Correlation, Regression & ANOVA: Statistical Analysis Guide

Lincoln and the Limits of Correlation Analysis

In the case of Abraham Lincoln, the researcher undoubtedly expected to find a correlation between education level and lifetime achievement. However, the researcher made two mistakes. The first was expecting such a correlation in the first place. In Lincoln's time, the correlation between education and achievement was weaker than it is today, and a man could become President without a strong formal education. Education was often informal, and much of the relevant learning for all citizens came outside of the formal education system. Thus, the variable of formal education was not a particularly useful variable during Lincoln's era. It is not unreasonable for a researcher to test the variable of education, but if that variable proves not to be useful, the researcher should not be surprised. The researcher should view the hypothesis as equally likely to be disproven as proven.

The second mistake the researcher made was to assume that there would be no outliers in the survey results. With any measure of correlation, it is reasonable to expect exceptions or outliers. For other individuals, the correlation between education and achievement may have been strong, but it is equally reasonable that the correlation is not universal. Thus, the researcher should not have been surprised at the presence of an outlier in the survey results.

The researcher had clearly expected a linear relationship. However, as the flaws in the thinking reveal, Abraham Lincoln did not exhibit that relationship. Furthermore, the researcher did not test for the assumptions of a linear relationship. An assumption was made, but it does not appear that the different variables in the study were tested against each other. Therefore, the researcher did not adequately test for the conclusion before reaching it.

Gas Prices and Passenger Mile Ratio

What the figures show is that the price of gas is highly correlated with the passenger mile ratio. There were three main relationships tested statistically. In addition to the relationship between price and passenger mile ratio, each of those variables was tested against "year," representing normal growth. From these figures, we can see that normal growth — the change in passenger mile ratio year over year — is less strongly correlated than the change in relation to gas price changes.

The model shows that gas prices increase as time increases, such that on average there is an increase in the gas price with each passing year. The model also shows an increase in the passenger mile ratio with each passing year. This indicates to management that there may be a correlation between gas prices and passenger mile ratio, and more clearly that the passenger mile ratio improves over time, on average. The data further shows an increase in passenger mile ratio with an increase in gas prices.

The strength of these correlations tells a more specific story. Although business improves each passing year, some of that improvement is attributable to natural growth and some is directly related to the increasing price of gas. The correlation between gas prices and passenger mile ratio is stronger and exhibits less volatility than the other correlations studied. This indicates that a certain portion of the increase in the passenger mile ratio is attributable directly to increases in the price of gas. This data therefore helps management understand the impact that gas prices have on their business growth potential by allowing them to separate the strength of that relationship from the strength of the relationship between time and the passenger mile ratio.