Conditional Probability

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. This statistical method helps uncover relationships between variables and provides new insights into the data. For this study, conditional probability tables are used to better understand how undergraduate enrollment and the school's geographic region impact completion rate. Conditional probability tables are a simple way of viewing all conditional probability permutations for a given scenario.

Undergraduate Enrollment (School Size)

First, given each possible school size, a conditional probability is calculated for each graduation rate. For brevity purposes, the medium graduation rate is not considered as the completion rates on the extremes provide more insight when making conclusions. The conditional probability table and subsequent conclusions are discussed in depth in the Results section below.

Geographic Region

Next, in the same fashion as the methodology described above, another set of conditional probabilities are calculated given the region where the school is located. The conditional probability table and subsequent conclusions are discussed in depth in the Results section below.

Combined Variable Analytics

Finally, using a very similar methodology as above, a third set of conditional probabilities are calculated given both the region and the size. By calculating the probability of a certain graduation rate given region and size combined, the conclusions reached by using the table are more targeted. The conditional probability table and subsequent conclusions are discussed in depth in the Results section below.

Results

Chi-square tests of independence can only determine whether variables are independent or dependent on each other; however, if the variables are dependent, further analysis can be done to understand the extent of this relationship. As seen above, completion rate is dependent on both school size and region. To gain a better understanding of the specifics of this relationship, conditional probability tables are created and shown below.

Undergraduate Enrollment (School Size)

To understand how school size affects graduation rates, a conditional probability table is created which gives the breakdowns of completion rates by school size. This is seen in the table below.

According to Table 4, very small schools, those with less than 1000 people enrolled, tend to have a lower graduation rate. Additionally, very large schools tend to have a higher graduation rate.

Geographic Region

To understand how region affects graduation rates, a conditional probability table is created which gives the breakdowns of completion rates by region. This is seen in the table below.

According to Table 5, Southern schools tend to have lower graduation rates versus schools in the Northeast. This is an important data point for someone looking to narrow a college search to schools with high graduation rates.

Combined Variable Analytics

Finally, to gain more specific insights than those seen in the single variable analysis, several conditional probability tables are created to analyze a combination of variables. Here the combination of variables are region and size and the combined effect on completion rates. These results could provide students with some general recommendations depending on what type of university the student is considering.

Table 6 indicates that large Northeast schools have a far greater chance of graduating more students than very small schools in the same region. According to the table, very small schools actually have the highest chance of graduating the fewest students.

According to the analysis presented in Table 7, a similar result was achieved for Southern schools as compared to Northeast schools. However, the chance that a very small school in the south has a lower graduation rate is much higher than the chance of a very small school in the Northeast.

Table 8 illustrates the results for completion rates for schools in the North Central region given their size. Again, the results and conclusions are similar to previous regions; however, this time, large schools are seen to have a higher chance of a higher completion rate.

Finally, in Table 9, there are conflicting results. According to the results, for schools in the West, medium schools have the highest percentage chance of both high and low completion rates. This isn't a very useful statistic on its own, however, looking at the rest of the probabilities, it is seen that schools in the west that are very large have a very low percentage chance of a low completion rate.