What is a Sampling Distribution? Video Lecture | Mastering R Programming: For Data Science and Analytics - Database Management

[Music]
hi I'm Mike Maron today we'll discuss
the concept of a sampling distribution
of the mean in many fields of study we
collect data these data are used to
calculate an estimate which we use to
understand our world usually we collect
only one random sample and then use it
to estimate something say the mean of a
much larger population but what if we
had collected another random sample
wouldn't the mean be different wouldn't
our estimate of the population mean
change so how can we know that our one
sample can be trusted to represent a
much larger population and how can we
know that our work as statisticians is
precise the concept of a sampling
distribution helps us to understand how
we can use one sample to make statements
about the mean of an entire population a
sampling distribution is an important
concept in the statistical sciences it's
foundational and it's crucial that you
learn it but what is it really a
sampling distribution is the probability
distribution of a given statistics based
on a random sample it describes the
what-if of all the possible estimates we
could have ended up with still not sure
what I'm talking about
stick with me today to help us
understand these concepts we will be
estimating the average length or mean
length of fish that live in this loop
the thing we are interested in
estimating today which we will call a
parameter is the population mean our
variable of interest is the length of a
fish we have our random sample of 25
fish from this lake here and we will
begin to measure the length of each
individual fish now in order to get the
true population mean of all the fish in
this lake
we'd have to measure every single fish
and that could be thousands and we don't
have that kind of time today so today
I'm going to entrust the skills of a
special friend of mine
[Music]
Neptune can you please help us calculate
the length of every single fish in this
lake the important thing to learn here
is that we must understand how samples
behave when we know well everything the
truth we can do this using probability
theory first we're going to build up an
understanding of how samples behave when
we know the true values for the entire
population then we'll be able to learn
how to use a sample to try and make
statements about the entire population
in the real world when we don't actually
know the truth without help from nekton
we do this using statistical inference
[Music]
so we now know the mean length and the
standard deviation of the length and
this probability distribution shows the
length of all fish in this lake every
single fish so we know that this
population has a mean length of 40
centimeters and a standard deviation of
10 centimeters thanks for your help
Neptune we now know what we can do when
we have the power of Neptune at our
disposal but what if we don't
what if we're back to being regular
statisticians what if we can't count
each and every fish we will now collect
a random sample of 25 fish and measure
their length and calculate the mean of
the 25 lengths we will call this our
sample mean or estimate of the
population it's important to remember
that this sample mean is just one of
many we could have gotten just by chance
we would have a different sample mean
each time we collect data because each
set of fish is slightly different from
one another look at this histogram it's
starting to show not one not two not
three not four but all of the possible
sample means we could have ended up
wouldn't it be great if we could see a
histogram of all the possible sample
means without having to collect sets of
25 fish over and over again forever what
would happen if we ended up with a
different sample of 25 fish you'll be
pleased to hear that there is a way to
know approximately what the shape of
this distribution would look like
without repeating the sampling process
thousands of times thanks to Neptune we
now know that the true mean of fish in
the lake is 40 centimeters and the
standard deviation is 10 centimeter and
we know that the distribution of the
lengths of all fish looks like this then
what's going to happen is we're going to
go out with our fishermen and we're
going to take a random sample of 25 fish
from this population and when we take
that sample of 25 fish we expect
our sample mean to be equal to 40
centimeters the true population mean
although we know that it's not going to
be exactly equal to 40 due to what we
call sampling variability now imagine
all the thousands of random samples we
could have taken each means deviate
slightly because not all samples have a
mean that equals Neptunes true mean of
40 the standard deviation of all
possible means is called the standard
error and we can work that out the
standard error is the standard deviation
of the lengths of individual fish
divided by the square root of the sample
size so our standard error is 2
centimeters and this tells us typically
how far the mean length of a sample of
25 fish deviates from the true mean so
typically we're going to end up with a
sample mean about 2 centimeters away
from the true mean of 40 even without
Neptune we have an idea of how close our
estimate will be to the true value the
sampling distribution of the mean will
be approximately normally distributed
under certain conditions in fact an old
rule of thumb tells us that
approximately 95% of sample means we
could get will say within about 2
standard errors of the true mean and be
between 36 and 44 centimeters that's
most of them not bad for a simple guy
with only a paper cut out of a fisherman
to help them out now check this out the
sampling distribution the distribution
of all the possible means we could get
is approximately normal or belching okay
so we think of the sample mean that
we're going to get as one of many
possible sample means we could get and
we can think of pulling it out of this
distribution here even if this here the
population of all individual fish
lengths is not normally distributed like
in a different lake with an unusual
proportion of large sea creatures this
year the set of all possible means would
still be normal in bell shape as long as
our sample size is large now that's
pretty cool now you may be thinking to
yourself why is Mike spending so much
time pretending
and talking about what is today well the
reason is that in everyday life we
generally don't know the true values for
an entire population and we must use a
sample to try and make statements about
the population we're sampling from
learning health samples behave when we
do know the true values for the entire
population allows us to build the theory
necessary to make statements about a
population when we do not know the true
values for the population in the videos
to follow we will discuss how the
sampling distribution of the mean can be
used to help us make statements about a
population as we learn how to construct
a confidence interval and test claims
using hypothesis test don't forget to
check out the statistics visualizations
that accompany this video you can find a
link in the description thanks for
watching
you
[Music]
I know I know I'll come back home I know
I know together
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