In the next preview assignment and in the next class, you will need to compute relative frequencies and find expected counts based on certain proportions.

Flight Frequencies

In the next in-class activity, we will be comparing distributions of a categorical variable for multiple populations. In particular, we will be looking at whether different airlines have the same distribution of flight status. Our categorical variable will be flight status, and the populations we are comparing are the flights for different airlines. The values of our categorical variable are On-Time, Delayed, Canceled, and Diverted. The following table is a two-way table(also called a contingency table), and it gives the counts for each value of the variable flight status for Delta Airlines and Southwest Airlines arrivals at the Atlanta airport in March 2021.^[1] Notice that each row gives the distribution of flight status for an individual airline.

For Delta Airlines, we can find the relative frequency of on-time flights, or the proportion of on-time flights, by looking at the ratio of on-time flights to the total number of flights: [latex]\frac{12,716}{13,651}≈0.9315[/latex] So, about 93.15% of Delta Airlines’ arriving flights in Atlanta in March 2021 were on time. We can performa similar computation for each value of the variable flight status to obtain the relative frequency distribution of flight status for Delta Airlines in terms of percentages, as shown in the following table.

	On-Time Percentage Flights	Delayed Percentage Flights	Canceled Percentage Flights	Diverted Percentage Flights	Total
Delta Airlines	93.15%	6.62%	0.17%	0.06%	100%

Now, let’s imagine that Delta Airlines and Southwest Airlines both had flight status proportions that matched the overall proportions. For example, Delta Airlines would have 92.246962% of its flights arrive on time. Since Delta Airlines had 13,651 flights arrive in Atlanta in total in March 2021, 92.246962% of 13,651=0.92246962×13,651≈12,592.633is the number of flights that we would expect to be on time. This is called the expected count of on-time flights if Delta Airlines’ distribution matched the overall proportions. Similarly, for Southwest Airlines, we would expect to have 0.92246962×2,562≈2,363.367 on-time flights if its distribution matched the overall proportions. Notice that these expected counts do not have to be whole numbers because they are theoretical values. Notice also that there were 14,956 total on-time flights for these two airlines in March 2021, so once we knew that Delta Airlines would be expected to have 12,592.633 on-time flights if its distribution matched the overall proportions, we could have found the expected number of on-time flights for Southwest Airlines by subtracting:

14,956−12,592.633=2,363.367

We see that we get the same expected count as we did when we used the percentage.