Chapter Eight

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Chapter Eight

• Correlation and Prediction

PowerPoint Presentation created by Dr. Susan R.
BurnsMorningside College
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Correlation

• Correlation is the extent to which two variables
  are related.
• If the two variables are highly related, then
  knowing the value of one of them will allow you
  to predict the other variable with considerable
  accuracy.
• The less highly related the variable, the less
  accurate your ability to predict when you know
  the other.

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The Nature of Correlation

• Often used as means for prediction, correlation
  tells us how related two variables are.
• However, note that even though two variables may
  be highly correlated, you should not assume that
  one variable causes the other.
• CORRELATION DOES NOT IMPLY CAUSATION.
• For example, there is the third variable
  possibility (i.e., there may be additional
  variable(s) that are causing the two things you
  are investigating to be related to each other).

Theres a significant NEGATIVE correlation
between the number of mules and the number of
academics in a state, but remember, correlation
is not causation
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The Scatterplot Graphing Correlations

• Also known as the scatter diagram, the
  scatterplot allows us to visually see the
  relation between two variables.
• One variable is plotted on the ordinate and the
  other on the abscissa.
• Although you can list either variable on either
  axis, it is common to place the variable you are
  attempting to predict on the ordinate.
• Positive correlations occur when both variables
  move in the same direction (e.g., as SAT scores
  increase, so to do GPAs).
• Negative Correlations occur when one variable
  increases, the other decreases (e.g., as age
  increases, the number of speeding tickets
  decrease).

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The Scatterplot Graphing Correlations
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The Pearson Product Moment Correlation Coefficient

• The correlation coefficient is the single number
  that represents the degree of relation between
  two variables.
• The Pearson Product-Moment Correlation
  Coefficient (symbolized by r) is the most common
  measure of correlation researchers calculate it
  when both the X variable and the Y variable are
  interval or ration scale measurements.
  Mathematically, it can be defined as the average
  of the cross-products of z-scores.
• The raw score formula for r is

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The Range of r Values

• The Range of r correlation coefficients can
  range in value from -1.00 to 1.00.
• A correlation of -1.00 indicates a perfect
  negative correlation between the two variables of
  interest. That is, whenever there is an increase
  of one unit in one variable, there is always the
  same proportional decrease in the other variable.

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The Range of r Values

• The Range of r correlation coefficients can
  range in value from -1.00 to 1.00.
• A zero correlation means there is little or no
  relation between the two variables. That is, as
  scores on one variable increase, scores on the
  other variable may increase, decrease, or not
  change at all.

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The Range of r Values

• The Range of r correlation coefficients can
  range in value from -1.00 to 1.00.
• Perfect positive correlation occurs when you have
  a value of 1.00 and as we see an increase of one
  unit in one variable, we always see a
  proportional increase in the other variable.
• The existence of a perfect correlation indicates
  there are no other factors present that influence
  the relation we are measuring. This situation
  rarely occurs in real life.

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Interpreting Correlation Coefficients

• Statistically significant results mean that a
  research result occurred rarely by chance.
• If the correlation you calculate is sufficiently
  large that it would occur rarely by chance, then
  you have reason to believe that these two
  variables are related.
• The standard by which significance in psychology
  is determined is at the .05 level.
• That is, a result is significant when it occurs
  by chance 5 times out of a hundred.
• Researchers who are more caution may choose to
  adopt a .01 level of significance.

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Effect Size

• Even though statistical significance is an
  important component of psychological research, it
  may not tell us very much about the magnitude of
  our results.
• Effect size refers to the size or magnitude of
  the effect an independent variable (IV) produced
  in an experiment or the size or magnitude of a
  correlation.
• Effect size calculation is important because,
  unfortunately, a research result can be
  significant and yet the effect size may be quite
  small.
• An example of this situation occurs as sample
  size gets larger, the critical value needed to
  achieve significance becomes smaller.

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Effect Size

• To calculate the effect size for the Pearson
  product-moment correlation, all you have to do is
  square the correlation coefficient.
• r2 is known as the coefficient of determination.
• Multiply the coefficient of determination by 100
  and you will see what percentage of the variance
  is accounted for by the correlation.
• The higher r2 becomes, the more variance is
  accounted for by the relation between the two
  variables under study.
• Lower r2 values indicate that factor, other than
  the two variables of interest are influencing the
  relation in which we are interested.

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Prediction

• Generally speaking, regression refers to the
  prediction of one variable from our knowledge of
  another variable.
• We label the variable that is being predicted as
  the Y variable and refer to it as the criterion
  variable.
• We label the variable that we are predicting from
  as the X variable and refer to it as the
  predictor variable.
• In other words, we use X to predict Y.

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Prediction

• The Regression Equation
• The regression equation is the statistical basis
  of prediction

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The Regression Equation

• The regression line is a graphical display of the
  relation between the values on the predictor
  variable and predicted values on the criterion
  variable. It is similar to the scatterplot used
  to display correlations.
• The calculation of b can be done a couple of
  ways

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The Regression Equation

• A second way to calculate involves the following
  formula

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The Regression Equation

• The calculation of a is as follows

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Constructing the Regression Line

• There are two ways to construct the regression
  line
• First, you could calculate several predicted
  values, plot these values, connect the points,
  and then extend the line to the Y intercept. This
  procedure works, but carries with it the
  potential for calculation errors and inaccurate
  points.
• The second procedure is less likely to have
  calculation errors
• Locate the Y intercept (a)
• Plot My and Mx
• The line that passes through these points is the
  regression line.
• Remember, the steepness of the regression line is
  known as the slope, whereas the point at which
  this line crosses the vertical axis is called the
  Y intercept.

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Constructing the Regression Line