yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Comparing with z-scores | Modeling data distributions | AP Statistics | Khan Academy


2m read
·Nov 11, 2024

Before applying to law school in the U.S., students need to take an exam called the LSAT. Before applying to medical school, students need to take an exam called the MCAT. Here are some summary statistics for each exam.

For the LSAT, the mean score is 151 with a standard deviation of 10. For the MCAT, the mean score is 25.1 with a standard deviation of 6.4. Juwan took both exams; he scored 172 on the LSAT and 37 on the MCAT. Which exam did he do relatively better on?

So pause this video and see if you can figure it out. The way I would think about it is you can't just look at the absolute score because they are on different scales and they have different distributions. But we can use this information.

If we assume it's a normal distribution, or relatively close to a normal distribution, with a mean centered at this mean, we can think about how many standard deviations from the mean did he score in each of these situations. In both cases, he scored above the mean, but how many standard deviations above the mean?

So let's see if we can figure that out. For the LSAT, let me write this down. On the LSAT, he scored 172. So how many standard deviations is that going to be? Well, let's take 172, his score, minus the mean. So this is the absolute number that he scored above the mean, and now let's divide that by the standard deviation.

So on the LSAT, this is what this is going to be: 21 divided by 10. So this is 2.1 standard deviations above the mean. You could view this as a z-score; it's a z-score of 2.1. We are 2.1 above the mean in this situation.

Now let's think about how he did on the MCAT. On the MCAT, he scored a 37. The mean is a 25.1 and there is a standard deviation of 6.4. So let's see; 37.1 minus 25 would be 12, but now it's going to be 11.9.

11.9 divided by 6.4. So without even looking at this, this is going to be approximately, well, this is going to be a little bit less than 2. This is going to be less than 2.

So based on this information, we could figure out the exact number here. In fact, let me get my calculator out. So I get the calculator. If we do 11.9 divided by 6.4, that's going to get us to one point. I'll just say one point, I'll say approximately 1.86.

So approximately 1.86. So relatively speaking, he did slightly better on the LSAT. He did more standard deviations; although this is close, I would say they're comparable. He did roughly two standard deviations if we were to round to the nearest standard deviation.

But if you wanted to get precise, he did a little bit better relatively speaking on the LSAT. He did 2.1 standard deviations here, while over here he did 1.86 or 1.9 standard deviations. But in everyday language, you would probably say, well, this is comparable. If this was three standard deviations and this is one standard deviation, then you'd be like, oh, he definitely did better on the LSAT.

More Articles

View All
Example of shapes on a coordinate plane
So we’re told here the four corners of a rectangle are located at the points (1, 1), (1, 6), (9, 6), and (9, 1). Plot the four corners of the rectangle on the coordinate plane below, and they gave us these four points. We can move them around with our mou…
How to Make it Through Calculus (Neil deGrasse Tyson)
Through it, I have a, I have a— I don’t quite call it elevated to the level of a parable, but it’s a story in my life that I reference all the time. Right now, I share it with you as short. I’m in high school, I’m a junior in high school and I want to ta…
Copán Ruinas Was a Thriving City - Until One Day, It Went Away | National Geographic
[Music] Copan Ruinas is one of the most mysterious and spectacular cities of the Maya civilization. At its height, between 250 to 900 AD, approximately 27,000 mile IFFT. Here, thereafter, the civilization mysteriously crumbled, and the Copan Ruinas were l…
Why Time Goes Faster As You Get Older
Close your eyes. Remember yourself as a child, playing with your friends, stressing out about spelling tests at school, coming home to snacks on the table, and asking for help with your homework. What do you feel? Maybe you’re suspended in a time when thi…
STOICISM | How Epictetus Keeps Calm
Even though they followed the same philosophy, Marcus Aurelius was an emperor and Epictetus was a slave. The fact that someone from the lowest class became one of the greatest Stoic philosophers indicates that Stoicism isn’t just for the elite: it’s for e…
2015 AP Chemistry free response 3a | Chemistry | Khan Academy
Potassium sorbate, and they give us its formula right over here, has a molar mass of 150 grams per mole. They put this decimal here to show us that these are actually three significant figures; even the zero is a significant digit. Here is commonly added …