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Hi all,

This week I want to introduce you to the concept of correlations. This is a statistical measure used to describe the relationship between two variables (x and y) ? and whether there even is one to begin with!

What I mean by relationship, is whether the score of one variable affects the other variable in a predictable manner, and vice versa. With this measure, you do not need to manipulate any variables ? you take the data as is. The correlation value will fall anywhere between 0 (meaning no correlation exists) to 1 (perfect correlation). It can be positive, meaning the two variables change in the same direction (i.e., an increase in x will cause an increase in y, and a decrease in x will cause a decrease in y). It can also be negative, meaning that the two variables change in opposite directions (i.e., an increase in x causes a decrease in y, and a decrease in x causes an increase in y).

Let?s illustrate this with an example. Let?s say we want to see whether or not a relationship exists between student final exam scores (x) and student satisfaction scores (y). We run the data in SPSS and get a correlation of -0.76. This tells us that the variables affect each other in opposing directions (negative correlation) and that we have a correlation of 0.76, which seems like a fairly strong correlation upon face value. What this number tells us, is that one variable partially predicts the other (we can say this about any correlation value above 0 but less than 1). To determine how strong of a correlation 0.76 really is, what we must do is square the correlation value. In doing that, we get 0.5776 ? round up to 0.58. What this now tells us, is that 58% of the y variable (in this case, student satisfaction scores) is attributable to the x value (in this case, final exam scores). Think of this as being like calculating the effect size for correlations!

There are a few different types of correlation statistics we can use. The most common one is the Pearson correlation, which measures the degree of a straight-line relationship between two variables that use interval or ratio scale of measurement (continuous data). There is also the Spearman correlation, which is great to use if you (a) have data measured on a rank-order scale and/or (b) may not have a linear relationship (but still have a consistent relationship). The other two are point-biserial correlation (one of the variables only has two values/dichotomous ? e.g., under 18 and above 18, student vs. Non student) and the phi-coefficient (both variables only contain two values/dichotomous).

So all of this is great information about correlations ? but why use them? In general, these are the types of things that correlations are helpful with:

Making predictions
If you two that two variables are correlated to one another, you can then go on to make predictions about one of the variables (like in an experimental study). In this situation, we can take data that we already have at Teton and run correlations on different data to see whether any type of relationship exists between them, and if there is, then we can make educated hypotheses and test them (running statistics like ANOVA or t-tests on the data we find!)
Confirming validity
Measuring whether two variables are correlated is a method to test for validity ? because if a test is said to measure something, it should correlate to other tests of the same measure. For example, we can run correlations on whether the final exam questions actually reflects student performance in the course.
Confirming reliability
Following the same logic as validity, correlation can be used to attest to the reliability of a measurement procedure. You can, for example, have the same set of students answer the satisfaction survey twice for a given course ? if the scores are highly correlated, it means that the survey collects data consistently (i.e., reliably).
Validating theories
This ties into prediction, as many theories make predictions about the relationship between two variables ? so correlation is a quick tool to validate this information. This use is mainly relevant when it comes to research and theory, so it is likely not applicable to Teton Grand, but useful to know this in general.

The final piece of information to keep in mind is four considerations when it comes to interpreting correlations:

While correlations can tell us that two variables are related, it does not give us information as to why they are related. So if you find that there is a correlation, for example, between instructor tenure and final exam scores, this is not proof enough that instructor tenure causes increases in final exam scores. We can use this information to look into this relationship further and conduct other statistical analyses to potentially make this conclusion, but a correlation on its own is not enough evidence to make this statement (even though it looks like it can be ? you must resist the temptation!).
The actual value of the correlation can be greatly affected by the range of scores selected. In this vein, it is important to include as much information as possible (rather than a smaller subset of the data). Think of it as taking a very small section of a large picture ? the smaller this section of the picture, the harder it is to actually be able to accurately tell what the bigger picture is.
The actual value of the correlation can also be greatly affected by outliers (i.e., extreme scores). For example, if one student skipped the final exam and got a score of 0 (compared to the rest of the class that all scored, for example, minimum 45), this would be considered an outlier. A tip to be able to identify outliers easily is to plot the scores on a graph ? outliers are those scores that fall far from the general trend of the majority of the data.
Be sure never to take the correlation value as is when trying to interpret the strength of it. Always use the square of the correlation to determine it?s ?effect size? as explained in more detail earlier in this communication.

While we have some great data, I do have some suggestions for new data for Teton Grand to collect to be able to glean more information that would be helpful for the company. In particular, because the survey data collected involves classes over the course of a year, it would be helpful to have a question to ask who the instructor was that the student is answering questions about, as well as the semester (spring, summer, fall, etc.).

Having this kind of information can enable us to more accurately assess whether they may be correlations between class semester and final exam scores, or student satisfaction scores.

We may also be able to assess specific instructors and their effectiveness in teaching the course (since tenure is only one facet that may impact teacher effectiveness). In fact, we create a table for all the professors, have a column for average final exam score and average satisfaction score results (we can tease out the questions specifically related to the instructor), and the tenure level they would fall into. Based on this, we can compare between each professor and validate whether their scores correlate with the scores for that tenure level. This way we can confirm whether tenure level is enough of an indication to consider, or if there needs to be more information on course instructors collected to assess effectiveness.

In terms of course semester, we can gather whether or not correlations exist between time of year a course is offered and final exam and student satisfaction scores. If there is a correlation, this can invite a closer look to see what may be causing the relationship. Perhaps we find that students perform better in certain courses at a specific time of the year, or students enjoy on-ground learning more in the summer and prefer e-learning in winter months. This way, we can design each course to maximize it?s effectiveness.

Another set of useful piece of information we can glean comes from the marketing perspective. Including information about when the student signed up for the course, where they heard about Teton Grand/the course from, this kind of information will help in marketing the courses and encouraging sign ups. We may find a correlation exists between time of student sign up and gender of students, which again could invite a closer look at the causation of this and what actions Teton Grand can take to promote their services more towards a population that you may not be reaching.