Correlation

and the strength of the relationship Definition Shows the direction and the strength of the relationship between two variables.

Scatter plot for correlational data

Positive correlation: when a small amount of one variable is associated with a small amount of another variable, and a large amount of one variable is associated with a large amount of the other .

Negative correlation: when a small amount of one variable is associated with a large amount of another variable, and a large amount of one variable is associated with a small amount of the other. WEIGHT 220 210 200 190 180 170 160 150 140 PULLUPS 120 110 100 90 80 70

Examples of positive and negative relationships Beer Coffee

{ } { } Perfect Correlation As X changes a unit, Y changes a specific increment. { } { }

STRENGTH WEIGHT example A perfect correlation David Peter Mark Jim Ken 200 190 180 170 160 150 140 STRENGTH 100 90 80 70 David Peter Mark Jim Ken A perfect correlation

Is this correlation perfect?

120 110 100 90 80 STRENGTH 70 140 150 160 170 180 190 200 210 220 WEIGHT

Zero correlation: when there is no association between two variables.

Example A zero correlation

4 120 15 7 8 13 10 17 110 9 5 20 100 19 IQ 2 90 12 1 14 80 6 18 11 16 3 70 140 150 160 170 180 190 200 210 220 WEIGHT

Three degrees of relationship Zero weak Perfect

Examples of different values for relationships

practice For each pair, tell whether r is high, moderate, low or zero.± 1- Correlation between math ability and shoe size in K-12 2- Height and intelligence in adult population 3- Crime rate and the number of churches 4- The academic degree and income The number of cars and the number of accidents. The height and age of k-12 students. k-12 students’ scores on a math test and a science test k-12 students’ scores on a math test and a PE test The birthrate and social economic level The length of the base of a square and the length of its diagonal.

Conclusion If A correlates with B, three possible causal relationship exist A causes B, B causes A, or C causes both A and B/

Interpreting correlations Correlation does not demonstrate causation Number of books at home and students’ academic achievement The faster windmills are observed to rotate, the more wind is observed to be The number of storks and birth rate in Denmark Earlier wake- up times are consistently related to higher GPA.

Real examples Correlation Confusion Eating chocolate, number of acnes. Drug use and income Crime rate and the number of death penalties Number of youth Joining terrorists and unemployment

Changing Together

Changing Together

Factors affecting correlation

Restricted range

Cognitive Development Fear of Death r = -.40 r =.52 r = .10 First graders Sixth Graders

Correlation of small sample and population

Influence of outlier on correlation

Types of correlation index

Types of correlation index

The Correlational Research Strategy Chapter 12

Correlational Research The goal of correlational research is to describe the relationship between variables and to measure the strength of the relationship.

3 characteristics A correlation describes three characteristics of a relationship. The direction (positive / negative)of the relationship. The form (linear/ nonlinear) of the relationship. The consistency or strength (magnitude) of the relationship.

Direction In a positive relationship, there is a tendency for two variables to change in the same direction. In a negative relationship, there is a tendency for two variables to change in opposite directions.

3 forms of correlation Linear correlation Data points in the scatter plot tend to cluster around a straight line. The size of increase in Y is consistently predictable (not accurately). (height and age)- Pearson Monotonic correlation The relationship is consistent and predictable, but not linear. (practice & Performance) Spearman Nonlinear correlation When all the points on the scatter diagram tend to lie near a smooth curve.

2 forms of correlation

2 forms of correlation Example?

chi- square If both variables are non- numerical, the relationship is typically evaluated by organizing the data in a matrix. The matrix shows the frequency or number of individuals in that cell and the data are evaluated using a chi- square hypothesis test

Chi- Square Drama Action Comedy Latino 4 12 10 Asian 8 White 19 18 14

APPLICATIONS OF THE CORRELATIONAL STRATEGY Prediction (SAT & GPA) Reliability and Validity (Test & Retest) Evaluating Theories (IQ and Math)

coefficient of determination The squared value of a correlation is called the coefficient of determination and measures the percentage of variability in one variable that is determined, or predicted, by its relationship with the other variable.

Sample size With a sample of two individuals, you will always obtain a perfect correlation of 1.00 As the sample size increases, it becomes increasingly more likely that the sample correlation accurately represents the real relationship that exists in the population.

Advantages Can identify variables and describe relationships between variables that might suggest further investigation using the experimental strategy to determine cause- and- effect relationships. Allow researchers an opportunity to investigate variables that would be impossible or unethical to manipulate. Correlational studies tend to have high external validity.

Weaknesses - Internal Validity Correlational studies tend to have low internal validity. A correlational study does not determine which variable is the cause and which is the effect. The third- variable & Directionality problems .

RELATIONSHIPS WITH MORE THAN TWO VARIABLES For studying multivariate relationships we use a statistical procedure known as multiple regression.