Bam! Mr. Tarrou. Okay, in this video we are going to do, actually a fair amount of review.
What we’re going to do is define two types of variables that you’ve already learned
how to deal with. Continuous random variables, which you have started working with in chapter,
oh maybe 2 or 3 in your textbook when you started learning how to do z-score calculations
and the normal distribution, and discrete random variables, which is the science that
we have in the probability chapter we just finished. Where maybe you roll the die and
there were 6 specific outcomes. So, um, I could verbally do that, or you could just
take your notes down. Okay. So, discrete and continuous random variables. A random variable
is a variable whose value is a numerical outcome of a random phenomenon such as if you flip
two coins, when you flip two coins each one has two outcomes, heads or tails, so that’s
2 times 2 or 4 different combinations, and they are listed here in our sample space.
A concept again in our probability chapter. Where you list all of the possible outcomes
of whatever your series of trials are. Now, that’s not a random variable though. So,
we define a random variable as whatever you know, interest we are looking at, in this
case we are going to look at the number of heads when we flip our two coins. Well, there’s
4 specific outcomes in the sample space, but now that our variable, our random variable
is defined as number of heads that will occur in these two tosses, we have 0 heads, which
would be, you know, getting two tails of course, 1 value of head, and that’s heads tails,
or tails heads, and then of course our last outcome which is getting two heads. So, we
have a sample space of all of the possible outcomes like if you’re rolling 2 die, that
would have 36 possible outcomes. But then your random variable is exactly what you are
looking for. And in this example, we are counting the number of heads out of two dice. Okay.
Well, what I’ve got here is an idea of a discrete random variable. Where there is a
countable number of possibilities. 0, 1, or 2. So that is a discrete random variable,
has accountable number of possible outcomes, and the probability distribution of a discrete
random variables lists all of the values of their possibilities. Basically, a pdf, the
probability distribution, and then the f stands for function. Probability distribution function.
That’s basically the same thing as the probability model, where you list all of the outcomes
and their probabilities. The name is just changed because we are going from just looking
at sample space to you know actually giving this formal definition of a random variable,
assigning it these numerical outcomes, and you know, so we are just slightly changing
the name, just slightly, to indicate that we are talking about basically a probability
model of a random variable, but now we are calling it a probability distribution function.
Just list off the outcomes and the probabilities. And I can do that with these three outcomes,
but I have another example here in the next screen, but I’ll verbally say it, the probability
of getting 0 heads is 1/4th, the probability of getting 1 head, well that’s two of the
outcomes here, out of the four possibilities, so that would be ½. And then the probability
of getting 2 heads, well that’s 1/4th. Alright? Moving on the to next screen. Or panel of
chalkboard. Na-na-na-na. There we go. Okay. So, we have a variable defined as, let’s
say I’m just making a random selection of people in the population, and I want to find
out how many TVs they have in their entire household. Maybe 1 in the bedroom, one in
the living room, or what have you. And my variable of interest is number of TVs per
household, so I have a pdf, a probability distribution function set up here, basically
looks exactly like the probability model, and, you can probably just use that interchangeably.
We have a list of outcomes, 0 through 6, and if you do what it says, they have so much
money that, I mean, we have like 2, but you know, even 6 TVs I think is excessive, and
if any more than that, give me break. I’m not even bothering with you. So, I am ignoring
all of the values or all of the answers of people saying they have more than 6 TVS. So,
I have a list of my outcomes, and a list of those probabilities, and for this to be valid,
those do need to add up to one. And indeed, they will. So here I’ve got a pdf, nice
organized, set of data representing a discrete random variable. So, note: only us the formulas
for discrete random variable when the data is displayed in a probability distribution.
And these are the formulas for finding the mean and standard deviation of a discrete
random variable. It says the mean of x, so this would be the mean number of TVs in the
household, is found by taking the sum of x sub I times p sub i. well, these subscripts
simply meaning where you are in the list. This would be x sub 1 p sub 1. X sub 2 p sub
2. X sub 3 p sub 3, x sub 4 p sub 4 and so on. So, you’re are just going to take the
individual outcomes, the individual probabilities, and then add all of those together. And the
standard deviation of x is equal to the square root of the summation of x sub i, again the
individual outcomes minus the overall mean of x, so you have to find the average before
you find the standard deviation. That difference is going to be squared. And then you multiply
the probability of each individual outcome together. I’m going to do an example of
how to work this out in you TI-84 and your inspire calculators in a later video, coming
up very soon after the thanksgiving holidays, after the pumpkin pie, uh. Let me just write
out a short string of steps here for the average If you wanted to find the average for x, it
is the individual outcomes multiplied by the probabilities, added together. So, it looks
something like this. 0 times .08 + 1 times .16 + 2 times .37 and so on, until you finish
adding up that product between the outcomes and the probabilities all along the pdf table.
