‘Pearson Correlation’ | Where is it used | How is it interpreted |

QUESTION

What do you understand by ‘Pearson Correlation’? Where is it used and how is it interpreted?

Course: Educational Statistics

Course code 8614

Level: B.Ed Solved Assignment 

ANSWER 

 The Pearson Correlation

The most commonly used correlation is the Pearson Correlation. It is also known as Pearson product-moment Correlation. It measures the degree and the direction of linear relationship of between two variables. It is denoted by r, and r = degree to which X and Y vary together / degree to which X and Y vary separately = co-variability of X and Y / variability of X and Y vary separately

To calculate the Pearson correlation r we use the formula

where SP is the sum of the product of deviation.

Two formulas (definitional and computational) are available to calculate the sum of square of product. Both formulas are given in the following box.

SS is sum of squares, SSx is the sum of squares of the variable X and SSy is the sum of squares of variable Y. In the following lines different formulas are given to calculate SSx and SSy. These formulas are categorized as definitional and computational. The definitional formulas for sum of squares of variable X are:

The computational formulas for sum of squares of variable X are

The definitional formulas for sum of squares of variable Y are:

The computational formulas for sum of squares of variable Y are:

It should be kept in mind that whichever formula one uses, it will yield similar result.

 

 Using and Interpreting Pearson Correlation

First let us have a brief discussion about where and why we use correlation. The 

discussion follows under following headings.

i)  Prediction

If two variables are known to be related in some systematic way, it is possible to use one variable to make prediction about the other. For example, when a student seeks admission in a college, he is required to submit a great deal of personal information, including his scores in SSC annual/supplementary examination. The college officials want this information so that they can predict that student’s chance of success in college. 

ii)  Validity

Suppose a researcher develops a new test for measuring intelligence. It is necessary that he should show that this new test valid and truly measures what it claims to measure. One common technique for demonstrating validity is to use correlation. 

If newly constructed test actually measures intelligence, then the scores on this test should be related to other already established measures of intelligence –  for example standardized IQ tests, performance on learning tasks, problem-solving ability, and so on. The newly constructed test can be correlated to each of these measures to demonstrate that the new test is valid. 

iii)  Reliability

Apart from determining validity, correlations are also used to determine reliability. A measurement procedure is reliable if it produces stable and consistent measurement. It means a reliable measurement procedure will produce the same (or nearly same) scores when the same individuals are measured under the same conditions. One common way to evaluate reliability is to use correlations to determine relationship between two sets of scores.

iv)  Theory Verification

Many psychological theories make specific predictions about the relationship between two variables. For example, a theory may predict a relationship between brain size and learning ability; between the parent IQ and the child IQ etc. In each case, the prediction of the theory could be tested by determining the correlation between two variables. 

Now let us have a few words on interpreting correlation. For interpreting correlation following consideration should be kept in mind.

i)  Correlation simply describes a relationship between two variables. It does not explain why two variables are related. That is why correlation cannot be interpreted as a proof of cause and effect relationship between two variables. 

ii)  The value of the correlation cannot be affected by range of scores represented in the data. 

iii)  One or two extreme data points, often called outliers, can have a dramatic effect on the value of the correlation.

iv)  When judging how good a relationship is, it is tempting to focus on the numerical value of the correlation. For example, a correlation of + 5 is halfway between 0 and 1.00 and therefore appears to represent a moderate degree of relationship. Here it should be noted that we cannot interpret correlation as a proportion. Although a correlation of 1.00 means that there is a 100% perfectly predictable relationship between variables X and Y; but a correlation of .5 does not mean that we can make 

a prediction with 50% accuracy. The appropriate process of describing how accurately one variable predicts the other is to square the correlation. Thus a correlation of r = .5 providesr²= .5²= .25, 25% accuracy. (The value r² is called coefficient of determination because it measures the proportion of variability in one variable that can be determined from the relationship with the other variable)