Describing Relationships: Scatterplots and Correlation

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Chapter 14

• Describing Relationships Scatterplots and
  Correlation

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Correlation
Objective Analyze a collection of paired data
(sometimes called bivariate data). A correlation
exists between two variables when there is a
relationship (or an association) between them.

• We will consider only linear relationships.
• - when graphed, the points approximate a
• straight-line pattern.

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Scatterplot

• A scatterplot is a graph in which paired (x, y)
  data (usually collected on the same individuals)
  are plotted with one variable represented on a
  horizontal (x -) axis and the other variable
  represented on a vertical (y-) axis. Each
  individual pair (x, y) is plotted as a single
  point.

Example
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Examining a Scatterplot

• You can describe the overall pattern of a
  scatterplot by the
• Form linear or non-linear ( quadratic,
  exponential, no
• correlation etc.)
• Direction negative, positive.
• Strength strong, very strong, moderately
  strong,
• weak etc.
• Look for outliers and how they affect the
  correlation.

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Scatterplot
Example Draw a scatter plot for the data below.
What is the nature of the
relationship between X and Y.
Strong, positive and linear.
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Examining a Scatterplot

• Two variables are positively correlated when high
  values of the variables tend to occur together
  and low values of the variables tend to occur
  together.
• The scatterplot slopes upwards from left to
  right.
• Two variables are negatively correlated when
  high values of one of the variables tend to occur
  with low values of the other and vice versa.
• The scatterplot slopes downwards from left to
  right.

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Types of Correlation
As x increases, y tends to decrease.
As x increases, y tends to increase.
Negative Linear Correlation
Positive Linear Correlation
No Correlation
Non-linear Correlation
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Examples of Relationships
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Thought Question 1
What type of association would the following
pairs of variables have positive, negative, or
none?

• Temperature during the summer and electricity
  bills
• Temperature during the winter and heating costs
• Number of years of education and height
• Frequency of brushing and number of cavities
• Number of churches and number of bars in cities
• Height of husband and height of wife

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Thought Question 2

• Consider the two scatterplots below. How does
  the outlier impact the correlation for each plot?
• does the outlier increase the correlation,
  decrease the correlation, or have no impact?

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Measuring Strength Directionof a Linear
Relationship

• How closely does a non-horizontal straight line
  fit the points of a scatterplot?
• The correlation coefficient (often referred to as
  just correlation) r
• measure of the strength of the relationship
  the stronger the relationship, the larger the
  magnitude of r.
• measure of the direction of the relationship
  positive r indicates a positive relationship,
  negative r indicates a negative relationship.

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Correlation Coefficient
Greek Capital Letter Sigma denotes summation or
addition.
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Correlation Coefficient

• The range of the correlation coefficient is -1 to
  1.

If r -1 there is a perfect negative correlation
If r 1 there is a perfect positive correlation
If r is close to 0 there is no linear correlation
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Linear Correlation
r ?0.91
r 0.88
Strong negative correlation
Strong positive correlation
r 0.42
r 0.07
Try
Weak positive correlation
Non-linear Correlation
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Correlation Coefficient

• special values for r
• a perfect positive linear relationship would have
  r 1
• a perfect negative linear relationship would have
  r -1
• if there is no linear relationship, or if the
  scatterplot points are best fit by a horizontal
  line, then r 0
• Note r must be between -1 and 1, inclusive
• r gt 0 as one variable changes, the other
  variable tends to change in the same direction
• r lt 0 as one variable changes, the other
  variable tends to change in the opposite direction

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Examples of Correlations

• Husbands versus Wifes ages
• r .94
• Husbands versus Wifes heights
• r .36
• Professional Golfers Putting Success Distance
  of putt in feet versus percent success
• r -.94

Plot
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Correlation Coefficient

• Because r uses the z-scores for the observations,
  it does not change when we change the units of
  measurements of x , y or both.
• Correlation ignores the distinction between
  explanatory and response variables.
• r measures the strength of only linear
  association between variables.
• A large value of r does not necessarily mean that
  there is a strong linear relationship between the
  variables the relationship might not be linear
  always look at the scatterplot.
• When r is close to 0, it does not mean that there
  is no relationship between the variables, it
  means there is no linear relationship.
• Outliers can inflate or deflate correlations.

Try
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Not all Relationships are LinearMiles per Gallon
versus Speed

• Curved relationship(r is misleading)
• Speed chosen for each subject varies from 20 mph
  to 60 mph
• MPG varies from trial to trial, even at the same
  speed
• Statistical relationship

r-0.06
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Common Errors Involving Correlation

• 1. Causation It is wrong to conclude that
  correlation implies causality.
• 2. Averages Averages suppress individual
  variation and may inflate the correlation
  coefficient.
• 3. Linearity There may be some relationship
  between x and y even when there is no linear
  correlation.

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Correlation and Causation

• The fact that two variables are strongly
  correlated does not in itself imply a
  cause-and-effect relationship between the
  variables.
• If there is a significant correlation between two
  variables, you should consider the following
  possibilities.
• Is there a direct cause-and-effect relationship
  between the variables?
• Does x cause y?

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Correlation and Causation

• Is there a reverse cause-and-effect relationship
  between the variables?
• Does y cause x?
• Is it possible that the relationship between the
  variables can be caused by a third variable or
  by a combination of several other variables?
• Is it possible that the relationship between two
  variables may be a coincidence?

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Example

• A survey of the worlds nations in 2004 shows a
  strong
• positive correlation between percentage of
  countries
• using cell phones and life expectancy in years at
  birth.
• Does this mean that cell phones are good for your
  health?
• No. It simply means that in countries where cell
  phone use is high, the life expectancy tends to
  be high as well.
• What might explain the strong correlation?
• The economy could be a lurking variable. Richer
  countries generally have more cell phone use and
  better health care.

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Example

• The correlation between Age and Income as
  measured on 100
• people is r 0.75. Explain whether or not each
  of these
• conclusions is justified.
• When Age increases, Income increases as well.
• The form of the relationship between Age and
  Income is linear.
• There are no outliers in the scatterplot of
  Income vs. Age.
• Whether we measure Age in years or months, the
  correlation will still be 0.75.

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Example

• Explain the mistakes in the statements below
• My correlation of -0.772 between GDP and Infant
  Mortality Rate shows that there is almost no
  association between GDP and Infant Mortality
  Rate.
• There was a correlation of 0.44 between GDP and
  Continent
• There was a very strong correlation of 1.22
  between Life Expectancy and GDP.

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Warnings aboutStatistical Significance

• Statistical significance does not imply the
  relationship is strong enough to be considered
  practically important.
• Even weak relationships may be labeled
  statistically significant if the sample size is
  very large.
• Even very strong relationships may not be labeled
  statistically significant if the sample size is
  very small.

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Key Concepts

• Strength of Linear Relationship
• Direction of Linear Relationship
• Correlation Coefficient
• Problems with Correlations
• r can only be calculated for quantitative data.