# Correlation coefficient relationship

### Statistics 2 - Correlation Coefficient and Coefficient of Determination

Correlation coefficients are used in statistics to measure how strong a relationship is between two variables. There are several types of correlation coefficient. Test for the significance of relationships between two CONTINUOUS variables. We introduced Pearson correlation as a measure of the STRENGTH of a. The most used correlation coefficients only measure linear relationship. It is therefore perfectly possible that while there is strong non linear relationship between.

Direction The sign of the coefficient indicates the direction of the relationship. If both variables tend to increase or decrease together, the coefficient is positive, and the line that represents the correlation slopes upward. If one variable tends to increase as the other decreases, the coefficient is negative, and the line that represents the correlation slopes downward.

The following plots show data with specific correlation values to illustrate different patterns in the strength and direction of the relationships between variables.

The relationship is positive because as one variable increases, the other variable also increases. The relationship is negative because, as one variable increases, the other variable decreases.

Consider the following points when you interpret the correlation coefficient: It is never appropriate to conclude that changes in one variable cause changes in another based on correlation alone. Only properly controlled experiments enable you to determine whether a relationship is causal.

**The Correlation Coefficient - Explained in Three Steps**

The Pearson correlation coefficient is very sensitive to extreme data values. A single value that is very different from the other values in a data set can greatly change the value of the coefficient.

## Correlation and dependence

These scores are normally identified as X and Y. The pairs of scores can be listed in a table or presented in a scatterplot. This student is represented in the scatterplot by high-lighted and labled "5" dot in the upper-left part of the scatterplot.

Note that the Pearson correlation explained below between these two variables is. Characteristics of a Relationship Correlations have three important characterstics. They can tell us about the direction of the relationship, the form shape of the relationship, and the degree strength of the relationship between two variables. The Direction of a Relationship The correlation measure tells us about the direction of the relationship between the two variables.

The direction can be positive or negative.

## Overview of Correlation

In a positive relationship both variables tend to move in the same direction: If one variable increases, the other tends to also increase. If one decreases, the other tends to also.

In a negative relationship the variables tend to move in the opposite directions: If one variable increases, the other tends to decrease, and vice-versa. The direction of the relationship between two variables is identified by the sign of the correlation coefficient for the variables. Postive relationships have a "plus" sign, whereas negative relationships have a "minus" sign.

The Form Shape of a Relationship: The form or shape of a relationship refers to whether the relationship is straight or curved. A straight relationship is called linear, because it approximates a straight line.

### Psychological Statistics

A curved relationship is called curvilinear, because it approximates a curved line. An example of the relationship between the Miles-per-gallon and engine displacement of various automobiles sold in the USA in is shown below. This is curvilinear and negative. In this course we only deal with correlation coefficients that measure linear relationship. There are other correlation coefficients that measure curvilinear relationship, but they are beyond the introductory level. The Degree Strength of a Relationship Finally, a correlation coefficient measures the degree strength of the relationship between two variables.

The mesures we discuss only measure the strength of the linear relationship between two variables.

Two specific strengths are: