Which correlation coefficient best represents a moderate relationship between variables?

Pearson’s correlation coefficient is the test statistics that measures the statistical relationship, or association, between two continuous variables.  It is known as the best method of measuring the association between variables of interest because it is based on the method of covariance.  It gives information about the magnitude of the association, or correlation, as well as the direction of the relationship.

Questions Answered:

Do test scores and hours spent studying have a statistically significant relationship?

Is there a statistical association between IQ scores and depression?

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Assumptions:

1. Independent of case: Cases should be independent to each other.
2. Linear relationship: Two variables should be linearly related to each other. This can be assessed with a scatterplot: plot the value of variables on a scatter diagram, and check if the plot yields a relatively straight line.
3. Homoscedasticity: the residuals scatterplot should be roughly rectangular-shaped.

Properties:

1. Limit: Coefficient values can range from +1 to -1, where +1 indicates a perfect positive relationship, -1 indicates a perfect negative relationship, and a 0 indicates no relationship exists..
2. Pure number: It is independent of the unit of measurement.  For example, if one variable’s unit of measurement is in inches and the second variable is in quintals, even then, Pearson’s correlation coefficient value does not change.
3. Symmetric: Correlation of the coefficient between two variables is symmetric.  This means between X and Y or Y and X, the coefficient value of will remain the same.

Degree of correlation:

1. Perfect: If the value is near ± 1, then it said to be a perfect correlation: as one variable increases, the other variable tends to also increase (if positive) or decrease (if negative).
2. High degree: If the coefficient value lies between ± 0.50 and ± 1, then it is said to be a strong correlation.
3. Moderate degree: If the value lies between ± 0.30 and ± 0.49, then it is said to be a medium correlation.
4. Low degree: When the value lies below + .29, then it is said to be a small correlation.
5. No correlation: When the value is zero.

Related Pages:

• Conduct and Interpret a Bivariate (Pearson) Correlation
• Correlation (Pearson, Kendall, Spearman)

The Correlation Coefficient: Definition

Bruce Ratner, Ph.D.

The correlation coefficient, denoted by r, is a measure of the strength of the straight-line or linear relationship between two variables. The correlation coefficient takes on values ranging between +1 and -1. The following points are the accepted guidelines for interpreting the correlation coefficient:

1. 0 indicates no linear relationship.
2. +1 indicates a perfect positive linear relationship: as one variable increases in its values, the other variable also increases in its values via an exact linear rule.
3. -1 indicates a perfect negative linear relationship: as one variable increases in its values, the other variable decreases in its values via an exact linear rule.
4. Values between 0 and 0.3 (0 and -0.3) indicate a weak positive (negative) linear relationship via a shaky linear rule.
5. Values between 0.3 and 0.7 (-0.3 and -0.7) indicate a moderate positive (negative) linear relationship via a fuzzy-firm linear rule.
6. Values between 0.7 and 1.0 (-0.7 and -1.0) indicate a strong positive (negative) linear relationship via a firm linear rule.
7. The value of r squared is typically taken as “the percent of variation in one variable explained by the other variable,” or “the percent of variation shared between the two variables.”
8. Linearity Assumption. The correlation coefficient requires that the underlying relationship between the two variables under consideration is linear. If the relationship is known to be linear, or the observed pattern between the two variables appears to be linear, then the correlation coefficient provides a reliable measure of the strength of the linear relationship. If the relationship is known to be nonlinear, or the observed pattern appears to be nonlinear, then the correlation coefficient is not useful, or at least questionable.

The calculation of the correlation coefficient for two variables, say X and Y, is simple to understand. Let zX and zY be the standardized versions of X and Y, respectively. That is, zX and zY are both re-expressed to have means equal to zero, and standard deviations (std) equal to one. The re-expressions used to obtain the standardized scores are in equations (3.1) and (3.2):

zXi = [Xi - mean(X)]/std(X)                                                            (3.1)

zYi = [Yi - mean(Y)]/std(Y)                                                            (3.2)

The correlation coefficient is defined as the mean product of the paired standardized scores (zXi, zYi) as expressed in equation (3.3).

rX,Y = sum of [zXi * zYi]/(n-1), where n is the sample size              (3.3)

For a simple illustration of the calculation, consider the sample of five observations in Table 1. Columns zX and zY contain the standardized scores of X and Y, respectively. The last column is the product of the paired standardized scores. The sum of these scores is 1.83. The mean of these scores (using the adjusted divisor n-1, not n) is 0.46. Thus, rX,Y = 0.46.  ( Related Article: When Data Are Not Straight )

Is r strong moderate or weak?

Is 0.7 A strong correlation?

What is a moderate correlation?

Is 0.4 A strong correlation?