Main content

## Z-scores

# Normal distribution problem: z-scores (from ck12.org)

## Video transcript

Here's the second problem
from CK12.org's AP statistics FlexBook. It's an open source
textbook, essentially. I'm using it essentially to
get some practice on some statistics problems. So here, number 2. The grades on a statistics
midterm for a high school are normally distributed with a
mean of 81 and a standard deviation of 6.3. All right. Calculate the z-scores for each
of the following exam grades. Draw and label a sketch
for each example. We can probably do it all
on the same example. But the first thing we'd
have to do is just remember what is a z-score. What is a z-score? A z-score is literally just
measuring how many standard deviations away from the mean? Just like that. So we literally just have to
calculate how many standard deviations each of these guys
are from the mean, and that's their z-scores. So let me do part a. So we have 65. So first we can just figure out
how far is 65 from the mean. Let me just draw one chart
here that we can use the entire time. So it's just our distribution. Let's see. We have a mean of 81. That's our mean. And then a standard
deviation of 6.3. So our distribution, they're
telling us that it's normally distributed. So I can draw a nice
bell curve here. They're saying it's normally
distributed, so that's as good of a bell curve as
I'm capable of drawing. This is the mean
right there at 81. And the standard
deviation is 6.3. So one standard deviation above
and below is going to be 6.3 away from that mean. So if we go 6.3 in the positive
direction, that value right there is going to be 87.3. If we go 6.3 in the
negative direction, where does that get us? What, 74.7? Right, if we add 6, it'll
get us to 80.7, and then 0.3 will get us to 81. So that's one standard
deviation below and above the mean, and then you'd add
another 6.3 to go 2 standard deviations, so on and so forth. So that's a drawing of
the distribution itself. So let's figure out
the z-scores for each of these grades. 65 is how far? 65 is maybe going to
be here someplace. So we first want to say,
well how far is it just from our mean? So the distance is, you just
want to positive number here. Well actually, you want
a negative number. Because you want your z-score
to be positive or negative. Negative would mean to the left
of the mean and positive would mean to the right of the mean. So we say 65 minus 81. So that's literally
how far away we are. But we want that in terms
of standard deviations. So we divide that by the length
or the magnitude of our standard deviation. So 65 minus 81. Let's see, 81 minus 65 is what? It is 5 plus 11. It's 16. So this is going to be
minus 16 over 6.3. We'll take our calculator out. And let's see, if we have
minus 16 divided by 6.3, you get minus 2 point--
oh, it's like 54. Approximately equal
to minus 2.54. That's the z-score
for a grade of 65. Pretty straightforward. Let's do a couple more. Let's do all of them. 83. So how is it away
from the mean? Well, it's 83 minus 81. It's two grades above the mean. But we want it in terms
of standard deviations. How many standard deviations. So this was part A. A was right here. We were 2.5 standard
deviations below the mean. So this is part A. 1, 2, and then 0.5. So this was A right there, 65. And then part B, 83, 83 is
going to be right here. A little bit higher,
but right here. And the z-score here, 83 minus
81 divided by 6.3 will get us-- let's see, clear
the calculator. So we have 83 minus 81
is 2 divided by 6.3. It's 0.32, roughly. So here we get 0.32. So 83 is 0.32 standard
deviations above the mean. And so it would be roughly 1/3
third of the standard deviation along the way, right? Because this as one whole
standard deviation. So we're 0.3 of a standard
deviation above the mean. Choice number C. Or not choice, part C, I
guess I should call it. 93. Well, we do the same exercise. 93 is how much above the mean? Well, it's 93 minus 81 is 12. But we want it in terms
of standard deviations. So 12 is how many standard
deviations above the mean? Well, it's going
to be almost 2. Let's take the calculator out. So we get 12 divided by 6.3. It's 1.9 standard deviations. Its z-score is 1.9. Which means it's 1.9 standard
deviations above the mean. So the mean is 81, we go one
whole standard deviation, and then 0.9 standard deviations,
and that's where a score of 93 would lie, right there. Its z-score is 1.9. And all that means is
1.9 standard deviations above the mean. Let's do the last one. I'll do it in magenta. D, part D. A score of 100. We don't even need
the problem anymore. A score of 100. Well, same thing. We figure out how far is 100
above the mean-- remember, the mean was 81-- and we divide
that by the length or the size or the magnitude of our
standard deviation. So 100 minus 81 is
equal to 19 over 6.3. So it's going to be a little
over 3 standard deviations. And in the next problem we'll
see what does that imply in terms of the probability of
that actually occurring. But if we just want to figure
out the z-score, 19 divided by 6.3 is equal to 3.01. So it's very close. 3.02, really, if
I were to round. So it's very close to 3.02. Its z-score is 3.02, or a grade
of 100 is 3.02 standard deviations above the mean. So remember, this was the
mean right here at 81. We go 1 standard deviation
above the mean, 2 standard deviations above the mean, the
third standard deviation above the mean is right there. So we're sitting right
there on our chart. A little bit above that, 3.02
standard deviations above the mean, that's where
a score of 100 will be. And you can see the
probability, the height of this-- that's what the chart
tells us-- it's actually a very low probability. Actually, not just a very
low probability of getting something higher than that. Because as we learned before,
in a probably density function, if this is a continuous, not a
discreet, the probability of getting exactly that is 0,
if this wasn't discrete. But since this is scores
on a test, we know that it's actually a discrete
probability function. But the probability is low of
getting higher than that, because you can see where
we sit on the bell curve. Well anyway, hopefully this at
least clarified how to solve for z-scores, which is pretty
straightforward mathematically. And in the next video,
we'll interpret z-scores and probabilities
a little bit more.