(upbeat music)
- Hello, I am professor Rick Jerz
and I welcome you to this session titled,
Continuous Probability Distributions.
We are going to continue exploring
probability distributions and probability calculations.
In our last session we learned
about discrete probability distributions.
In this session, we're going to switch our focus
to continuous data and continuous probability distributions.
Our focus will be on one of the most common
continuous probability distributions,
the normal probability distribution.
However, to get us in the groove, we will look briefly
at the uniform probability distribution,
which is simpler than the normal distribution.
We will continue to ask about likelihood of events
and explore area under the curve.
Yes, more equations and tables in this session.
But don't worry, we will continue using Excel.
I think you're going to enjoy this session.
Let's get started.
We have a number of goals for this session.
We want to understand the difference between
discrete and continuous distributions.
We want to be able to compute the mean,
standard deviation and probabilities
for a uniform distribution.
We want to be able to list the characteristics
of the normal probability distribution.
We want to be able to define and calculate the z values.
We want to determine the probability
of an observation being between two points
on a normal probability distribution.
And we want to determine the probability
an observation is above or below a point,
or is between two points
on a normal probability distribution.
Let's begin by reviewing what we already know.
We know that we can create a frequency distribution
of data, as we did with our green M&Ms in a bag.
We also know that we can convert a frequency distribution
into a probability distribution
by dividing frequency by the total count.
We also did this for our green M&Ms.
Being more precise, we call this
a discrete probability distribution
because we cannot have a bag with,
oh let's say, two and a fourth green M&Ms in it
or four and a half green M&Ms.
We can only have discrete or whole numbers.
Here is another discrete probability distribution
that we saw earlier for tossing two dice.
On the left scale, I show the count
and on the right side, I show the percent or probability.
This too is a discrete probability distribution
because we cannot throw two dice
and have them add up to, oh, let's say, 9.75.
We only get whole numbers.
From an earlier session I mentioned
that probabilities involve the area under the curve.
For a relative probability distribution,
the total area under the curve is equal to one.
We can do this, even with
a discrete probability distribution,
if you consider that the width of each bar
is equal to one unit.
Remember, area of a rectangle
is equal to length times width.
For these discrete probability distributions,
it's exactly the same as adding up the heights of each bar.
For example, if the probability of one bar is .167,
if we multiply this by the width of one,
we end up the area of .167.
The total of all the areas of these bars equals one.
You have to appreciate that
for a discrete probability distribution,
the curve that I refer to is just a series of these bars.
So you might consider this to be a special case of a curve.
When the experiment is simple,
creating these probability distributions is also simple.
But we also learned that for some experiments,
adding up all the possibilities
becomes a little bit more difficult.
This is what we saw with the binomial distribution.
Remember for the binomial distribution,
we actually have a family
of discrete probability distributions
based upon how many times we repeat the experiment.
If somebody was asking about the probability
of an event for a binomial distribution,
and if the event of interest involves several values,
in other words, a cumulative binomial distribution question,
it would get a little bit hard to add up the heights
of all of these bars.
This is why some people use an equation
for the binomial distribution.
However, we did it in Microsoft Excel.
My Excel model allows you to put in
the probability of successes,
the total number of events,
the highest and lowest number of successes,
and it calculated the probability of success.
This was also plotted for us in my bar chart shown in red.
Now let's turn our attention
to continuous probability distributions.
Here is what a continuous probability distribution
might look like.
Notice the continuous line moving up and down.
For any continuous probability distribution,
we still have the concept of central tendency.
Where is the balance point of this curve?
There is also a concept of dispersion
and we can still ask probability questions.
We would answer the probability questions
by calculating the area under the curve
between the two values of interest,
just as we've done for discrete probability distributions.
Before we look at the normal distribution,
which is a little bit more complex,
let's look at a simple, continuous distribution
called the uniform distribution.
The uniform probability distribution
is perhaps the simplest distribution
for continuous random variables.
This distribution is rectangular in shape
and is defined by its minimum and maximum values.
The total area of this uniform distribution,
we can define, still, as equaling one.
A rectangle is a very simple shape
and you might recall that the area of a rectangle
is equal to length times height.
So our total area must still equal one.
If you look at this rectangle,
its width is equal to b minus a.
Width times height has to equal one.
So we can actually solve for its heighth
and find out that the heighth of the rectangle
must be equal to one over b minus a.
If we multiply the heighth times the width, we get one.
The author has provided us
with the mean and standard deviation
of this uniform distribution and it's height,
equal to one divided by b minus a.
Given that this is a continuous probability distribution,
you have to appreciate that the probability
of any single value occurring
is the width times the height of that value.
But the width of any single value is very narrow,
so the probability of any single value happening
is essentially zero.
Well, from this observation we may conclude that
probabilities cannot be determined
for a uniform distribution, but they can.
This is a very important point that I'm about to make.
For a continuous probability distribution,
we can only calculate the probability between two values.
We can no longer do it for one value.
It must be between two values.
Recall that for discrete probability distributions
we were able to ask about the probability of a single value.
If you look at this rectangle once again,
given two values, we can subtract the top from the bottom
to give us a width and then multiply it by its heighth.