And then, working this formula out, this formula will need to be worked out, or will probably
be worked out, using your STAT edit list, you know where you do your L1, L2, L3 type
place in your calculator. We’ve put data in that screen when we were doing scatterplots,
like the Xs would be in in list 1 and the Ys would be in List 2. Well, you can use some
calculator commands at the top of those columns to help sort of simplify or make more efficient
use of your calculating that standard deviation formula. There is no calculator command that
will find the standard deviation for you automatically that I know of. Like 1-var stats will not
run this calculation formula for you. So, I kind of do a little bit of calculator assisted
work using the stat edit, those List 1 list 2 columns that we have like in the TI-83 or
TI-84. And I will be showing you how to do that with an inspire as well, that will be
in a later video, and like I said after I take off for holidays. So, about a week from
now if you are watching these videos as I am you know developing this channel. If data
is given as a long list of numbers, then just simply put those into the calculator and do
1-var stats. Now you know you would find these formulas on your AP formulas sheet under the
probability section, not somewhere where it says discrete random variable. You only use
these formulas like I’ve stated if you’ve got this data organized into a pdf table already.
If I just said “hey, I went and asked how 50 people how many TVs are in their household,
and these are the 50 numbers that I got.” Now, even though I can count he fact that
I’m only accepting 7 possible answers, if you just give me 50 of these numbers, I’m
not going to necessarily take the time to make the pdf table. I’ll just put those
numbers into list 1 and do what you learned how to do in either chapter 2 or 3 in your
textbook, which is simply do 1-var stats with your calculator, and let it simply add up,
let’s say, all 50 numbers and then divide by 50. And the standard deviation of my sample
again is going to, you know, 1-var stats in your calculator is going to run this formula
out which is the square root of 1 divided by n-1 times the sum of x sub i minus the
mean of x squared. Now, I’ve got to do another panel of notes, but I feel like I really want
to stop and, yeah I’m probably going to run out of time anyways. Let me make this
a two-part video and remind you of some of the characteristics, I’m not sure that’s
the word I’m looking for, but characteristics of standard deviation. Like what is this n
minus 1? You know, just, you could go back watch my earlier videos, but just as a reminder,
you might be getting close to your semester 1 exam, don’t forget that n minus 1, the
reason that we are divided by n minus 1 when we are finding the standard deviation of a
sample, is that’s degrees of freedom, that’s how many pieces of data are allowed to vary
freely. If you take a sample of a population, and that sample let’s say, I just tell you
that the average is going to be 10. Well, if it’s a sample of 5 numbers that are going
to have an average of 10, four of those numbers could be anything, like you could just make
them up. Four of those numbers are able to vary freely, and it’s only that fifth number,
that last number that has to be fixed so that when you add them all up and divide by 5,
you get that average that you’re supposed to get, maybe like 10. And remember, sample,
results from a sample, where there’s a sample mean or a sample proportion, you know, that’s
supposed to be equal to the population mean. Now, you know that it won’t be exactly equal
because we have this thing called sampling variability. Even if you flip a coin 10 times,
you’re not going to get heads 50% of the time, but you’re going to get heads around
50% of the time. So, if you were to build a sampling distribution, then you know flip
the coin ten times record the percent of heads, flip the coin 10 times record the percent
of heads, and repeat, and repeat, and repeat. If you were to take those results and make
a histogram out of it, that would be called your sampling distribution, some old concepts
I’m bringing up now. The center of that sampling distribution, if it is unbiased,
is supposed to have the same mean as the population mean. And that degree of freedom therefore,
idea, again, if you know that this, or if a collection of numbers is going to have a
particular average, all but one those numbers can vary freely. So, n minus 1 is degree of
freedom. The distance between each piece of data and the mean, the sum of those differences
is always going to be zero remember? That’s when we have to square them before we add
them so we don’t keep getting a sum of 0. Don’t forget that standard deviation is
the average distance each piece of data is from the mean. Okay? And calculation standard
deviation is not appropriate unless your population distribution or your sampling distribution
is approximately normal. So, we will be talking about that in a second. I’ve only got one
panel of notes, but I’m going to do a lot of verbal reminding you, a lot of verbalization,
(chuckling), I’m losing my words as I’m trying to think of what to say, but I’ll
verbally remind you of a lot of concepts you learned a lot earlier in your statistics book
as we go over this last panel of notes. Let’s see. Standard deviation is only appropriate
when your data is normal. Don’t forget that any calculation that uses the mean as part
of its calculation is not a resistant measure. Remember that resistant measures are those
that are not strongly affected by outliers. And pretty much the only thing that it is
resistant is the median, and any calculation that uses median. And there’s not many,
maybe IQR, the distance between Q3 and Q1. All of the other calculations that we have
like standard deviation, like correlation which is r, or r squared, any calculation
that uses the mean, and part in its formula is NOT going to be a resistant measure, it
will going to be strongly affected by outliers, and the unit of measure for standard deviation
is whatever the unit of measure is that your data is. If you’re measuring something in
miles, then standard deviation is in miles. If you’re measuring something in pounds,
then standard deviation is pounds. So, I’ve used up my couple of minutes I had left to
remind you of some, you know, connects about standard deviation that you’ll need to know,
like say to correctly answer some multiple-choice questions. I’m Mr. Tarrou, I’ll be right
back with part 2. Bam!