide Properties... to edit.]

• Yes• No

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Conclusion Based on Sample Numbers and Graphs

• Easy -- Based on the samples, the populations are different—no statistical jargon

• But I must have a p-value for my boss, for my paper, and so on.

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Formal Tests

• The classical approach in determining whether two populations are the same is to test to see whether the two population means are equal.

• But first we check to see whether the two population variances are equal:

•

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2 2:o A EH :o A EH

continued...

Formal Tests

• We use t-test Two Sample.

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Test 2

Test 1

Section 1.3

Simple Linear Regression

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.

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MPG Example

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Weight = 3000

1

21

3

23

2

22

4

24

Weight = 2600

Weight = 2900Weight = 2300

Take a sample of size 1 from each population!

Data

• We 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”?

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Survey

• Name the population parameters.

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Essential Part and Leftovers• We want to “model” the data as follows:

• MPG = Essential Part + Leftover• or• MPG = Mean + Leftover

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“Know” or Assumptions• First, 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?

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Poll: How many unknowns are there?Poll: How many unknowns are there?• [PlaceWare Multiple Choice Poll. Use PlaceWare > Edit Slide Properties... to edit.]

• 1• 2• 3• 4• 5• n

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Graph

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Observed, Essential Part, Leftover

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The Official Regression Model

• or

• or

• or

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The errors are “known” to be normal with mean 0 and variance . 2

Main Assumptions

• The means of the populations fall on a straight line.

• All of the variances are equal ( ).

• The errors are “known” to be normal with mean 0 and variance .

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2

2

Assumptions for Simple Linear Regression

Appendix A

• This demonstration illustrates the fundamental concepts of simple linear regression.

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View/Application Share: Demo: View/Application Share: Demo: Linear.docLinear.doc

• [PlaceWare View/Application Share. Use PlaceWare > Edit Slide Properties... to edit.]

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How Can We Estimate the Unknown Parameters?

• The Principle of Least Squares:

• or

• or

• Now, choose a and b so that is as small as possible.

• or• Minimize .

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OUTPUT_0

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OUTPUT

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OUTPUT_1

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OUTPUT_2

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OUTPUT_3

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OUTPUT_4

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Missing Values

• Suppose that we want to estimate the mean mpg when weight=2500.

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

• Why does this work?

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Survey

• Can anyone explain why this works?

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Conclusion

– Simple linear regression is very powerful.

– But it is based on assumptions (what we “know”).

– We need to check assumptions (residual analyses).

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