Let's take an example.
Southwest Arizona State University provides bus service
to students while they are on campus.
A bus arrives at the North Main Street
and College Drive stop every 30 minutes
between 6:00 a.m. and 11:00 p.m. during weekdays.
Students arrive at the bus stop at random times.
The total time that a student waits
is uniformly distributed from zero to 30 minutes.
In this example, the author simply states
that this data is uniformly distributed.
But how might we really know this?
Well, we could give somebody an assignment
to sit out at the bus stop and collect data.
Then we would plot this data
and observe what the actual distribution is.
If the actual distribution looked uniform
then we could apply this theory.
We want to be able to draw a graph of this distribution
and answer how long will a student
typically wait for the bus.
In other words, what is the mean waiting time?
What is the standard deviation?
We also wanna be able to determine,
what is the probability a student will wait
for more than 25 minutes?
You have to appreciate that more than 25 minutes
means between 25 and 30.
What is the probability a student will wait
between 10 and 20 minutes?
This chart shows how we can calculate
the heighth of this rectangle.
It is equal to .0333.
The mean and standard deviations
can be calculated from the equations.
You might be able to appreciate that the mean
of a uniform distribution is simply the middle.
In this problem, the mean is at 15 minutes.
To answer the first question,
that a student waits more than 25 minutes,
we have to figure out the area under the curve.
The width is equal to 30 minus 25,
times its heighth of .0333,
gives us an area of .1667.
The probability of waiting between 25 and 30 minutes
is .1667 or 16.67%.
For the second problem,
the probability of waiting between 10 and 20 minutes,
we again calculate the area under the curve.
The width is equal to 20 minus 10, or 10,
multiply this by the heighth of .0333
and we get the probability equaling 33.33% or .3333.
We are looking at this uniform distribution
before the normal distribution
just to appreciate this concept of calculating probabilities
by calculating the area under the curve of interest.
The uniform distribution is rectangular in shape
so calculating areas is pretty easy.
Having seen our first continuous probability distribution,
let's more on to the normal probability distribution.
This curve looks quite a bit more complex
than the uniform distribution curve.
We have this much more complex equation
to calculate the heighth of this curve at any point.
Before we do any calculations,
let's understand some characteristics
of the normal probability distribution.
It is bell-shaped and has a single peak
at the center of the distribution.
The arithmetic mean, median and mode are all equal.
The total area under the curve is still one.
Half of the area under the normal curve
is to the right of its center point
and the other half is to the left.
It is symmetrical about the mean.
It is also asymptotic.
The curve gets closer and closer to the x-axis
but never actually touches it.
To put this another way, the tails,
or the two sides of the curve,
extend indefinitely in both directions.
The location and its shape are determined
by the mean and standard deviation of our data respectively.
This last point is very subtle.
If we go back to the equation of this curve,
notice that the only variables that we have to calculate
are the mean and standard deviation
and then we could calculate probability.
To some of you, this equation might look like
it has some other variables, namely this pi and e values.
But pi is approximately 3.14159
and e is approximately 2.71828.
So these two letters are actually values.
Well, gee, this equation and concept looks pretty complex.
Why should we even care about the normal distribution?
Well it turns out that a lot of data in life
is normally distributed.
For example, exam scores, our heights, our weights,
the amount of coffee dispensed from a coffee machine.
There are many kinds of natural situations
where if you collect the data, plot it on a curve,
it'll come up looking like this normal distribution.
For those of you who are wondering
why this curve is called bell-shaped,
let me show you a little video
that will help you understand why.
I did this animation using my CAT software,
and quite honestly, I've never seen anybody else
do the same thing.
Finally, here is our bell.
Can you remember back
to the binomial probability distribution
where we said that there was not
a single binomial probability distribution
but rather a family of these.
The same thing is true for the normal distribution.
We have an infinite family of normal distributions
defined by the mean and standard deviation
shown here graphically.
In an early session we talked about the Empirical Rule
and how it applied to normally distributed data.
Now we're really gonna get into this.
About 68% of the area under the normal curve
is within one standard deviation of the mean.
About 95% is within two standard deviations of the mean.
And practically all within three standard deviations.
If I were to say this a little bit differently,
the area under the curve within in plus or minus
three standard deviations of the mean
is about 100% of the area under this curve.
Let's take a look at another quick example
of the Empirical Rule.
As part of its quality assurance program,
the Autolite Battery Company
conducts tests on battery life.
For a particular D-cell alkaline battery,
the mean life is 19 hours.
The useful life of the battery
follows a normal distribution
with a standard deviation of 1.2 hours.
Answer the following questions.
About 68% of the batteries failed between what two values?
How about for 95%?
And how 'bout for virtually all batteries?
The graphic to the right shows that 68% of the failures
will occur between 17.8 and 20.2 hours.
95% will occur between 16.6 and 21.4.
And practically all failures should occur
between 15.4 and 22.6 hours.
If I wanted to make a probability statement,
I would just rearrange the words
and say this a little bit differently.
Let me give it a try.
