Bam! Mr. Tarrou! Okay! We have learned so far this year how to handle a single piece
of data like a, let’s say, what’s the probability that I find somebody that’s
over 5’10”? And that population of whatever we’re talking about, adult males, is normally
distributed, you do a z-score calculation. Then we learned about discrete random variables,
in those settings, you can count the number of outcomes. And there’s formulas for discrete
random variable, mean, and standard deviation, but we don’t use those formulas unless we
have a PDF table, where all the specific outcomes, and all of those probabilities are given.
Then we just learned about the binomial setting which is when we have two possibilities, there’s
a set number of observations, there’s independence and the constant probability of success, and
we can deal with binomial settings when they’re recorded as a number of successes from a set
number of trials, or a percent of successes out of a set number of trials. That was the
binomial setting, binomial count. And what we just studied which was binomial proportions.
Woo! Okay. Well now we’re going to… we also have the geometric setting, where you
went to your first success, and everything else out of the same is the binomial setting.
Well, now we’re going to work with sample means. And if your analyzing multiple observations,
and each one has more than just two outcomes, like pass fail or yes no, then you may be
setting up sample mean variable. Like GPA, blood pressure, shoe size, etc. Let me think
on the shoe size that I give here. When you measure somebody’s shoe size, most shoe
sizes are in half sizes, 6, 6 ½, 7, 7 1/2, 8, and so on. So, I can actually state or
count those outcomes. There is a minimum shoe size, and a maximum shoes size. You know you
could have like your really huge sizes like from the world’s tallest man, or basketball
players, you know shoe sizes of 17, 20, 25, I don’t know how big those record breaking
sizes are, but it is a countable variable. So, let’s stick on that idea. We randomly
select 30 male adults. And we measured the size of their foot, or hence their shoe size.
That’s a discrete random variable. The individual measurements. But if I then say, well if I
take 30 men, I measure the length of their feet, or measure their shoe size, and therefore,
maybe say, well what is the probability that if take the sample size of 30 men, their average
shoe size, not individual shoe sizes, but average shoe sizes, are above a 10. Well,
when you take that discrete random variable of shoes size, I forgot what I said already,
you take 30, something, whatever amount you collect, you add all of those up and you divide,
you of course are going to get a decimal. And the larger the sample is, the more your
decimals are probably going to run on. And thus, we go from a variable that is individually
a discrete random variable, to an average of a collection. Maybe the average of 30 measurements.
That becomes therefore a continuous random variable. So, if I said well what is the probability
that a randomly selected adult male has the shoe size of over 10? Well if I have a pdf
table of all the sizes and all the individual probabilities, maybe I would just add up all
the probabilities for 10, 10 ½, 11, and so on until I get to the end of the pdf table.
Whereas, if I said, well if I take the collection of 30 men, what’s the probability that the
average of all 30 men’s shoes sizes is over 10? Well then that is going to be a calculation,
a z-score calculation, or possibly a t-score calculation. You will not hear this in today’s
lecture, but when you’re working with means, there’s going to be sometimes when you do
a z-score, which is all my students know at this point, and then occasionally be doing
something called a t-score. That’s a concept that is going to be coming up in my class
very, very soon. That’s when you’re working with averages, and you don’t know basically
anything about the population. All you have is the data being collected from your sample.
So, you have a sample standard deviation, as opposed to a population standard deviation,
and we’ll get into that more later, but for now, let’s just stick with z-scores.
So, characteristics of sample means. Means of random samples are less variable than individual
observations. Take the shoes sizes, some people have really, really tiny feet, some people
have huge feet. Take the money market, not money market, but like the stock exchanges.
