In this video we're going to introduce the idea of the normal distribution
again it's a theoretical probability distribution which allows us to go from
if we know the truth for an entire population learning what can show
up when we collect the sample of data; so we're going to take this example here
let's live in the pretend world and suppose that we know that the heights of
an entire population are normally distributed meaning they have a
distribution: histogram of heights that is bell-shaped and
perfectly symmetric about its mean and we know that the mean is 175 centimeters
and the standard deviation is 10 centimetres. In other words we know that
the distribution of heights looks like this in the population. It is worth
mentioning now often when working with a normal distribution people tend to start
leaving out the y axis here and not labeling it as describing the
probability okay so let's learn some properties of the normal distribution
the mean describes where the normal distribution centers itself and the
standard deviation tells us something about how spread out or how variable it
is. As the standard deviation gets larger the normal distribution gets wider as it
gets smaller the normal gets narrower the normal distribution is well
described by its standard deviation. So there's this very handy 68, 95, 99.7 rule
or one two three standard deviation rule then again this is just handy for doing
some simple ballpark calculations so what this tells us is if we go from the
mean and reach out one standard deviation on each side so if we go plus
one standard deviation, minus one standard deviation approximately 68% of
observations are contained within this interval here approximately 68% of
individuals will have a height between 165 and 185 centimeters
if we reach out two standard deviations going up to 195 and down to 155,
again reaching out two standard deviations above and two (standard deviations) below the mean
approximately 95% of the population will be in this interval and I won't draw it in
because we don't want the picture to get too messy but approximately 99.7%
most of the individuals stay within three standard deviations of the mean
again for a normal distribution.
Now for our normal distribution we're gonna want
to work out if we know the truth for a
population how likely are certain things to show up will we collect some data so
let's think about the example of if we randomly select an individual from this
population what's the probability that that individual's height say is greater
than 190 centimeters okay and again the main point of doing these exercises is
for us to learn how do samples behave. To work out something like this essentially
what we want to do say the mean is 175 if we reach in this population randomly
select one individual how likely is it that that they will be 190 centimeters
or more. and we'd like to work out this area here and again to think of it
conceptually what we're going to be doing is you can think of what you've
done earlier when learning these concepts
with a histogram if we wanted to know what percent of our sample is 190 or
more we'd simply add up these bars here okay so conceptually that's what we want
to do let's get to talking about how we can do these calculations so to do this
we're going to work with the concept of standardizing
and again the way I'm going to discuss this is introducing it as being simply a
unit conversion and the reason I like to do this is when we think of unit
conversions a lot of things become intuitive so let's say for example if we
moved from Heights being recorded in centimeters to Heights being recorded in
inches do you think that the shape of the distribution would change instead of
looking normal would it start to look skewed? of course not
right we take these units here and we replace them with whatever 175
centimetres is in inches or 190 centimetres is in inches; okay so when
you think of converting from one scale to another it helps you think about
what's going to happen to the distribution itself, so again let's think
of using the standard deviation as a unit of measurement we want to focus on
this here being 190 centimeters and if we think of how far is 190 from that
mean of 175 it's 15 centimeters above the average so we can refer to this
individual as being 190 centimeters or as being 15 centimeters above the
average and now remember we've learned about the standard deviation and we've
learned that essentially this is telling us what's the average deviation so if we
look at this height in terms of the average deviation they are 15 centimetres
above the average and if we divide by the standard deviation of 10 centimeters
we find that there one and a half standard deviations above the average
okay so we can refer the same individual who is 15centimetres above average as
being one and a half of the average deviations above the average so this
idea here that's what we label with a Z
Okay or Z for you Americans, we look at how far is an observation from the mean
in terms of a standard deviation.Or how far is X from Mue in terms of Sigma
the standard deviation. and now as we said thinking of this idea of
standardizing is being simply a unit conversion it becomes more intuitive
that if Heights are normally distributed at the population level then the
standardized heights should also be normally distributed so standardizing,
all we're really doing is changing this scale here so we found that the height
of 190 centimeters is 15 centimeters above average or one and a half standard
deviations above average. So let's just quickly draw that out here so here's
looking at the heights' mean of 175 and we want to find out what's the
probability of an individual being 190 or more again this is on the scale of
centimetres and as we noted we can think of this as simply being a unit
conversion to standard deviations or the mean of 175 is zero and the height of
190 centimeters is 1.5 standard deviations above average okay so the
probability of someone being 190 centimeters or more is the same as the
probability of being one and a half standard deviations above average or
more or if you like to picture a little bit cleaner we can redraw this here
rather than on the scale of height on the standardized height
and we want to find out what's the probability of being one and a half
standard deviations above average or more we can find that this here comes up
to be 0.067 or 6.7% so we can think of this as
being that if we randomly select an individual from the population there's
about a 6.7% chance of them being 190 centimetres or more or we can think of
it that in the entire population 6.7%
of people will be 190 centimetres or more. Now I've just jumped
to throwing this probability here we can find it using a Z-table or we can find it
using software (like R). the reason I've left it out here is I want to focus this on the
concepts rather than how to find this number from a table or how to find it in
a piece of software because that distracts us from what is it we are
actually trying to do!