The probability of a battery lasting
between 17.8 hours
and 20.2 hours is 68%.
The probability of it lasting
between 16.6 hours and 21.4 hours is 95%.
You might have to look at this graphic to the right
to appreciate these probability statements.
Remember how we converted frequency data
into relative frequency data?
Frequency data involved the actual count
of how many times an event occurred within a category.
When we converted this frequency diagram
into a relative frequency diagram,
we ended up with the area under the curve equaling one.
We can do the same thing for normally distributed data.
This graphic shows the conversion of normal data
into what is called standard normal data,
where the mean is equal to zero,
the standard deviation is equal to one,
and the area under the curve is equal to one,
just like we did for frequency data.
When we have a standard normal distribution,
we can find the probability of certain events happening
by look at the area under the curve.
But, we have a challenge,
how do we figure out the area under the curve?
With the uniform distribution
where we had rectangular shapes,
the area under the curve was pretty simple to calculate.
But for the standard normal distribution,
calculating areas is much more difficult.
Let me give you a quick little example.
Pretend we made a cake in the shape
of this standard normal distribution
and we had four children in our family.
And I told you to cut this cake into four equal pieces.
The first cut might be pretty simple,
you'd cut it right in the middle.
But where would you cut the right side of this cake
so that the remaining two children
would have exactly half of this half,
or 25% of the cake?
Well, we could give you that complex equation
to crunch through every time
I was asking a question about probabilities.
We can't use the Empirical Rule to help us out either
because it only defines
one, two or three standard deviations from the mean,
at 68%, 95% or all.
It says nothing about 25%,
but many years ago statisticians created a simpler way
to find out the area under the curve.
They created a table
based upon the normal standard distribution
and created another value called the z-value
to help us figure out area under the curve.
The z-value is a distance between a selected value,
designated by x and the population mean,
divided by the population standard deviation,
shown here in this formula.
This z-value measures how many standard deviations
away from the mean is your data of interest.
Once again, the Empirical Rule described
one, two or three standard deviations from the mean.
In other words, z equals one, two or three.
But what do you do if you're curious about
what is the probability one and a half standard deviations
away from the mean?
Or 2.58 standard deviations away from the mean?
At the back of most statistics books,
the authors provide a z table,
which provides area under the curve or probabilities.
In this graphic, if the z-value was equal to 1.5,
the probability would be .4332.
Since the normal distribution is symmetrical,
most statistics book only give you the area
under one side of this normal curve,
and they leave it up to you to realize and then calculate
what the probability would be
if the values were on both sides of the mean.
This is the way that all textbooks approach probabilities
for normally distributed data.
But I have to tell you that I do not like this approach.
In fact, I hate table approaches.
You might be wondering why I make such a bold statement
about this approach.
But remember back to our session
on our binomial probability distribution.
My model that I created in Microsoft Excel
was both easier and faster
when we wanted to calculate the probability
for binomial distributed data.
We did away with the need of referring
back to tables in the back of the book.
Well, guess what, I'm going to give you
another one of my Excel models
that will make calculating probabilities
for normally distributed data easy.
We will never have to calculate these z-values.
However, for you to appreciate this z-value approach,
let's look at a couple examples.
The weekly incomes of shift foremen in the glass industry
follow the normal probability distribution
with a mean of 1000 and a stand deviation of 100.
What is the z-value for the income,
let's call it x, of a foremen who earns $1100 per week?
How about for a foreman who earns $900 per week?
This problem is challenging you to calculate the z-value.
Once you get the z-value,
you'll be able to read the probability
from the table at the back of the book.
But I don't like this example
because in real life we don't care about these z-values.
We only care about the probability of something happening.
The z-value is an intermediate calculation,
not the final result.
This power point shows a graphic
describing the area of interest for this problem.
Visually we can see the area
but we have to go back to that table in the back of the book
to determine what the area actually is.
And yes, Microsoft Excel
gives us a number of functions
relating to the normal probability distribution.
In my next session I will show you
my Excel normal distribution model
and we're going to try to solve
these remaining two examples.
Let me end by telling you that
when I took my first course in probability and statistics,
I was taught this z-value approach.
Being an engineering student,
I had no problem looking at equations.
I had no problem reading tables from the back of the book.
And I was able to do these calculations
and come up with the correct answer to problems.
Realistically, I didn't know what the heck I was doing.
I never realized that all these z-value calculations
were intermediate calculations
that were going to be used to calculate
the probability of something happening.
Oh well, I got As in my statistics courses.
Now you have to appreciate
that when I was an undergrad student,
we did not have Microsoft Excel
or even personal computers.
In fact, we didn't have calculators
until my second year in college.
So the table approach was the only realistic way
to do these calculations.
But modern computers and powerful software like Excel,
I believe, has eliminated the need for tables.
Some day I think all statistics books
will do away with the z-value.
But until then, we still get exposed to it.
In the future, we will have a few more sessions
dealing with normally distributed data.
For now, we have covered the basics.
(upbeat music)
Thanks for taking time
to spend some moments with me.
I hope that you have enjoyed what I have shared with you.
If you ever have any questions or need my help,
please let me know.