The degree to which two quantitative variables, X and Y, are related is quantified by the correlation coefficient (abbreviated as r) in statistics. A positive correlation exists when high values of X are linked to high values of Y. A negative correlation emerges when high X values are linked to low Y values. In general, scatter plots may reveal a:

1. A positive correlation (high values of X are associated with high values of Y) 
2. Negative correlation (high values of X are associated with low values of Y)
3. No correlation (values of X are not at all predictive of values of Y)

Correlation coefficients (abbreviated as ‘r’) are statistics that measure the relationship between two variables in unit-free terms. A correlational investigation can yield three outcomes: a positive correlation, a negative correlation, or no correlation.

• The stronger the positive correlation, the closer r is to +1.
•  The stronger the negative correlation, the closer r is to -1.

Pearson correlation coefficient

• Normally, three separate sums of squares are required to calculate a correlation coefficient (SS). The sum of the squares for variables X and Y and the sum of the cross-product of XY.

Positive correlation

A positive correlation is a two-variable association where both variables move in the same direction. As a result, when one variable increases while the other decreases, or when one variable decreases as the other decreases.

• Height and weight are an example of a positive association. 
• Taller people are typically heavier.

Negative correlation

A negative correlation is a link between two variables in which an increase in one variable causes the other to drop.

• Height above sea level and temperature is an example of a negative association. 
• It gets colder as you climb higher up the mountain (decrease in temperature).

Zero correlation

When there is no correlation between two variables, the correlation is 0.

• For example, there is no link between the amount of tea consumed and intelligence level.

Confounding variable

While variables may be associated because one causes the other, it is also possible that another factor, such as a confounding variable, is generating the systematic movement in our variables of interest.

As a third variable may be involved, correlation does not always indicate causality. 

Being a hospital patient, for example, is linked to death, but this does not always imply that one occurrence causes the other, as a third variable could be involved (such as diet, level of exercise).

Advantages of correlations

• Correlation allows researchers to look at naturally occurring variables that would be impractical or immoral to assess experimentally. Conducting an experiment to see if smoking causes lung cancer, for example, would be unethical.
• Correlation allows the researcher to see if there is a relationship between variables quickly and efficiently. This can then be represented graphically.

Limitations of correlations

• Correlation does not imply causality and cannot be construed as such. Even if two variables have a substantial correlation, we cannot infer that one causes the other.

For example, assume we discovered a link between watching violence on television and violent behavior in youth. It’s possible that a third (extraneous) element causes both of these, such as growing up in a violent environment, and that both TV watching and violent conduct are the results of this.

• We can’t go beyond the data that is provided through correlation.

For example, assume it was discovered that there was a link between the amount of time spent on homework (half an hour to three hours) and the number of GCSE passes (1 to 6). It would be wrong to conclude from this that doing homework for 6 hours will result in 12 GCSE passes.

Conclusion

Correlation simply determines the association between two variables (X or Y). A relationship exists between events, things, or mathematical or statistical variables that tend to vary, be related, or occur together in a way that is not predicted by chance alone. The apparent and strong link between systolic (SBP) and diastolic (DBP) blood pressures is a classic example (DBP). If both variables are continuous (scale) variables, a bivariate correlation (e.g., systolic and diastolic pressures) can be displayed on a scatter plot diagram. The most crucial point to remember is that correlation does not always equal causation. As the popularity of ice cream grows, so does the number of drowning deaths and the frequency of forest fires. These events occur at the same time, yet they are unrelated.