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Statistics 528Data Analysis I Lecture #2June 22, 2006 Overview of Today’s Lecture ^ Example of use of median in aproduction process ^ IPS Sections 1.3 and 2.1^ ^ The Normal Distribution^ ^ Scatterplots
Production Process – Median Example ^ Two groups of workers, each with fivepeople. ^ Group A is trained using one method, andGroup B is trained using another method. ^ Over the next 5 days, we monitor howmany completed products each workermakes on each day. ^ We want to know if the method used totrain Group A results in more output thanthe method used to train Group B. Production Process – Median Example
Production Process – Median Example
Group A^
Group B
Review: Strategy for Exploring Data ^ Plot the data^ ^ Categorical data – bar chart or piechart^ ^ Quantitative data – stem-and-leaf,histogram, time plot, etc. ^ Look for the overall pattern anddeviations from that pattern ^ Calculate numerical summaries todescribe the center and spread
Next step: Apply a Mathematical Model ^ Why would you dothis?^ ^ Most obvious – easier tosummarize theinformation this waythan reporting all thevalues^ ^ More useful – If the dataare representative of alarger group, themathematical model isuseful for describing thelarger group. Density Curves ^ Density curves are the mathematical models usedto represent the distribution of data. ^ The link between the curve and the histogram isthe^ proportion
of data that falls between two values^ Area =.^
Area =.
Standard Deviation of a Density Curve ^ The concept – approximately theaverage distance from the mean. ^ Difficult to approximate by eye, butcan be calculated mathematically. Notation: Observation summaries vs.Density properties ^ For observations of a variable:^ ^ Mean =^ ^ Standard Deviation = ^ For a density curve:^ ^ Mean =^ ^ Standard Deviation =
x μ^
s σ
Normal Distribution (Density) ^ The normaldensity is asymmetric,bell-shapedcurve that isuseful fordescribingmany types ofdata
(^2122) (^1) ), 2 |(
− −
μ σ πσ σμ
x e xf
),|( σ μ xf ),|( σ μ xf
Why is it important? 1.^ Good description of real data 2.^ Good approximation to the resultsof chance outcomes 3.^ Statistical inference proceduresrely heavily on the normaldistribution.
68-95-99.7 rule Example: Heights of women age 18 to 24 ^ The distribution is approximately normal.Measurements are in inches. ^ How tall would a woman (18-24) need tobe to be in the top 5% of heights?
) (^25). (^6) , (^5). (^64) ( ),( (^2) ~ N NX =^ σμ
Standard Normal ^ If a variable follows a normal distribution,then z = (x – μ)/σ follows a standardnormal distribution: z ~ N(0, 1) ^ This fact is very useful for finding areasunder a normal curve other than the onesexactly at the 1, 2, and 3 SD marks. ^ When an observation is transformed bysubtracting the mean and dividing by thestandard deviation, the resulting value iscalled the z-score. Example: IQ scores ^ IQ scores are normally distributedwith a mean of 100 and a standarddeviation of 10.
X~N(100,100)
^ What fraction of people have an IQscore under 85?^ 1.^ Draw a picture.^ 2.^ Shade the region of interest.^ 3.^ Look up the areas you need in TableA.
Normal Quantile Plot: Examples
C Frequency
(^6420) Histogram of C7 200 150 100 50 0 -2-4-6-
99.99^ Mean^99958050 Percent^2051 0.01^ 5.02.50.0-2.5-5.0-7.5^ C
-0.03750StDev 1.339N (^1000) A D 6.605<0.005P-Value Probability Plot of C7^ Normal - 95% CI C Frequency
(^1412108) Histogram of C8 1401201008060402006420
99.99^ Mean^99958050 Percent^2051 0.01^151050 -5^ C
2.850StDev 2.311N (^1000) AD 32.046<0.005P-Value Probability Plot of C8^ Normal - 95% CI
Relationships Between Variables
Regression Density Curve Models
Correlation Center,spread NumericalSummaries
Scatterplots Histograms,dotplot, etc. GraphicalSummaries
(Chapter 2) (Chapter 1)
2 Variables 1 Variable
Exploring the Relationship ^ Generically, we call two variables Xand Y ^ Are the variables
associated
?^ When
the value of one increases does theother increase?
When the value of
one increases does the otherdecrease? Scatterplot ODJFS child care data:^ ^ X – Full-time weekly rate for infants^ ^ Y – Full-time weekly rate for toddlers
Infant_FTW Toddler_FTW
(^260240) (^220200) (^180160) Scatterplot of Toddler_FTW vs Infant_FTW 240220200180160140120100140120
Association or Explanation ^ In some cases, we are only interested inunderstanding whether the variables areassociated (ODJFS is a good example) ^ In some cases, one variable is thought toexplain another^ ^ Example: Pressure treatment on plastic^ ^ Response Variable (dependent variable) –Migration of chemical after 24 hours^ ^ Explanatory Variable (independent variable) –Pressure level for treament ^ Note: Do not equate explanation withcausation! Examples: ^ Time spent studying vs. grade onexam ^ Height of husband vs. height of wife ^ Percent of districts voting majorityRepublican in 2000 vs. percent ofdistricts voting majority Republicanin 2004.
What to look for in a scatterplot ^ Overall pattern – deviations from thepattern ^ Form of relationship (linear, curved, etc) ^ Direction and strength of relationship^ ^ Positively associated – increase in X is seenwith increase in Y^ ^ Negatively associated – increase in X is seenwith decrease in Y^ ^ Do the points closely follow this pattern, orloosely? ^ Outliers Pattern - Linear
Outliers Adding categorical variables ^ Use different colors or symbols to add acategorical variable to a scatterplot – don’tforget to label.
Infant_FTW Toddler_FTW
(^260240220) (^200180160) 240220200180160140120100140120
PROGRAM TYPECO MBINATIONFULL-TIME Scatterplot of Toddler_FTW vs Infant_FTW
A note of caution: lurking variables ^ Factors other than the main ones ofinterest may have an effect. Categorical Explanatory Variables ^ Side-by-side boxplots^ ^ For just a few measurements, we could plotthe actual values (previous example) ^ Back-to-back stem plots ^ For nominal variables, it makes no senseto talk about positive or negativeassociations. ^ For ordinal variables, we can make astatement about positive or negativeassociations.
Example: Boxplots for an ordinalvariable vs. a continuous variable.
