Scatterplots, Association, and Correlation

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Chapter 7

• Scatterplots, Association, and Correlation

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From single variables to 2

• We have been interested so far in the
  distribution of single variables, or perhaps
  comparing the distributions of 2 similar
  variables
• top speed for males and females
• Now How do we look at the relationship between 2
  Quantitative variables?
• Is there a trend or pattern between variables?
• Does one predict another?
• Are there outlying cases?
• Are the data grouped in an interesting way?

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A Plot between 2 Variables.

• To see associations between variables, we use a
  type of plot youre probably very familiar with
  already.
• What is it?
• A Scatterplot!
• The best way to start observing the relationship
  between two quantitative variables.

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Keys to the Scatterplot

• Shows relationship between 2 Quantitative
  Variables.
• Can I make a scatterplot of car model and number
  stolen?
• Both variables must be measured on the same
  individuals
• Will a scatterplot of height of 100 women vs
  weight of 100 men work?
• One variable per axis
• Each individual represented by single pt on plot.
• Centered on data (remember the midrange?)

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DataDesk and Scatterplots

• Lets look at how to make a scatterplot on
  DataDesk.
• But first, ALWAYS ask
• Who? (individuals)
• What? (vars, units)
• Class Data, Travel Distance / Travel Time

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What goes where?

• Which variable should go on which axis?
• Vocabulary
• Explanatory/Predictor variable If one variable
  explains or predicts the other, it is the EV
• Place on X axis.
• Response variable If one variable responds to
  the other, it is the response variable,
• Place on Y axis.
• What if no clear influence?
• Your choice!

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Examples

• Which is the EV, which RV?
• Daily high temp, Cooling costs
• Degree of pain relief, Drug dosage
• Score on test, Time spent studying
• Height, Weight
• Shoe size, Grade-point average

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Whats important in a Scatterplot?

• What do you think? What do you see?
• Lets look at Monopoly data.
• Price, Rent
• Which one X, which one Y?

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Describing Scatterplots

• 3 Things to look for in a scatterplot
• Direction
• Positive Increasing, as one variable increases,
  the other one does too.
• Negative Decreasing, as one variable increases,
  the other decreases.
• Form What is the overall pattern of the plot?
• Linear
• Curved (Parabolic, Exponential, Bent)
• Others (Fan, etc)
• NO DATA RE-EXPRESSION IN THIS CLASS!

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Describing Scatterplots

• Strength How strong is the relationship?
• How closely do the variables conform to the
  described form?
• Good strength (low scatter)- follows form
  closely, tightly clustered in a single stream
• Poor strength (high scatter)- points form a
  cloud with barely any visible pattern.
• Also, look for the unusual!
• Outliers points well away from the pattern
• Clusters Maybe you have groups, and need to
  split data.

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Some examples

• Lets look at some datafiles, and describe the
  relations.
• Before graphing- what do you expect to see?
• CEO Salaries and Ages
• Airfares
• Oil Change
• Colleges

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How strong?

• Assessing strength of linear association by eye
  is difficult, eyes are easily misled.
• Back to Colleges.
• Scaling can change how an association appears.
• We also want to be able to measure strength
  independent of units. ( or pesos? Years or
  months?)
• What do we know that is independent of units?
• Z-scores

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Z-scores and Scatterplots

• If we plot our variables as z-scores, the new
  scatterplot will be
• Centered at (0,0)
• Axes will be same scale
• Independent of original units of the data
• Gives a neutral way of drawing the scatterplot,
  fairer impression of strength.
• Class Data again

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Z-scores and Correlation

• Correlation, r, is the statistic used to give a
  numerical indication of strength, independent of
  scale.
• Points in Quadrants I and III will have a
  positive product of z-scores, points in II and IV
  will have a negative product. Points on axes
  dont get to vote (0 product).

Do Not Calculate by hand- use Technoloy!
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About Correlation

• Does Correlation (r) have any units?
• Will changing the units on a variable change the
  correlation?
• What are the requirements for finding a
  correlation?
• 2 quantitative variables
• Approximately linear form
• Be aware of outliers- they can affect r a lot.
• Ie Check the scatterplot FIRST!

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All about Correlation

• r will vary between 1.0 and -1.0 ONLY.
• Sign of r gives the direction
• The closer r is to -1 or 1, the stronger the
  linear relation
• r -1.0 perfect linear association, negative
  slope
• r 1.0 perfect linear assoc., positive slope
• r 0 no LINEAR association
• There is no distinction between X and Y in
  calculating.
• Like Mean and SD, sensitive to outliers.
• Outliers can create a false good correlation
• Can hide a true good correlation
• Go to ActivStats, Chapter 7 Scatterplot Tool

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Data Desk and Correlation

• Estimating the correlation is hard.
• Even Statisticians have a hard time doing it.
• We will always let Technology calculate r for us.
• Class Data

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How strong is strong?

• How high does r have to be to be good?
• It depends!
• Education study might be thrilled with r 0.6
• Engineering study might find r0.9 too low!

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Correlation or Causation?

• Correlation is shows an association.
• We can use one variable to predict another.
• Does this mean that one variable causes the
  other?
• Shoe size, reading level
• Be careful- there might be lurking variables!
• A variable, hidden or not considered, that
  influences both variables simulataneously.
• Age

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Example 18

• Drug Use
• Find Correlation
• Describe association
• Do results confirm that marijuana is a gateway
  drug, that it leads to the use of other drugs?