SSF1063: Statistics for Social Sciences

LU11: Linear Correlation 31st March 2008

CorrelationCorrelation To examine the relationship between two

(bivariate) continuous variables Some commonly used jargons:

Linear and Non-linear Relationship Correlation Scatterplot diagram Correlation Coefficient Pearson (Product-Moment r)

A descriptive statistics Describe the degree and direction of

linear correlation between two quantitative variables

Class size and students’ test performance Hours of work and income

CorrelationCorrelation

The General Idea About The General Idea About CorrelationCorrelation Two variables may correlate to each other in

3 possible ways: Positive RelationshipPositive Relationship: Both variables vary in the same same direction - as

one goes up, the other goes up. Eg. salary and years of education are

positively correlated because the people who make the highest salaries tend to be the ones who have gone to school the longest

Negative RelationshipNegative Relationship: Two variables vary in the oppositeopposite

direction - as one goes up, the other goes down.

For example, the longer the hour of working mother works outside the home, less time is available to spend with children.

The General Idea About The General Idea About CorrelationCorrelation

The General Idea About The General Idea About CorrelationCorrelation Zero RelationshipZero Relationship: Two variables vary with freely without any

kind of rules or relationship. For example, the relationship between

personality fluctuations and the movement of distant stars has a zero correlation. Bona fide studies have found that there is absolutely no correlation between the position of stars at the time you are born and the true nature of your personality

The degree of correlation between two variables can be established using two methods:

ScatterplotScatterplot – a graph with plotted values for two variables being compared

Correlation CoefficientCorrelation Coefficient

Degree of Correlation: How Strongly Are Variables Correlated?

Pearson correlation only measures relationships in variables that are linearly related.

The correlation coefficient shows the slope of straight line passing through the data cloud. It ranges between –1 r +1.

r shows the magnitude and direction of a relationship

Watch up A correlation coefficient of zero does not

mean that there isn’t any relationship between two variables. It is possible that the two variables are strongly related in nonlinear way.

ScatterplotScatterplot

Steps to get scarterplot diagram:

Graph Scatter Simple

Define Select the appropriate variables to the test variable cell OK

Correlation CoefficientCorrelation Coefficient

CorrelationCorrelation characterizes the existence of a relationship between variables

Correlation coefficientCorrelation coefficient is a quantitative indicator of the degree of relationship between variables

There are several measures of correlations depending to the nature of variables being investigated. The most commonly used measure of correlation is called Pearson product moment Pearson product moment correlation correlation (denoted by rr)

Each measure of correlation uses different kind of rules and modeling

CORRELATIONCORRELATION

Computing The Correlation Computing The Correlation CoefficientsCoefficients

Pearson Product Moment (r)Pearson Product Moment (r)

The Pearson correlation r can be computed using the formula:

Eg 1.Eg 1.

Compute the correlation between the men’s Heights (in inches) and Weights (in pounds) for the following data:

ManManHeight Height

(X)(X)

Weight Weight

(Y)(Y)

M.P. 72 190

T.D. 66 135

C.Q. 69 155

C.Y. 72 165

D.P. 71 155

Note that the paired data are in the form of interval-interval. Thus, the most suitable correlation measure would be Pearson product moment.

Computation of Computation of rr ::

Correlation from SPSS – Analyse Correlate Bivariate

Correlations

1 .880**

.000

474 474

.880** 1

.000

474 474

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

Current Salary

Beginning Salary

Current SalaryBeginning

Salary

Correlation is significant at the 0.01 level (2-tailed).**.

The correlation value, 0.767, indicate that there exist positive linear relationship between the body length and body weight in the samples of lizards.

Correlations

1 .767**

. .000

25 25

.767** 1

.000 .

25 25

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

BODYL

WEIGHT

BODYL WEIGHT

Correlation is significant at the 0.01 level(2-tailed).

**.

Pearson Correlation 0.8335Sig. (2-tailed) 0.0795N 5

Pearson Correlation -0.9499Sig. (2-tailed) 0.0000N 10

Pearson Correlation 0.1047Sig. (2-tailed) 0.8669N 5

Correlation Coefficient -0.0952Sig. (2-tailed) 0.8225N 8

Pearson Eg.1Pearson Eg.1

Pearson Eg.2Pearson Eg.2

Pearson Eg.3Pearson Eg.3

Spearman Eg.1Spearman Eg.1

When is a correlation ‘strong enough’?When is a correlation ‘strong enough’?(Guilford’s rough guide only)(Guilford’s rough guide only)

< 0.2< 0.2 slight; almost negligible relationship

0.2 – 0.40.2 – 0.4 low correlation; definite but small relationship

0.4 – 0.70.4 – 0.7 moderate correlation; substantial relationship

0.7 – 0.90.7 – 0.9 high correlation; marked relationship

> 0.9> 0.9 very high correlation; very dependable relationship

Words of CautionWords of Caution

Examine your data distribution (i.e. using scatterplot) before you do anything with the correlation and make sure you know the do’sdo’s and don’tsdon’ts with correlation coefficients!Correlation coefficient is just an index of relationship which tells nothing about the ‘cause and effect’cause and effect’ of the relationship! The correlation coefficient tells you nothing about the reasons of the relationship, if any.Limit yourself to linearlinear relationship if you don’t have adequate statistical background!