April 30, 2019

How Fast Will My Score Go Up?

Using my students' past score increases, I made the tables below. They roughly predict how much your score is likely to increase after five or ten tutoring sessions.

If you compare the ACT and SAT scores from the tables, you'll find that improvements tend to happen more quickly on the ACT. All else being equal, the ACT is an easier test to prep for.

Subject Tests include Math Level 2, SAT Chemistry, SAT Physics, SAT Biology, and SAT Literature.

Remember that these are average score increases. Your scores may go up slower or faster based on your background knowledge, how hard you're able to study, and how well-rested you are on the day of your official ACT or SAT.

Since the difficulty of earning each successive ACT point ramps up rapidly as scores approach 36, I came up with a technique to compare score increases from student to student even though they start at different initial scores.

I went through my students' records and eliminated any that didn't have final test scores from a real SAT or ACT sitting. I used the highest practice SAT or ACT score each student received before working with me as that student's initial score, with the final score being the one from the official test sitting.

That sample potentially biases the results: students who stopped tutoring before taking the test may have chosen not to e-mail their scores to me. This survivorship bias could make my results look better because students with low scores might choose not to tell me about them.

However, I'm also setting a very high bar by taking each student's highest score before working with me but only using an official test result as the final score. I could easily have used average scores for both the initial and final scores instead.

I also decided not to remove the scores of students who didn't complete all of my assigned homework. This has the effect of moving the bar even higher because those students received smaller score increases.

There's no way to know to what extent these biases affect the results. Without more information, I made the assumption that they cancel out.

Now for the fun bits: ACT scores can get to 36 but no higher, so it makes sense to use a function with a horizontal asymptote. I reflected an exponential decay function across the x-axis and moved the asymptote up to 36.5. (Since a 35.5 rounds to 36, and since the ACT's curve allows students to get some questions wrong and still get 36 on some sections, a 36 isn't truly a "perfect" score with respect to getting all of the questions right. That's why I'm using 36.5.)

I modeled ACT scores using the transformed exponential decay function (with number of sessions as the independent variable) and then took the log10 of both sides. You can derive the decay function yourself using the information in this post. If you get it right, I'll give you a free session.

Because the relationship is now linear, the data give us the regression below, which has a reasonably nice fit (R2 = 0.47) given the small sample and a couple of outliers.

I then plugged the slope of the best-fit line back into the original exponential decay function to generate the ACT score increase table at the beginning of this post.

(If you're a chemistry nerd as well as a math nerd, you may have used a similar technique to plot the data for a first-order reaction, which is also modeled by exponential decay. Finance nerds will recognize that a logarithmic axis is used to chart stock prices, which are modeled using exponential growth.)

I then modified the function so that it would asymptote to 1600 and generated the table of SAT score increases.

Here's a similar regression for my Subject Test students. Note the steeper slope and the larger value of R2, which suggest that Subject Tests are easier to prep for.

Because I work personally with every student and teach a variety of subjects, including ACT, SAT, Subject Tests, and AP tests, the samples used for this analysis aren't large. Remember that these are just averages across small pools of students, so your own results may be quite different.

If you'd like to find out more, please contact me using the form at the bottom of my tutoring information page.