# Variable Associations and Correlation Coefficients

Updated on October 20, 2014

What are the different types of associations found among variables? What tools can be used to display these relationships?

Correlation is the “association between scores on two variables” (Aron, Coups, & Aron, 2013, p. 440). There are five main types of correlation among variables; linear correlation, curvilinear correlation, positive correlation, negative correlation, and no correlation. Linear correlation is the relationship between two variables this association shows up when the dots on a scatter plot roughly follows a straight line. Curvilinear correlation is the association between two variables that can be seen on a scatter diagram however unlike linear correlation this correlation does not follow a straight line, but instead follows a systematic pattern. Positive correlation is an association where when the amount of one variable increases, the amount of a second variable also increases (About Education, 2014). Negative Correlation is the almost opposite of positive correlation; high scores go with low scores, mediums with mediums, and lows with highs. Negative correlation appears on a scatter diagram when the dots do follow a straight line, but the line slopes down and to the right. The last type of correlation is no correlation this happens when there is “no systematic relationship between two variables” (Aron, Coups, & Aron, 2013, p. 446). The best tool to show the different types of associations found among variables is the graphical representation know as a scatter diagram or a scatter plot.

## Reference

Aron, A., Aron. E., Coups. E. (2014). Statistics for Psychology Pearson Education Inc. 2014.

What are the four types of correlation coefficients? After reviewing quantitative research, how can the correlation coefficients be used to draw conclusions about the relationship of the data?

A correlation coefficient is the “measure of degree of linear correlation between two variables ranging from -1 (a perfect negative linear correlation) through 0 (no correlation) to +1 (a perfect positive correlation)” (Aron, Coups, & Aron, 2014, p. 453). There are four main types of correlation coefficients they are: Pearson’s product-moment correlation, Spearman rank-order correlation, point-biserial correlation coefficient, and the phi coefficient.

Pearson’s product-moment correlation (r), also known as the Pearson correlation coefficient, is a measurement of the degree of a linear relationship between two variables, typically labeled X and Y (Stockburger, 2014). Spearman rank-order correlation (, also signified by rs) is based on rank-order scores and is less affected by outliners; it is normally used in situations where the scatter diagram shows a curvilinear relationship between two variables (Aron, Coups, & Aron, 2014, p. 476). A point-biserial coefficient (rpb) relates observed item responses to a total scores; it is specifically used when one set of the data is dichotomous in nature (Measured Progress, 2014). The phi coefficient is a measure of the degree of association between two binary variables. This measure is similar to the correlation coefficient in its interpretation. The phi coefficient is a measure of the degree of correlation between two binary variables; the interpretation is similar to the correlation coefficient (Pmean, 2014). The formula for the phi coefficient is .

Correlation coefficients are commonly used to draw conclusions about the relationship of the data. The correlation coefficient can be used to determine if the data that results from quantitative research is statistically significant. When the data is not statistically significant it means that the data occurred from chance alone and the results cannot be consistently replicated. To test the statistical relationship of the data the correlation coefficient is used in a hypothesis test. The formula for this hypothesis test is H0: p=0 HA: p≠0 ; H0 is rejected if T≥ tα/2, n-2 or if T≤ -tα/2,n-2. If the null hypothesis is rejected then the conclusion is that correlation coefficient is not equal to zero and that there is statistically significant linear correlation between the two variables. However if the null hypothesis is not reject then the conclusion is that the correlation coefficient is equal to zero and that either there is no linear relationship between variables.

## References

Aron, A., Aron. E., Coups. E. (2014). Statistics for Psychology Pearson Education Inc. 2014.

Correlation Coefficients. (n.d.). AEA 267 Assessment. Retrieved October 10, 2014, from https://www.aea267.k12.ia.us/assessment/statistics/descriptive-statistics/correlation/correlation-coefficients/

Discovering the Point Biserial. (n.d.). Measured Progress. Retrieved October 10, 2014, from http://www.measuredprogress.org/learning-tools-statistical-analysis-the-point-biserial

Spearman's Rank-Order Correlation. (n.d.). Laerd Statistics. Retrieved October 10, 2014, from https://statistics.laerd.com/statistical-guides/spearmans-rank-order-correlation-statistical-guide.php

Stockburger, D. (n.d.). Correlation . Introductory Statistics: Concepts, Models, and Applications . Retrieved October 10, 2014, from http://www.psychstat.missouristate.edu/introbook/sbk17.htm

## Chapter 11: Correlation

Correlation

Association between scores on two variables

e.g., age and coordination skills in children, price and quality

Graphing Correlations
The Scatter Diagram

Steps for making a scatter diagram

1. Draw axes and assign variables to them

2. Determine range of values for each variable and mark on axes

3. Mark a dot for each person’s pair of scores

Graphing Correlations
The Scatter Diagram

Graphing Correlations: The Scatter Diagram

Patterns of Correlation

Linear correlation

Curvilinear correlation

No correlation

Positive correlation

Negative correlation

Degree of Linear Correlation
The Correlation Coefficient

Figure correlation using Z scores

Cross-product of Z scores

Multiply Z score on one variable by Z score on the other variable

Correlation coefficient

Average of the cross-products of Z scores

Degree of Linear Correlation
The Correlation Coefficient

General formula for the correlation coefficient:

Positive perfect correlation: r = +1

No correlation: r = 0

Negative perfect correlation: r = –1

Correlation and Causality

Three possible directions of causality:

1. X Y

2. X Y

3. Z

X Y

Correlation and Causality

Correlational research design

Correlation as a statistical procedure

Correlation as a research design

Issues in Interpreting the Correlation Coefficient

Statistical significance

Proportionate reduction in error

r2

Used to compare correlations

Restriction in range

Unreliability of measurement

Curvilinearity

Spearman’s rho

Power for Studies Using
Correlation Coefficient
(.05 significance level)

Table indicated below here

Approximate Sample Size for
80% Power for Correlation Studies (.05 significance level)

Table indicated below here

Correlation in Research Articles

Scatter diagrams occasionally shown

Correlation matrix

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4

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