Agenda Correlation

CORRELATION Co-relation 2 variables tend to “go together” Does knowing a person’s score on one variable give you an idea of their score on other variable? Predict the degree of co-occurrence or association between 2 variables

Correlation A measure of association between –Two ordinal variables –An ordinal and an interval/ratio variables –Two variables Correlation analysis examines if one variable changes by a certain amount, how much the other variable would change in which direction

Example: Test Score  Ten students took a Chemistry class and a Biology class together  Compared final exam scores in two classes J I H G F E D C B A BiologyChemistry

Example: Scatter Plot  Students who get higher score in the Chemistry class also get higher score in the Biology class

Positive Correlation When scores of two variables move together in the same direction, we say that these variables are positively (or directly) correlated There is a positive correlation between between the chemistry final score and the biology final score When two variables are positively correlated, the scatter plot shows a trend line that runs from lower-left to upper-right

Example: Test Score  The same students also took the Art class  Compared final exam scores in Chemistry and Art J I H G F E D C B A ArtChemistry

Example: Scatter Plot  Students who get higher score in the Chemistry class get lower score in the Art Classc

Negative Correlation When scores of two variables move in opposite directions, we say that these variables are negatively (or inversely) correlated There is a negative correlation between between the chemistry final score and the art final score When two variables are negatively correlated, the scatter plot shows a trend line that runs from upper-left to lower-right

Example: Test Score  The same ten students also took an English class together  Compare the English final score with the Chemistry final score J I H G F E D C B A EnglishChemistry

Example: Test Score  Score in Chemistry and Score in English are not related

No Correlation When the change in one variable does not affect the change in another variable, we say these variables have no correlation There is no correlation between the chemistry final score and the English final score When two variables have no correlation, the scatter plot shows the dots scattered throughout the grids

SIGN 0: No relationship Positive: + As one variable gets bigger, so does the 2nd Negative: - As one variable gets bigger, the 2 nd gets smaller

Should your height in inches have anything to do with…. How much you weigh? What size shoe your wear? How old you are? What portion of your tuition you pay? Your college GPA? Your skill in basketball? Your attractiveness to the opposite sex? Your eye color?

TASK 1 Get in groups of 8-10 Line up in order of height NameHeight

Line up in order of shoe size (men, add 2 sizes) NameShoe size Line up in order of how much $ you spent the last time you bought something Name$ spent Where any of the 3 orders similar?

Do they go together? Draw a scatterplot Each person has 2 scores, 1 on each variable Plot each person’s score Is there a pattern? E.g. as one variable gets bigger, does the other get bigger too? (Or smaller?) A strong relationship makes a diagonal line

Draw a line through the dots

r = -.4 p =.01 Fall 2002 Midterm score Minutes

Correlation between midterm score and time to finish test Fall 2002

Correlation Co-efficient A number that indicates how strongly and in which direction 2 variables are correlated with each other A correlation co-efficient varies –1 to +1 Indicated as r r = +1: Perfect positive correlation –If one variable increases by x%, another variable also increases by x% r = - 1: Perfect negative correlation r = 0: No correlation

Correlation Co-efficient +10 NegativePositive Stronger Weaker Perfect None

Significance Test Correlation co-efficient also comes with significance test (p-value) p=.05:.05 probability of no correlation in the population = 5% risk of TYPE I Error = 95% confidence level If p<.05, reject H 0 and support Ha at 95% confidence level

Chemistry & Biology ChemistryBiology A 6265 B 7068 C 7269 D 7682 E 7883 F 8085 G 8883 H 8990 I 93 J 9498 r =.939 p =.000 Significant, Positive and Strong Correlation

Chemistry & Art ChemistryArt A 6290 B 7080 C 7274 D 7678 E 77 F 8077 G 8875 H 8974 I 9073 J 9471 r = p =.002 Significant, Negative and Strong Correlation

Chemistry & English ChemistryEnglish A 6285 B 7060 C 7290 D 7675 E 7880 F 90 G 8875 H 8960 I 9089 J 9476 r = p =.714 Non- Significant Correlation

SIZE / STRENGTH Ranges from –1 to or close to 0 indicates NO relationship +/ weak +/ moderate +/-.>6 -.8 strong +/-.>8 -.9 very strong +/ perfect Negative relationships are NOT weaker! USE THIS FOR YOUR ANALYSIS!

Types of Correlation r Use Spearman rho’s correlation if one or both of your variables are ordinal Use Pearson’s r correlation if both of your variables are interval or ratio You can interpret both kinds of correlation in the same way

STATISTICAL SIGNIFICANCE How different from 0 must r be for there to be some kind of relationship? Depends on size of sample, other factors IF statistically significant, safe to conclude there is a relationship If p <.05 –May also be indicated by * next to the correlation

Limitations of r (correlation) Correlation does NOT equal cause C AB Linear relations only AB BA  “Third variable” ? ? “Problem of direction”

TASK 2 w ith your neighbor: Write down 3 examples variables expect to have no association variables expect to have positive association variables expect to have negative association. Write down 2 examples variables expect a weak to medium association variables expect a strong or very strong association.

TASK: Graph job aptitude vs. performance Graph self-esteem vs. depression John 91420Jake 84621Milly 83822Homer 67725Leslie 55827June 09829Bob Jane Joe DepressionSelf-EsteemPerformance Job Aptitude Name

r =.84, p <.005

r = -.92, p <.001

Example from class data Height in inches, weight, age, GPA, # semesters completed, height in cm, who pays tuition (high score = self), where sit in class (high score = towards front) For which pairs do you expect no relation? Positive? Negative? Strongest? Weakest?

*.544 * 1.00 * Hgt cm * Sems GPA *.725 *.016 Shoe *.108 Wt Ht Age Hgt cm Sems compGPAShoeWt.Ht.Age