Comparing with z scores

- [Instructor] Before applying
to law school in the US,

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.

So the LSAT, the mean score is 151

with a standard deviation of 10.

And 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.

So 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 meet, centered at this mean,

we can think about, well,
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.

So on the LSAT, let's see,
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,

deviations

above the mean,

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 gonna be 11.9,

11.9

divided by 6.4.

So without even looking at this,

so this is going to be approximately,

well, this is gonna be a
little bit less than two.

This is going to be less than two.

So based on this information,

and we could figure out
the exact number here.

In fact, let me get my calculator out.

So you get the calculator.

So if we do 11.9

divided by 6.4,

that's gonna get us to one
point, I'll just say one point,

I'll just 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.