Scatterplots, Association and Relationships

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Scatterplots, Association and Relationships

• Chapter 7

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Variables

• Response Variable measures the outcome of a
  study (y variable, dependent variable, the
  result)
• Explanatory of Predictor Variable attempts to
  explain the observed outcomes (x variable,
  independent variable, the cause)

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Scatterplot

• Shows the relationship between two variables
  measured on the same individual.
• Graph the explanatory variable on the horizontal
  axis. (x list)
• Graph the response variable on the vertical axis
  (y list)

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Interpreting Scatterplots

• Overall pattern
• Direction (increasing or decreasing ?)
• Form (linear, exponential?)
• Strength of relationship (narrowness of curve)
• The scatter or width of the hallway
• Outliers
• What does not fit the pattern
• Falls outside the usual values or either variable

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Direction (Think Slope)

• Two variables are positively associated when the
  above average values of one are associated with
  the above average values of the other.
• Two variables are negatively associated when the
  above average values of one are associated with
  the below average values of the other

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Form

• Form is the general shape of the dots in the
  scatterplot
• Linear, exponential, logarithmic, . . .
• Correlation is ONLY relevant with linear data.
• Curved data must be straightened before we
  can use correlation

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Strength

• How much is the data lined up.
• The closer to a straight line the stronger the
  relationship
• Correlation is a measurement of the strength of
  the relationship between predictor and response
  variables

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Correlation

• How strong is the relationship

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Calculation of Correlation(The Theory, thank TI
again)

• Calculate the means and standard deviations of
  both the xs and ys
• Standardize both the x value and y value for each
  data point (calculating z-scores)
• Multiply the z values of each point
• Add up all of the products of the zs
• Divide the grand total by n 1 where n is the
  number of points

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Calculation Example
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Correlation

• The average of the product of standardized xs
  and ys

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Another way to calculate correlation coefficients
?
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Correlation Facts

• It makes no difference which variable is the x
  and which is the y
• Positive r indicates a positive association
  between the variables and negative r indicates a
  negative association
• -1 r 1 Values near zero indicate a weak
  association. Values near 1 or -1 indicate strong
  association

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Correlation facts 2

• There are no units on r so it is immune to
  changes when units change.
• Correlation measures the strength of ONLY linear
  relationships. Correlations do not describe
  curved relationships
• Correlation is greatly affected by extreme values

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Correlation Facts 3

• Outliers can make a strong relationship look weak
• Outliers can make a weak relationship look strong
• Report the correlation both with and without the
  outlier(s)

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Conditions for Correlation

• The quantitative condition Both variables must
  be quantitative.
• The linear condition The form of the
  relationship must be linear.
• The outlier condition Outliers greatly affect
  correlation. Report the r value both with and
  without the outliers factored in
• Correlation Demo

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Some things to watch for

• Dont say Correlation when you mean
  Association
• Association is a vague term used when describing
  the relationship between two variables
• Correlation is a very precise term describing a
  linear relationship between quantitative
  variables
• Beware of Outliers
• Dont confuse correlation with causation
• The explanatory variable does not cause the
  response to change

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More to watch for

• Lurking Variables
• Lurking Variables are Hidden Variables other than
  the explanatory and response variables that we do
  not see on the scatterplot.