Introductory Statistical Concepts

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Introductory Statistical Concepts

• F. Michael Speed, Ph.D.
• Department of Statistics
• Texas AM University

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Disclaimer

• I am not an expert SAS programmer.
• Nothing that I say is confirmed or denied by
  Texas AM 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 work for?

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• 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
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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

Click Here for more information on Mode Mean
Median Click Here for an applet
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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
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Air Quality IndexWhat 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 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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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 47)/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

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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 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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Uniform-Overlay
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Classical Distributions-Chi-Square
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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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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 Example

• Suppose we want to simulate mpg for a car that
  weighs 3000 lbs.
• Let us assume that the mean mpg24.
• Let us assume that the standard deviation1 mpg.
• We will generate a number from the normal
  distribution with mean 0 and standard
  deviation1.
• We will then add (subtract) that number from 24.

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MPGComposition
Observed
Essential Part 24
Leftovers N(0,1)
/-

Let us generate 1000 mpg.
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SAS Code 3
data mpg1 do i1 to 1000 mean 24
lo normal(-1) mpg mean lo
output end run
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Simulated MPG
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MPG-Histogram
Compare with true values !
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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
  mean24 and standard deviation1.
• You can practice this after class.

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After Class Practice

• Simulate 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 with Mean 0 and
Standard Deviation of s1.5
data mpg1 s1.5 mean 24 do i1 to
1000 lo snormal(-1) mpg mean lo
output end run
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Simulating Data SAS_Code4.sas

• This demonstration illustrates how to simulate
  data for a given population.

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View/Application Share Demo Simulation

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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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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 mean30mpg?

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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
Sample mean Sample variance
S
Inference Estimates Test of hypotheses
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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
continued...
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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 Sampling
Distributions Applet

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http//www.ruf.rice.edu/lane/stat_sim/sampling_di
st/index.h...

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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

What does this mean?
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Test That the Population Mean 30 mpg

• Use t-test ? One Sample t-test
• Requirements for running this test
• Large n gt 35
• Or leftovers are normal
• What is the p-value or sig value?

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Testing Mean 30
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Conclusions of the Test

• Choose an alpha level, usually alpha.05.
• If sigltalpha, 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?

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Requirements for Doing This Test

• Large n ? n gt 35
• Or leftovers are normally distributed.
• Use Histogram to test for normality.

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Testing Hypotheses

• This demonstration illustrates the testing of
  hypotheses using the data set cars_american.

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View/Application Share Demo Testing Hypotheses

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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?

American Cars Mpg Distribution Mean Variance
European Cars Mpg Distribution Mean Variance
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Poll How many populations are there?

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

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Parameters
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Analyses

• We want to look at the distributions.
• We want to estimate the parameters.
• 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.

American Cars Mpg Distribution Mean Variance
European Cars Mpg Distribution Mean Variance
Sample
Sample
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Results from the Sample
continued...
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Results
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Box Plots
American
European
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Histograms
American
European
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Poll Are the populations the same?

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• Yes
• No

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Conclusion Based on Sample Numbers and Graphs

• Easy -- Based on the samples, the populations are
  differentno 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

continued...
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Formal Tests

• We use t-test ? Two Sample.

Test 2
Test 1
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Comparing Two Populations

• This demonstration shows how to compare two
  populations using the data set car_am_eu.

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View/Application Share Demo Comparing Two
populations

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Example

• Run summary statistics.
• Ask for histogram and box plot.
• What do you get?

data temp1 x 1 output run
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Summary

• The building blocks of analyzing data using
  statistical techniques are
• population parameters
• population parameters
• population parameters.
• Population parameters are very important.

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Section 1.3

• Simple Linear Regression

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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.

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MPG Example
Weight 3000
Weight 2600
Take a sample of size 1 from each population!
Weight 2300
Weight 2900
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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?

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• 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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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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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 Linear.doc

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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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David Lanes Applet

• http//www.ruf.rice.edu/lane/stat_sim/reg_by_eye/

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Regression Applet Reg_by_eye

• This demonstration references David Lanes applet
  at Rice University.

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View/Application Share Demo David Lane's Applet

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http//www.ruf.rice.edu/lane/stat_sim/reg_by_eye

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  Properties... to edit.

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Output of SAS Regression cars_american

• This demonstration illustrates the use of
  regression in SAS Enterprise Guide.

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View/Application Share Demo Output of SAS
Regression

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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 weight2500.
• Predicted (Estimated) Mean MPG 44.05 -
  .0078weight
• 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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