When the relationship between two variables is perfect and inverse, what is the value of r?

Values of the Pearson Correlation

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PPA 696 RESEARCH METHODS
CORRELATION
What does a positive r value mean?
What is the level of correlation if the value of r?
Is an inverse relationship positive or negative?
What does r 1.00 indicate quizlet?

David M. Lane

Prerequisites

Introduction to Bivariate Data

Learning Objectives

Describe what Pearson's correlation measures
Give the symbols for Pearson's correlation in the sample and in the population
State the possible range for Pearson's correlation
Identify a perfect linear relationship

The Pearson product-moment correlation coefficient is a measure of the strength of the linear relationship between two variables. It is referred to as Pearson's correlation or simply as the correlation coefficient. If the relationship between the variables is not linear, then the correlation coefficient does not adequately represent the strength of the relationship between the variables.

The symbol for Pearson's correlation is "ρ" when it is measured in the population and "r" when it is measured in a sample. Because we will be dealing almost exclusively with samples, we will use r to represent Pearson's correlation unless otherwise noted.

Pearson's r can range from -1 to 1. An r of -1 indicates a perfect negative linear relationship between variables, an r of 0 indicates no linear relationship between variables, and an r of 1 indicates a perfect positive linear relationship between variables. Figure 1 shows a scatter plot for which r = 1.

Figure 1. A perfect positive linear relationship, r = 1.

Figure 2. A perfect negative linear relationship, r = -1.

Figure 3. A scatter plot for which r = 0. Notice that there is no relationship between X and Y.

With real data, you would not expect to get values of r of exactly -1, 0, or 1. The data for spousal ages shown in Figure 4 and described in the introductory section has an r of 0.97.

Figure 4. Scatter plot of spousal ages, r = 0.97.

Figure 5. Scatter plot of Grip Strength and Arm Strength, r = 0.63.

The relationship between grip strength and arm strength depicted in Figure 5 (also described in the introductory section) is 0.63.

Please answer the questions:

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PPA 696 RESEARCH METHODS

CORRELATION

Correlation coefficients are statistics which can help to describe data sets which contain variables measured at the interval and ratio levels. Correlation coefficients are measures of association between two (or more) variables.

Correlation is a measure of association that tests whether a relationship exists between two variables. It indicates both the strength of the association and its direction (direct or inverse). The Pearson product-moment correlation coefficient, written as r, can describe a linear relationship between two variables.

For example is there a relationship between:
the budget of the police department and the crime rate?
the hours of batting practice and a player's batting average?

The value of r can range from 0.0, indicating no relationship between the two variables, to positive or negative 1.0, indicating a strong linear relationship between the two variables.

Value of r	Indications
0.0	No linear relationship between the two variables
+1.0	Strong positive linear relationship; as X increases in value, Y increases in value also; or as X decreases in value, Y decreases also.
-1.0	Strong inverse linear relationship; as X increases in value, Y decreases in value; or as X decreases in value, Y increases in value.

SCATTERPLOTS

It is useful to obtain a plot of the joint distribution of the values of the two variables, X and Y. These are called scatterplots. The values of X are displayed on the lower, or horizontal axis (called the X-axis) and the values of Y are displayed on the upper or vertical axis (called the Y-axis).

If small values of X are associated with small values for Y, and large values of X are associated with large values of Y, then the data will stretch from the lower left hand corner of the plot to the upper right hand corner of the plot. This indicates a positive relationship.

If small values of X are associated with large values for Y, and large values of X are associated with small values of Y, then the data will stretch from the upper left hand corner of the plot to the lower right hand corner of the plot. This indicates an inverse relationship.

If there is no discernible pattern to the distribution, then the two variables probably are not related in a linear fashion. There may be a strong, non-linear relationship between the two variables (for example, think of the normal curve) but it cannot be detected by r.

When there are only a few data points, it is fairly easy to estimate the strength of the relationship by eyeballing the data. However, with many data points statistics are needed to summarize the strength and direction of the relationship.

The Pearson r assumes that the variables are measured at the interval or ratio level. If the variables are measured at the ordinal level, however (for example, a Likert-type scale), then the Spearman rank correlation can be used. Neither Pearson nor Spearman are designed for use with variables measured at the nominal level; instead, use the point-biserial correlation (for one nominal variable) or phi (for two nominal variables).

The formula for r is as follows:

What does a positive r value mean?

The correlation coefficient r ranges between -1 and +1. A positive r values indicates that as one variable increases so does the other, and an r of +1 indicates that knowing the value of one variable allows perfect prediction of the other.

What is the level of correlation if the value of r?

Where does the r value come from? And what values can it take?.

Is an inverse relationship positive or negative?

Inverse relationship and negative correlation are synonymous. Both can be used to describe any two variables that reliably move in opposite directions. When an inverse relationship is measured, the result will be a negative number.