Curvilinear Relationship Analysis Essay

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  • Introduction

    After becoming familiar with your data using descriptive statistics, you are probably itching to find out if and how certain variables are related to one another. So now we foray into bivariate statistics, meaning we are now dealing with analyzing how two variables are related to one another. Correlation, sometimes known as Pearson’s correlation, is a bivariate statistic that measures how two variables are related to or associated with each other. As this is AP Statistics, you know by now you’re going to need to know more than that to be well prepared for the exam. So what is correlation exactly and how do we interpret it?

    In statistics, we are not only interested in describing observations, but how observations are related to each other. Correlation describes the strength and direction of a linear relationship for bivariate quantitative data. It is typically given as a correlation coefficient, usually known as Pearson’s r. This r value reflects the strength and direction of a linear relationship between two variables, or bivariate data. Specific to correlation, we are interested in describing the quality of a linear relationship. It is important to keep in mind that correlation is only appropriate when there is a linear relationship between two variables of interest.

    The correlation coefficient r ranges from -1 to +1. An r coefficient of -1 is a very strong linear negative relationship, while at the other end a +1 is a very strong linear positive relationship. A positive correlation is one in which both variables increase. On the other hand, in a negative correlation, one variable increases while the other decreases. A scatter plot will show a positive correlation as an uphill slope, while a negative correlation is a downhill slope.

    It is also possible to get a score of 0, which reflects the absence of a linear relationship. The absence of a linear relationship, an r of 0, could mean that there may be some other kind of relationship between two variables. Again, to emphasize, an r of 0 does not mean that there isn’t a relationship; rather it means that a linear relationship does not exist. Instead, it is possible that while r is 0, there may be a curvilinear relationship between the two variables. The further the r coefficient is from -1 and +1, or closer to 0, the weaker the linear relationship between the two variables.

    As a general rule r values between -.3 and +.3 are considered weak correlations. Moderate correlations are between -.3 and -.7, and between .3 and .7. Strong correlations are greater than .7 or less than -.7.

    When it comes to correlation, keep in mind that it is a measure of linear relationship. In other words, a correlation is only appropriate when there is a straight line that depicts the relationship between two variables. If the relationship is curvilinear, you will get a correlation coefficient that is close to 0. Thus, in order to calculate correlation it is important to do some exploratory data analysis using a visual display. In particular, you will want to explore the relationship between two variables using a scatterplot to ensure that there is a straight line relationship. Again, an r score is only meaningful if there is a linear relationship. When a scatterplot shows a curve, you are likely to get an r of 0. An r of 0 doesn’t mean a relationship is nonexistent, but that it isn’t linear.

    It is also important to keep in mind that the correlation coefficient is influenced by extreme values. Extreme values can influence a positive and strong relationship, so it is important as well to be mindful of outliers when interpreting correlation. When you have a scatterplot, it can be easy to spot observations that are far removed from the rest of scatter pattern. These outliers can drag the shape of the correlation line to be positive or negative, to be weak or to be strong.

    You might see scatterplots associated with an r coefficient. Usually, statistical analysis programs will calculate r alongside a scatterplot. However, you will never be asked to calculate an r coefficient from looking at a scatterplot alone. It’s difficult for a person to do this, so you’ll be given other information if you need to find an r coefficient.

    How to Interpret Correlation?

    When you come across an r statistic, you want to interpret what it reflects about the strength and direction of the linear relationship between the two variables.

    Let’s go through some examples to practice interpreting correlation.

    Example 1

    In the first example, let’s examine the correlation between murder rate and poverty rate. The Pearson correlation is .63.

    We would interpret this r to reflect a moderately strong, positive, linear relationship between two variables. This means that as poverty rate increases, so does the murder rate. Note that this doesn’t mean that murder causes poverty, or that poverty causes murder. Rather, it could be that both poverty and murder are influenced by some other factor, like geographical location or amount of time spent away from the home. We can’t make positive statements about what causes something without further information.

    Example 2

    In this example, we examine the relationship between crime and ice cream sales. The Pearson r is .4.

    We would interpret this r to reflect a moderately strong correlation with a positive, linear slope relationship between ice cream sales and crime. As ice cream sales increase, so does crime.

    This is also a good example of how two seemingly unrelated variables can be correlated. This correlation is probably not by chance. Rather, ice cream sales might be a proxy, or a reflection, of weather. Hotter temperatures might be positively related to more crime being committed, as there is more opportunity to commit crime when people are out and about. Again, we’d have to do more research to figure out the exact reason for the correlation.

    Example 3

    For a final example, we have a correlation coefficient for age and income. The Pearson r is .10.

    This r coefficient is considered to be fairly weak, as it is near 0. Does it mean that there is no relationship between age and income? This is certainly possibly. However, we could also reason that the relationship between age and income is not linear in character. That is, income does not necessarily increase as age increases. In fact, income may decrease after a certain age. Retirees, for example, may see a great drop in their income after they stop working. Thus, it is possible that the relationship between age and income is actually a curve, in which after a certain age, say after 55 years, income takes a downward turn. There’s a correlation here, but it’s not linear, so a scatterplot isn’t helpful in analyzing it.

    Summary

    Correlation is an important first step in data analysis when we want to know how two variables are related to one another. Typically, statisticians will calculate correlation after already having done a lot of descriptive statistics and identifying outliers. Statisticians will typically explore the relationship between two variables graphically using a scatterplot to ensure that a correlation is appropriate to calculate.

    The value of Pearson’s r will always be between -1 and +1. Keep in mind that the correlation coefficient is about the direction and the strength of a linear relationship. However, not all bivariate data will be related in a linear pattern, so correlation is specific to linear relationships. Furthermore, correlated variables are important for multivariate analyses where strong correlations can lead to biased estimates.

    Now that we have reviewed correlation, you should feel confident when you encounter correlation questions on the AP Statistics exam. Be sure to check out other Statistics resources, such as Albert.io, and good luck on your exam!

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