I proposed a situation to my senior-level Statistics students a few weeks ago. Please critique or correct any of my methods. If you don’t like math, you may not like this article. However, I thought it was interesting to see the improbability of meeting AYP in Illinois using statistics. Here is the situation and any background information you may need.
In the state of Illinois, adequate yearly progress (AYP) is measured in high schools using the ACT test. This is a normalized test with a mean of 18 and a standard deviation of 6. However, these numbers vary from year to year, depending on the population taking it. I have been told many times that, if a student earns an 18 or better on the ACT, they will be considered proficient and meet or exceed expectations set forth by the No Child Left Behind Act of 2001 (NCLB).
Last year, 515 juniors took the ACT at my school. In order to meet AYP, 85%, or 438 students would have to earn an 18 or better. What is the probability of this occurring?
The first things students had to do was calculate the z-score. This is done using the formula: . Using this , students found a z-score of 0, which gives us a probability of 0.5. This makes sense because the ACT is a normalized test with a mean of 18. This means half of the scores will be less than that, half of the scores will be greater than that.
After finding this, students then used a binomial probability distribution in order to determine the probability of 438 students earning an 18 or better and 77 students earning below that. The formula for a binomial probability distribution is: where n = the number of trials, x = the number of successes, p = the probability of a success, and q = the probability of a failure. In our situation n = 515, x = 438, p = 0.5 and q = 0.5. So, this would look like: .
As students put this into their calculators and got answers, they started to look around as if they did something wrong. I saw students input the numbers again. Then, I started to hear about it. “I got zero, what did you get?” “I got zero too!”
Yes, the probability of 438 out of 515 students earning an 18 or better on the ACT is statistically 0. Yet, I see the media report over and over on this story. Is the media also doing stories on the fact that someone who bought a Powerball ticket didn’t win either? They are statistically equivalent, why not?
The problem is, people are uninformed about the math behind the ACT and AYP, including legislators. I believe that, had legislators sat in a statistics classroom full of seniors for 5 minutes and allowed students to do the math, they would have realized the ridiculousness of using the ACT in order to measure AYP. However, this is the same group of legislators who allowed Illinois to have a $43.8 billion deficit last year, so math is obviously not their strong suit.