Taking your SAT exam was probably the most stressful time of your junior year. A recent incident has brought something to light; you can end up getting test answers wrong even after you actually get them right. What does that mean? Well, we are talking about an SAT mathematics problem that even the College Board didn’t get right!
This was the case for about 50,000 high school students across the country who took the May 2019 SAT exam. Because of a sharp-eyed student and his tutor, the test scores were increased after the results had been announced. This happened because a question was disputed and the College Board had to change the accepted answer officially.
Compass Education Group has pointed out that the question was posed at the end of the marathon exam during the final 55-minute math phase where students are allowed to use calculators. The question reads, ‘The histogram summarizes the distribution of a data set composed of 50 integers. The first bar represents the number of integers that are at least 0 but less than 5. The second bar represents the number of integers that are at least 5 but less than 10, and so on. What is a possible value of the median of the data set?’
Notice that the question asks for a possible value. Bruce Reed is the Executive Director at Compass Education Group and wrote on his blog, ‘the test makers must then account for every possible correct answer when scoring the test. And that is where the mistake was made.’ David Linkletter is a Pure Mathematics Ph.D. candidate at the University of Nevada, Las Vegas and carrying out research in Set Theory – Large Cardinals. He is also a veteran SAT tutor. He says, ‘histogram can be nice for large data visualization but awkward for specific little questions about points of data, as this problem illustrated.’
Bruce Reed talked through about how the median problems should be answered. He writes, ‘Most students understand that the median is the middle value when all of the values are sorted. The histogram means that students can’t know the exact number, but they can still find the middle. Almost. When there is an odd number of values, one of them will be in the middle. Fewer students know the rule that, with an even number of items, the two middle values must be averaged to find the median. In the question above, we want the average of the 25th and 26th list items. These should be in the third bar, since the first three have 12, 9, and 9 items. The third bar has integers greater than or equal to 10 but less than 15 (i.e., 10, 11, 12, 13, 14). These could individually be any combination of numbers inside this range. Since we’ll be averaging two, we could get any of these values and any of the half values between them. This is where the developers took a wrong turn. They assumed that the two middle values would be the same. But the 25th and 26th values did not have to be identical. That is, the 25th and 26th items could be any of the pairs 10/10, 10/11, 10/12, 10/13, 10/14, 11/11, 11/12, 11/13, 11/14, 12/12, 12/13, 12/14, 13/13, 13/14, 14/14. These result in median values of 10, 10.5, 11, 11.5, 12, 12.5, 13, 13.5, 14.’
Linkletter says, ‘Math-minded kids love to mention that medians can end in .5. Half the time, I expect that to be part of the difficulty of the problem.’ He further says that he would have recommended that his students write ‘12, 13, or something in the middle not playing with the .5 possibility, just to be safe. But that’s supposed to be safety against your own misreading,” Linkletter says, “not against [the] College Board forgetting how medians work.’
The College Board released the full answers to the May 2019 SAT a couple of weeks after the test and only listed five values as the right answers; 10, 11, 12, 13, and 14. A student was insistent along with their Compass tutor that their response of 12.5 was also correct. The tutor while wrote an email to Compass HQ, ‘….The acceptable answers are all integers, but there are 50 items in the set, so the median would be the average of items 25 and 26, which could be 12 and 13, so I’m not sure why [my student’s] answer of 12.5 isn’t acceptable. What am I missing? It’s probably something obvious, but I can’t figure it out.’
Reed reached out to the College Board for clarification, and the College Board quickly recognized its mistake and raised the scores for the students who had previously got the answers wrong.