Section 3.1 Scatterplots

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Section 3.1Scatterplots
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Two-Variable Quantitative Data

• Most statistical studies involve more than one
  variable.
• We may believe that some of the variables explain
  or even cause changes in the variables. Then we
  have explanatory and response variables.
• Explanatorylike the independent variable, it
  attempts to explain the observed outcomes.
• Responselike the dependent variable, it measures
  an outcome of a study.

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Examples

• Identify the explanatory and response variables
• Alcohol causes a drop in body temperature. To
  measure this, researchers give several different
  amounts of alcohol to mice, then measure the
  change in their body temperature after 15
  minutes.
• If an object is dropped from a height, then its
  downward speed theoretically increases over time
  due to the pull of gravity. To test this, a ball
  is dropped and at certain intervals of time, the
  speed of the ball is measured.

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Scatterplots

• Used for two-variable quantitative data!
• Explanatory variable goes on the x-axis
• Response variable goes on the y-axis
• The explanatory variable does not necessarily
  CAUSE the change in the response variable.

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• Displaying Relationships Scatterplots
• Make a scatterplot of the relationship between
  body weight and pack weight.
• Since Body weight is our eXplanatory variable, be
  sure to place it on the X-axis!

• Scatterplots and Correlation

Body weight (lb) 120 187 109 103 131 165 158 116
Backpack weight (lb) 26 30 26 24 29 35 31 28
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Interpreting Graphs
One Variable Quantitative Data Two-Variable Quantitative Data
Center Form Linear? Clusters? Gaps?
Shape Direction Positive? Negative?
Spread Strength Strong? Weak? Moderate?
Outliers Outliers
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In sentence form

• There is a (strong/weak), (positive/negative),
  (linear/non-linear) relationship between (your
  two variables).

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• Interpreting Scatterplots

• Scatterplots and Correlation

• There is a moderately strong, positive, linear
  relationship between body weight and pack weight.
• It appears that lighter students are carrying
  lighter backpacks.

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Adding Categorical Variables to Scatterplots

• You can use different plotting symbols or
  different colors to designate a categorical
  variable.
• You still have two quantitative variables, but
  you can add a category to these variables.

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Some quick tips for drawing scatterplots

• Choose an appropriate scale for the axes. Use a
  break if appropriate.
• Label, Label, Label
• If you are given a grid, try to use a scale that
  will make the scatterplot use the whole grid.

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Section 3.2 CorrelationWe are not good judges!

• We shouldnt just rely on our eyes to tell us how
  strong a linear relationship is.
• We have a numerical indication for how strong
  that linear relationship is its called
  CORRELATION.

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• Definition
• The correlation r measures the strength of the
  linear relationship between two quantitative
  variables.
• r is always a number between -1 and 1
• r gt 0 indicates a positive association.
• r lt 0 indicates a negative association.
• Values of r near 0 indicate a very weak linear
  relationship.
• The strength of the linear relationship increases
  as r moves away from 0 towards -1 or 1.
• The extreme values r -1 and r 1 occur only in
  the case of a perfect linear relationship.

• Scatterplots and Correlation

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Facts About Correlation

• It does not require a response and explanatory
  variable. Ex. How are SAT math and verbal scores
  related?
• If you switch the x and the y variables, the
  correlation doesnt change.
• If you change the units of measurement for x
  and/or y, the correlation doesnt change.
• Positive r values indicate a positive
  relationship negative values indicate a negative
  relationship. Remember not cause.

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More Facts

• Correlation measures the strength of the LINEAR
  relationship. It doesnt measure curved
  relationships.
• Correlation is strongly affected by outliers.
• r does not have a unit.

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Homework

• Chapter 3
• 11, 13, 14, 15, 17, 20, 22, 26