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Descriptive Statistics: Correlation. Describes the relationship between two or more variables. Describes the strength of the relationship in terms of a number from -1.0 to +1.0. Describes the direction of the relationship as positive or negative. Types of Correlations.
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Descriptive Statistics: Correlation Describes the relationship between two or more variables. Describes the strength of the relationship in terms of a number from -1.0 to +1.0. Describes the direction of the relationship as positive or negative.
Types of Correlations • Variable X increases • Variable Y increases Positive Correlation Value ranging from .00 to 1.00 Example: the more you eat, the more weight you will gain
Types of Correlations • Variable X decreases • Variable Y decreases Positive Correlation Value ranging from .00 to 1.00 Example: the less you study, the lower your test score will be
Types of Correlations • Variable X increases • Variable Y decreases Negative Correlation Value ranging from -1.00 to .00 Example: the older you are, the less flexible your body is
Types of Correlations • Variable X decreases • Variable Y increases Negative Correlation Value ranging from -1.00 to .00 Example: the less time you study, the more errors you will make
Correlation Strength • .00 - .20 Weak or none • .20 - .40 Weak • .40 - .60 Moderate • .60 - .80 Strong • .80 - 1.00 Very strong
Positive or Negative? • IQ and reading achievement • Anxiety and test scores • Amount of calories consumed and weight gain. • Amount of exercise and weight gain • Reading achievement and math achievement • Foot size and math ability
Caution! • Correlation does not indicate causation. • Correlation only establishes that a relationship exists; it reflects the amount of variability that is shared between two variables and what they have in common. Examples: • Amount of ice sold and number of bee stings. • SAT scores and GPA in college.
A Picture of Correlation • A scattergram or scatter plot visually represents a correlation • The X axis is on the horizontal • The Y axis is on the vertical.
Correlation: IQ and GPA • IQ GPA • 110 2.5 • 140 4.0 • 80 1.0 • 100 2.0 • 130 3.5 • 90 1.5 • 120 3.0 • 70 .5
Correlation: IQ and Errors • IQ Errors • 80 14 • 120 6 • 100 10 • 90 12 • 130 4 • 110 8 • 140 2 • 70 16
Correlation: IQ and Weight • IQ Weight • 120 170 • 100 160 • 70 120 • 140 130 • 90 200 • 130 110 • 80 150 • 110 140
Caution • Do not interpret the coefficient of correlation as a percent! • If you want to know the percentage of variance in one variable that is accounted for by the variance in the other variable, compute the coefficient of determination
Coefficient of Determination • Square the coefficient of correlation. • r = .50 • r 2 = .25 or 25 % • Twenty five percent of the variance in one variable can be accounted for by the variance in the other variable.
Example: Coefficient of Determination • The correlation between IQ and reading at its highest level: r = .60 • r2 = .36 or 36 % • Thirty six percent of reading achievement is related to IQ. Reading achievement and IQ share 36% of the variance.
Factors Influencing Correlation • When interpreting the correlation coefficient, always consider the nature of the population in which the two variables were observed. • The correlation coefficient will vary from one population to another.
Factors Influencing Correlation • The relationship of variables may differ from population to population. • Example: Physical prowess and age are correlated between the ages of 10 and 16. • Example: Physical prowess and age are not correlated between the ages of 20 and 26.
Factors Influencing Correlation • Higher correlations are expected in a heterogeneous population than in a homogeneous one. • Example: In elementary and high school, there is a positive correlation between height and success in basketball. • Example: In the pros, there is no such correlation.
Factors Influencing Correlation • There may be a correlation between two variables not because there is a relationship between them but because both are related to a third variable. • Example: Average teacher salary for 20 years and the cost of hard liquor.
Choosing Correlation Formulas • X is nominal data • Y is nominal data Correlation Formula: Phi coefficient Example: Correlation of sex (male/female) and choice of car color (red, black, blue, white, silver)
Choosing Correlation Formulas • X is nominal data • Y is ordinal data Correlation Formula: Rank biserial coefficient Example: Correlation of race and rank in school
Choosing Correlation Formulas • X is nominal data • Y is interval data Correlation Formula: Point biserial Example: Correlation of sex and GPA
Choosing Correlation Formulas • X is ordinal data • Y is ordinal or interval data (interval data must be converted to ordinal) • Correlation Formula: Spearman rank coefficient • Example: Correlation between rank and GPA
Choosing Correlation Formulas • X is interval • Y is interval • Correlation Formula: Pearson correlation coefficient • Example: Age and the number of minutes it takes to solve a problem