In correlation analysis, we estimate a sample correlation coefficient, more specifically the Pearson Product Moment correlation coefficient. The sample correlation coefficient, denoted r, Show ranges between -1 and +1 and quantifies the direction and strength of the linear association between the two variables. The correlation between two variables can be positive (i.e., higher levels of one variable are associated with higher levels of the other) or negative (i.e., higher levels of one variable are associated with lower levels of the other). The sign of the correlation coefficient indicates the direction of the association. The magnitude of the correlation coefficient indicates the strength of the association. For example, a correlation of r = 0.9 suggests a strong, positive association between two variables, whereas a correlation of r = -0.2 suggest a weak, negative association. A correlation close to zero suggests no linear association between two continuous variables. It is important to note that there may be a non-linear association between two continuous variables, but computation of a correlation coefficient does not detect this. Therefore, it is always important to evaluate the data carefully before computing a correlation coefficient. Graphical displays are particularly useful to explore associations between variables. The figure below shows four hypothetical scenarios in which one continuous variable is plotted along the X-axis and the other along the Y-axis. 
A study of a random sample of 100 Americans summarizes the relationship between alcohol consumption and age with a correlation coefficient r= 0.03. The value of r tells us: Example - Correlation of Gestational Age and Birth WeightA small study is conducted involving 17 infants to investigate the association between gestational age at birth, measured in weeks, and birth weight, measured in grams. Infant ID # Gestational Age (weeks) Birth Weight (grams) 1 34.7 1895 2 36.0 2030 3 29.3 1440 4 40.1 2835 5 35.7 3090 6 42.4 3827 7 40.3 3260 8 37.3 2690 9 40.9 3285 10 38.3 2920 11 38.5 3430 12 41.4 3657 13 39.7 3685 14 39.7 3345 15 41.1 3260 16 38.0 2680 17 38.7 2005 We wish to estimate the association between gestational age and infant birth weight. In this example, birth weight is the dependent variable and gestational age is the independent variable. Thus y=birth weight and x=gestational age. The data are displayed in a scatter diagram in the figure below. Each point represents an (x,y) pair (in this case the gestational age, measured in weeks, and the birth weight, measured in grams). Note that the independent variable, gestational age) is on the horizontal axis (or X-axis), and the dependent variable (birth weight) is on the vertical axis (or Y-axis). The scatter plot shows a positive or direct association between gestational age and birth weight. Infants with shorter gestational ages are more likely to be born with lower weights and infants with longer gestational ages are more likely to be born with higher weights. The correlation requires two scores from the same individuals. These scores are normally identified as X and Y. The pairs of scores can be listed in a table or presented in a scatterplot. Example: We might be interested in the correlation between your SAT-M scores and your GPA at UNC. Here are the Math SAT scores and the GPA scores of 13 of the students in this class, and the scatterplot for all 41 students: The scatterplot has the X values (GPA) on the horizontal (X) axis, and the Y values (MathSAT) on the vertical (Y) axis. Each individual is identified by a single point (dot) on the graph which is located so that the coordinates of the point (the X and Y values) match the individual's X (GPA) and Y (MathSAT) scores. For example, the student named "Obs5" (in the sixth row of the datasheet) has GPA=2.30 and MathSAT=710. This student is represented in the scatterplot by high-lighted and labled ("5") dot in the upper-left part of the scatterplot. Note that is to the right of MathSAT of 710 and above GPA of 2.30. Note that the Pearson correlation (explained below) between these two variables is .32. 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.
In the example above, GPA and MathSAT are positively related. As GPA (or MathSAT) increases, the other variable also tends to increase. 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. 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. 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: There are strengths in between -1.00, 0.00 and +1.00. Note, though. that +1.00 is the largest postive correlation and -1.00 is the largest negative correlation that is possible. Here are three examples: Weight and Horsepower The relationship between Weight and Horsepower is strong, linear, and positive, though not perfect. The Pearson correlation coefficient is +.92. Drive Ratio and Horsepower The relationship between drive ratio and Horsepower is weekly negative, though not zero. The Pearson correlation coefficient is -.59. Drive Ratio and Miles-Per-Gallon The relationship between drive ratio and MPG is weekly positive, though not zero. The Pearson correlation coefficient is .42.
For example, we require high school students to take the SAT exam because we know that in the past SAT scores correlated well with the GPA scores that the students get when they are in college. Thus, we predict high SAT scores will lead to high GPA scores, and conversely. Is the association positive or negative?If one variable increases as the other variable increases, there is said to be a positive association. If one variable increases as the other variable decreases, there is said to be a negative association. If there is no relationship between the variables, then the points in the scatterplot have no association.
What does it mean to say that two variables are negatively associated?A negative, or inverse correlation, between two variables, indicates that one variable increases while the other decreases, and vice-versa.
What is positive association variables?Two variables have a positive association when above-average values of one tend to accompany above-average values of the other, and when below-average values also tend to occur together.
How to determine the direction of the association between two variables?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.
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