Percents Count

Two farmers entered in a contest to see who could grow a nicer stand of wheat with less weeds.  One farmer was very lazy and let his field overgrow with weeds while the other farmer, known to be a good farmer, meticulously kept an eye on his field.  After the harvest both farmers sent grain samples in to be examined for how much weed material they contained.  Surprisingly, when the results came back it showed that the good farmer had more weed material in his grain sample.  Thinking that something was strange, the good farmer stopped on by to see the test results for himself.  Sure enough the lazy farmer had 1 pound of weed material in his sample while the good farmer had 1.5 pounds of weed material.  But then something caught his eye, the total sample weight for the lazy farmer was 5 pounds, while it was 150 pounds for the good farmer.  Seeing this result the good farmer knows he had really won since his sample only contained 2% weed material while the lazy farmer had 20%.  The total weed weight difference was due to different sizes of grain samples.

Often when we want to compare two groups we would like to compare them on equal footing, and often times it is difficult (or impossible) to get samples of the same size.  In these cases we should consider reporting an overall percent (or proportion or rate) as opposed to counts so that the comparisons for each group are not influenced by the size of the sample taken.

Say we are looking at using a new curriculum to teach awareness about gambling to elementary school children.  We take two classes and pilot a different curriculum at each one.  After the program is over we test the students on their knowledge.

A blue dot means that a student passed the test while a red dot means a student didn't pass the test.  If we just count the number of passes we see Classroom A had 10 students that passed while Classroom B had 9 students that passed.  However, Classroom A has 40 students but Classroom B has 18.  Due to the unequal sample size we should not compare straight counts here, instead we should look at the percent of students that passed (the number that passed divided by the overall number of students in each class, then multiply the result by 100).  So for Classroom A we have 10/40=0.25 which means 25% of the students passed, and for Classroom B the results were 9/18=0.5 which is 50% of the students.  Even though more students passed in Classroom A a higher percent of students passed in Classroom B.  So we have evidence that the curriculum in Classroom B is better overall. 

Remember that when we want to make comparisons between groups do not let the size of each group skew the result.  Taking percentages can help us compare what is actually important and keep us pointed towards the truth