If you turn the stock markets on every day, you’ll see those values gyrate widely. But
if you then choose, well, it shows you the 30-day moving average, where it shows you
the average of the precious 30 days of values, then you get this nice, smooth curve, that
is easier to see what the trend is instead of all this jagged up and down movement. So,
averages will sort of like shave off or wear down the max and the min values, and gives
you a smoother, less variable statistic. You’re going to learn a lot about something called
the central limit theorem. That explains how the distributions of means are more normal
than individual observations. I’ll get to more of this later, but the central limit
theorem is a hugely important concept. Do more than just watch this video, read your
book, and take notes on that central limit theorem. It’s what allows so many settings
in an intro to statistics course to be done with a normal approximation, or a t or z-score,
because the central limit theorem will allow you to pull numbers out of a distribution
that’s clearly not normal, but yet the distribution of averages will be, I’ll get to more of
that at the end of this video. Alright? Page 2. Woo! [nananana] Bam! Okay. In the binomial
setting, we had four checks, two outcomes that were observations, independence, and
constant probability. Well, if you’re going to decide, or if you’ve figured out, or
you’ve concluded that you’re working with a sample mean, or you’re working with means,
and not proportions, then there’s three checks. Random selection from and SRS. No
matter what kind of test we’re doing, or what type of variable we think we are working
dealing with in intro to statistics, all of the data must have come from an SRS. Independent
observations. It’s often, but not always appropriate, but it is very often to simply
check that, yes an SRS was taken, and the population is over 10 times the sample size.
This must be true to use any of the standard deviation formulas. I’m talking about the
population being over 10 times the sample size. Now, there will be some settings in
these applications, that just simply saying it was an SRS from the population size, is
over 10 times the sample size, were just not appropriate. If you are an archeologist, and
you’ve found some fossils. You can’t just do an SRS of the population, you can only
work with what you have there in the ground. Well if there’s any possibility that this
large collection of fossils being sort of like a family unit, maybe you’re studying
some kind of animal or pre-historic humans that were nomadic in nature, we’ll have
to maybe have to draw conclusions off of those fossils that we find, you know, draw a conclusion
about the entire population. But the whole idea of the SRS, and especially the independence,
might be put into question because maybe there’s some sort of family genes that are shared
between those fossils that have been found, but this is really common, and usually appropriate.
You can just simply say that normally is stated or once again bring up this idea of central
limit theorem, which we will talk about on the last page. With a large sample of N, the
sampling distribution of the means, okay well we’re not talking about raw data, but the
means with have an approximately normal distribution of N, and this is, remember your shorthand
notation for normal distributions, capital M, and then mean and standard deviation, that’s
how you describe normal distribution, the mean and standard deviation. Or, the mean
of a sample mean, is the same as the population mean, that is true for any statistic or variable
that is unbiased, the law of large numbers says that this happens, and the mean of your
sample’s mean, or the standard deviation of your sample’s mean, is equal to the population’s
standard deviation divided by squared of n. Now, in intro. to statistics, a lot of times
we do use z-scores, and thus call things sigma or standard deviation of a population, but
a lot of times, in the real world especially, when you’re not in the textbook, we do use
means a lot. A lot of settings are not binomial, a lot of measurements that we take do not
just have 2 outcomes. So, when you’re working with these sample means, a lot of times we
just have the info. that we collected from our sample, and if that’s the case, when
you’re working with means, and you only have the sample standard deviation, which
is not what I have written here, I’m just giving you a preview of what is coming up,
then you would be doing something called a t-score instead of a z-score. And if you have
the sample standard deviation, and that was it, then this formula would be called standard
error of the mean, just giving you a preview again of what’s coming up. At this point,
in my textbook we’re still being given the population standard deviation, even though
in the real world, that’s not usually known. Just to kind of simplify and ease you into
these statistical analysis tests. Alright? Last page and we’ll finally give you, not
a demonstration, but an explanation of what central limit theorem is really trying to
discuss or show you. Bam! [nanana] Okay, so we talked about sample sizes, and what’s
appropriate for the central limit theorem, to ensure the average is the normal. Here’s
kind of a guideline. Required size samples of N for the central limit theorem, to ensure
normality for the distribution of the averages, is the sample size of less than 15 is going
to be okay, If you’re distribution of your raw data, or your population is close to normal,
if there’s some skewness to the raw data, or if there’s some skewness to your sample,
or if you know there’s some skewness to your population, then you want to sample size
between 15 and 30, and if you know that your population standard deviation is skewed, or
if you see a strong skewness with just the sample distribution, not the distributions
of means, but the sample’s distribution, then you need a sample size of 30 to make
sure the central limit theory will say or create this normal distribution for you for
the averages. Now let me give you a, at least an example or discussion of what I am talking
about here. Now, we know that questions involving money, I’m sure you could find an example,
but pretty much questions that always involve money are going to be right skewed. There’s
a lot of like low workers, and then just a few CEOs in the company. Or a lot of people
on the team, but there’s that two or three like star players that get all the press,
and all the money, and all the endorsements. So, money question will pretty much always
be right skewed. So, if we have a right skewed distribution about money, and then somewhere
there’s some kind of population mean, you know I could not say, well what if you pulled
one person out of the population, and you asked them you know, how much do you make.
