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## AP®︎/College Statistics

### Course: AP®︎/College Statistics > Unit 9

Lesson 1: The normal distribution, revisited# Probabilities from density curves

AP.STATS:

VAR‑6.A (LO)

, VAR‑6.A.2 (EK)

, VAR‑6.A.3 (EK)

, VAR‑6.B (LO)

, VAR‑6.B.1 (EK)

, VAR‑6.B.2 (EK)

Examples finding probabilities from probability distributions for continuous random variables.

## Want to join the conversation?

- at3:20how did you get the area as 68 %?(14 votes)
- hope this helps:

For datasets that have a normal distribution the standard deviation can be used to determine the proportion of values that lie within a particular range of the mean value. For such distributions it is always the case that 68% of values are less than one standard deviation (1SD) away from the mean value, that 95% of values are less than two standard deviations (2SD) away from the mean and that 99% of values are less than three standard deviations (3SD) away from the mean.(15 votes)

- how do we solve this using a z-table?(10 votes)
- It's been a year since the question, but others might wonder the same.

We know that 170 is one standard deviation away from the mean:`170 - 150 = 20`

In a z-table, look up the z-score 1.00 (my table gives me 0.84134). That is the probability`P(H<170)`

since z-scores work toward the left.

We want the opposite,`P(H>170)`

, so subtract that value from 1:`1 - 0.84134 = 0.15866`

, or roughly 16%

I hope this helped.(4 votes)

- How did Sal know this will be a normal distribution?(2 votes)
- In this case, it was a part of the problem. "A set of .. heights are normally distributed"(11 votes)

- How do we find the lower bound if it is not given?(2 votes)
- In a Normally Distributed Data-Set there is a special rule called the "Empirical Rule" that tells us what percent parts of the bell-curve data-set covers.

Empirical Rule:

68-95-99.7

Here is a Wikipedia example with an image to help you understand;

https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule(1 vote)

- what does the Y-axis of the density curves mean? like in the first example it was 0.25, what does it mean? is it relative frequency or percentile or density?. If it means density then what does it mean by 'density'?(1 vote)
- If the probability density at X = x is 0.25, then this means that the limit as h goes to 0 of P(x-h < X < x+h) / (2h) = 0.25. Loosely speaking, the probability that X is in a small interval containing x is about 0.25 times the length of that interval.(2 votes)

- how did he get the 68% ? and what's the point of using the mean in this problem, is there any other method ?(2 votes)
- How did he get 84% and 34%?(1 vote)
- How and when can we confidently assume that the mean is going to be exactly dividing the bell curve into two symmetrical regions? What if the bell curve was intrinsically
*asymmetric*? What if the median and mean were**not**the same?*I am fairly convinced that the median divides the area under the curve equally, but how do you know where the mean is if one of the***tails**of the curve is elongated or something like that? Won't the mean be affected by that attribute?

I'm confused.(1 vote)- Hey Ramana@

The definition of a normally distribution dataset (in this case a density curve) is that the median splits the data set into two equal sets.**Now**, many times that is not**exactly**the case so Mathematicians will use a rule to decide if it is close enough. Later you'll learn how they decide if this is so but for now this should hopefully help you understand.

Hope this helps,

- Convenient Colleague(1 vote)

- Is ther any video from khan academy regarding the values of standard deviation?(1 vote)
- what if this wasn't normally distributed ?(1 vote)

## Video transcript

- [Instructor] Consider
the density curve below and so we have a density curve that describes the
probability distribution for a continuous random variable. This random variable can take
on values from one to five and has an equal probability of taking on any of these
values from one to five. Find the probability
that x is less than four. So x can go from one to four. There's no probability that
it'll be less than one. So we know the entire area
under the density curve is going to be one. So if we can find the fraction of the area that meets our criteria then we know the answer to the question. So what we're gonna look at is
we wanna go from one to four. The reason why I know we can start at one is there's no probability,
there's zero chances that I'll get a value less than one. We see that from the density curve and so we just need to think
about what is the area here? What is this area right over here? Well, this is just a rectangle
where the height is 0.25 and the width is one, two, three. So our area's going to be 0.25 times three which is equal to 0.75. So the probability that x
is less than four is 0.75 or you could say it's a 75% probability. Let's do another one of these with a slightly more
involved density curve. A set of middle school students' heights are normally distributed with
a mean of 150 centimeters and a standard deviation
of 20 centimeters. Let H be the height of a randomly selected
student from this set. Find and interpret the probability that H, that the height of a randomly
selected student from this set is greater than 170 centimeters. So let's first visualize
the density curve. It is a normal distribution. They tell us that the
mean is 150 centimeters. So let me draw that. So the mean, that is 150 and they also say that we
have a standard deviation of 20 centimeters. So 20 centimeters above the mean, one standard deviation
above the mean is 170. One standard deviation
below the mean is 130 and we want the probability of if we randomly select from
these middle school students, what's the probability that
the height is greater than 170? So that's going to be this area under this normal distribution curve. It's going to be that area. So how can we figure that out? Well, there's several ways to do it. We know that this is the area above, one standard deviation above the mean. You could use a Z-table or you could use some
generally useful knowledge about normal distributions
and that's that the area between one standard
deviation below the mean and one standard deviation above the mean. This area right over here is roughly 68%. It's closer to 68.2%. For our purposes, 68 will work fine and so if we're looking
at just from the mean to one standard deviation above the mean, it would be half of that. So this is going to be approximately 34%. Now, we also know that
for a normal distribution, the area below the mean
is going to be 50%. So we know all of that is 50% and so the combined area below 170, below one standard
deviation above the mean is going to be 84% or approximately 84% and so that helps us figure out what is the area above one
standard deviation above the mean which will answer our question. The entire area under this density curve, under any density curve is
going to be equal to one and so the entire area is one. This green area is 84% or 0.84. Well, then we just subtract that from one to get this blue area. So this is going to be one minus 0.84 or I'll say approximately and so that's going to
be approximately 0.16. If you want a slightly more precise value, you could use a Z-table. The area below one standard
deviation above the mean will be closer to about 84.1% in which case this would
be about 15.9% or 0.159 but you can see that we got pretty close by knowing the general
rule that it's roughly 68% between one standard
deviation below the mean and one standard deviation above the mean for a normal distribution.