Correlation: Finding Statistical Relationships

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Section 2.2

• Correlation Finding Statistical Relationships

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Placing Correlation in Context

• Step one Create a Scatterplot
• Interpret the Scatterplot
• Form, Direction, Strength as well as Outliers
• Create a Numerical Summary
• Mean of X and Y
• Standard Deviation for X and Y
• Correlation
• Interpret the results with a model

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Introduction

• Linear relationships (straight lines) occur
  fairly often in practice (e.g., ACT vs. GPA)
• A linear relation is strong if the points lie
  close to a straight line. Conversely a linear
  relationship is weak if the points are widely
  scattered about a line
• Warning human eyes can be fooled by changing
  the plotting scales or the amount of white space
  displayed in a scatterplot.
• Hence, we use a numeric measure called correlation

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Defining Correlation

• Correlation(r)
• Correlation is a quantity used to measure the
  direction and strength of the linear relationship
  between two quantitative variables.
• Formula

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Equation

• Calculating r
• Let x, y be any two quantitative variables for n
  individuals
• The values for the first individual are and
  , the values for the second are and ,
  etc.

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Does a graphic help?
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Correlation

• Remember are standardized
  values of variable x and y respectively. Think
  about how the z-score standardized values.
• You can think of the correlation (r) as an
  average (d.f. divisor) of the products of the
  standardized values of the two variables x and y
  for the n observations. Remember, each
  observation comes in a pair.

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Properties of r

• Correlation makes no distinction between
  explanatory and response variables
• Correlation requires that both variables be
  quantitative.
• r does not change when we change the units of
  measurements of x, y, or both. Why?
• r has no units of measurements. r is simply a
  number
• For example, changing the unit of x from
  millimeter to kilometer will not change r.
  Likewise, changing the unit of y from grams to
  decagrams will not change r.

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Properties of r

• The correlation r is always a number between -1
  and 1
• values of r near 0 indicate a very weak linear
  relationship
• the strength of the linear relationship increases
  as r moves away from 0 toward either -1 or 1
• values of r close to -1 or 1 indicate that the
  points in a scatterplot lie close to a straight
  line
• the extreme values r -1 and r 1 occur only in
  the case of a perfect linear relationship
• Correlation measures the strength of a linear
  relationship.

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Properties of r

• r gt 0 then there is a positive correlation
  between two variables.
• r lt 0 then there is a negative correlation
  between two variables.
• Once again, correlation only measures the
  strength of a linear relationship.

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r gt 0 (positive)

• An example of a positively correlated X and Y

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rlt0 (negative)

• Negatively correlated X and Y

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r near zero

• Little if any correlation between X Y

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Non-linear?

• r is not useful for non-linear data
• The data have a clear relationship a curve.
  But the correlation will be near 0.

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Properties of r

• Correlation measures the strength of the linear
  relationship between two variables
• Correlation does not describe curved
  relationships between variables, no matter how
  strong they are
• This is why we identify form first
• Like the mean and standard deviation, the
  correlation is not resistant to outliers
• r is strongly affected by a few outlying
  observations
• This makes sense by looking at the formula

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Correlation is not resistant

• Original Graph Outlier Graph

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Other Notes

• At times, scatterplots may disguise the true
  correlation.
• Correlation is not a complete description of
  two-variable data, even when the relationship
  between the variables is linear.
• Also report the means and standard deviations of
  both x and y as well as the correlation.
• Reminder 22 Correlation only measures the
  strength of a linear relationship.

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Example ACT (x) vs. GPA (y)
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• Form
• Linear
• r 0.622
• Which tells us
• Direction
• positive
• Strength
• moderate

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Text Example

• Data

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The Result
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Example -- Correlation Calc.

• Two managers evaluate a performance of 5
  candidates

• n5
• Mean score of X is mean score of Y
  is
• Standard deviations

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Example Cont.

• Now plug in data to the formula (magnifying
  glass?)

R0.8 Two evaluations are highly correlated.
Managers agree on the score level for each
candidate, even though A tends to give higher.
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Correlation.

• Identify the flaws
• We found a high correlation between eating celery
  and adult height.
• We found a correlation of r 1.02 between years
  of age and weight for the experimental subjects.
• The correlation between eating x pounds of potato
  chips and y pounds gaining weight is r 0.55
  pounds.

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Section 2.2 Summary

• The correlation (r) measures the strength and
  direction of the linear association between two
  quantitative variables x and y. Although you can
  calculate a correlation for any scatterplot, r
  measures only straight-line relationships.

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Section 2.2 Summary

• Correlation indicates the direction of a linear
  relationship by its sign r gt 0 for a positive
  association and r lt 0 for a negative association.

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Section 2.2 Summary

• Correlation always satisfies -1 r 1 and
  indicates the strength of a relationship by how
  close it is to -1 or 1. Perfect correlation, r
  /- 1, occurs only when the points on a
  scatterplot lie exactly on a straight line.

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Section 2.2 Summary

• Correlation ignores the distinction between
  explanatory and response variables. The value of
  r is not affected by changes in the unit of
  measurement of either variable. Correlation is
  not resistant, so outliers can greatly change the
  value of r.