What would be the probability that they would say over $25,000? Or any value for that matter.
You would not be able to answer that question because you’re pulling that one person out
of a right skewed distribution, and we don’t have any techniques for dealing with probabilities
of a skewed distribution, at least not if we’re going to do some kind of normal approximation.
Maybe if you have a geometric PDF, or a binomial PDF distribution table, you can just pull
numbers out of that, and maybe would look right skewed if you were to graph it, but
If you were to do z or t-score calculations, to approximate these normal probabilities
that’s not possible. So, maybe you took instead of just one person out of the population,
you took maybe 5 people out of the population. So, you took the sample size of 5, and you
took those 5 incomes that people stated they made, and you averaged them together. Well
that, you know if you were to take another 5, and another 5, and another 5, and another
5, and, you get the idea, and to take all of those averages and display a distribution
of the average of 5 people’s response, it’s not just one by one, then that distribution
of averages might look something like this. I’m trying to show a little bit of right
skewness here still, if your sample is unbiased, the mean of your sample is supposed to be
the same as the mean of a population, but you might still have, with the sample size
of only 5, you might still have some right skewness, but the shape is starting to change.
Not the shape of the raw data, it’s still going to be very right skewed, but the shape
of the distribution of the average of 5 people’s incomes will start to appear to be a little
bit more normal. I’m not loving my graph there, but it’s less skewed. Then, if you
come in, and let’s just do two pictures because I’m not really loving my skill of
drawing there, and let’s say I took a sample of 40 people. So, I average 40 scores, 40
income values together, and I made a distribution of that. That’s again, asking 40 people
how much they made, take an average, you know add all of those up and divide by 40, find
an average. And then, that is what you would do in the real world, but as far as talking
about distributions, you know, maybe you do another sampling of 40, and you find another
average, and another sample of 40, and you find another average, and you take another
sampling of 40, and so on until you have many, many, many, many samples of that, and you
can make a distribution of those average values. That distribution is going to be very, very
normal, that’s what the Central Limit theorem says. So, that would allow us to answer question,
well, if I asked 40 people what their incomes are, what’s the probability that the average
of those 40 individuals incomes are greater than say 25 or 35,000 dollars or less than
10,000 dollars or more than 100,000 dollars, the point is, I couldn’t ask you the probability
that one person makes over x numbers of dollars, because it’s a right skewed distribution,
but I could, because the distribution of averages are going to be normal, due to the central
limit theorem say, well if I sample 40 people, what’s the probability that their average
of those 40 people’s incomes are over a certain number of dollars. That could be done,
that’s the power of the central limit theorem, that’s why normality is so important, and
why we talk about it so much in this book, because when you are working with averages,
the central limit theorem will ensure that distribution of those average values, is going
to be approximately normal, even if the raw data is skewed, and it just allows us to work
with a lot more settings than otherwise we would be able to. It’s also again, what
turn that original example of shoe size, which was a discreet random variable, and allows
to work with it as a continuous random variable because it is the average of a lot of discreet
values because you get decimals and that sort of turns in a continuous random variable.
So, I’m Mr. Tarou, that’s the best I can do to describe the sampling mean idea, and
central limit theorem. Let’s do some examples. Woo! BAM! Go do your homework, or please watch
my next video.