Inferential Statistics - Motivation Video Lecture | What is Data Analytics? - Data & Analytics

hello and welcome to our lecture and
inferential statistics in this
particular lecture we're going to be
talking more giving you more of a
motivation for why we need inferential
statistics and we'll be talking a little
bit about what kinds of problems you can
solve using this technique using these
techniques and and you know where you
would apply it and when you would apply
in the subsequent lectures we will talk
a little bit about how you would apply
it and also why you do some of the math
that you do why you do some of the
computation so understanding it a little
bit more on the nuts and bolts level is
something that will follow but in this
lecture we are going to focus on why you
need inferential statistics in the first
place and we're going to do this through
the lens of something called hypothesis
testing
so hypothesis testing is a very widely
used and accepted tool for a lot of data
analysis and if you understand
inferential statistics through
hypothesis testing many other concepts
in inferential statistics just fall in
place so things like confidence
intervals and so on which you might have
heard become fairly easy to understand
and process so having said that let me
jump into the subject the idea behind
inferential statistics is to make some
inference about the population from the
sample just to jog your memories I think
we've spoken about population and sample
a couple of times but this is different
from what you would have done with
descriptive statistics right with
descriptive statistics you don't care
about population or sample in the end of
the day you had some data set and for
simplicity sake assume that was a sample
that you got from population and this
sample you were very content in
descriptive statistics with describing
that data set with describing that
sample in case you have a sample but
here the kinds of problems that you're
more interested with inferential
statistics are ones where you have a
population
and you're getting us just a sample from
that but from this sample I don't want
to just say something about the sample I
don't want to talk about the mean of the
sample I don't want to talk about the
variance of the sample I don't want to
talk about centrality dispersion my goal
is to say something about the population
I only have data which is the sample I
don't have the data associated with the
population but with this sample can I
say something about the population so
that's the core idea and just to kind of
jog some of your memories the idea
behind population and sample is fairly
simple
we've looked at it with a couple of
examples but you can think of the
population in one of two ways
the first and the more obvious way is
that there exists this really large data
set associated with a phenomena so let's
say the phenomena was the height of all
boys who are in 10th standard
in public schools so that's so that
could be a very large data center let's
see let's say that was for India right
so it's a very large data set there are
a lot of children who are in the 10th
standard
who are in public schools across India
and you could think of that as the
population and you want to say something
about that population and you might not
have the data so you go take a sample so
you select five schools or use or ten
schools or you select 200 students
through a census process randomly and
take that as a sample so you've taken a
sample of you know some subset of
students from the population that's one
way of thinking of population sample and
another way could be that the population
itself is more of a theoretical abstract
concept so you could not have an actual
data set but it could be something like
I've created this new machine and this
new machine is going to start making
certain products and let's say you're
very interested in the product dimension
so let's say the diameter of a product
this machine makes so you put in raw
material into this machine the machine
spits out some finished product and this
finished product should have a diameter
I mean has some diameter here you don't
because it's a new machine you might not
actually have
a population that is unruly large data
set that exists somewhere the population
here is the concept that if this machine
were to create infinite such products
without any change in time or space the
dimensions of these products would be
the population and ultimately you might
say hey let's just for the first time
run this machine and make ten products
and these ten products that I physically
have and I've measured and so on is the
sample so here is a case where I still
want to use a sample to say something
about the machine in general not just
the ten products the machine is churned
out but the concept of a population here
is not an actual finite large data set
in my hands it's more of a concept okay
now having revisited population and
sample here let's again say the
statement which is that inferential
statistics your makes you want to say
something about the population from the
sample okay so as I said the major aim
of this lecture is to motivate you to
see why inferential statistics is
important so I felt that the best way to
for instance do that might be to give
you some examples so that's what we're
going to do and I'll start with some
simple examples I've broken them down
into one sample and two-sample examples
to sample examples and pretty soon that
will become clear what that distinction
is so let's start with the first example
which is a one sample example on the
upper left of your screen so the idea
here and just so you know we have in
this part and the idea here is that
let's say we were interested in noting
the average phosphate levels in our
blood and I don't have a medical
background or anything so so don't look
at the medical aspect of this example
but let's say that your doctor or
doctors in general or you know public
health advocates say that the average
phosphate levels in blood should be less
than four point eight milligrams per
deciliter
so again
in respect of the unit so the whole idea
is that this number which you can get if
you go measure your blood should be on
average less than 4.8 the key here is an
understanding that this should be less
than 4.8 on average so that means the
doctors or the public health advocates
understand that sometimes it could be
greater than 4.8 and that perhaps in
this particular case is not a cause for
alarm
again don't focus on the medical aspect
I don't know if it is or not but this is
the situation I'm creating right so but
the important thing the doctors have
told you is that on average it should be
less than 4.8 okay so let's say you say
and you know this obviously variation so
it really depends on what you ate that
day it depends on what time of the day
you take the measurement it depends on
what instrument you use to take the
measurement it depends on how much water
you had so let's say there are a lot of
factors that you don't seek to control
and that's the whole idea behind this
but you want to take a set of
measurements and you want to take a set
of measurements and answer the question
as to whether the average phosphate
levels in your blood in general right
not just on the sample that you've taken
you don't mean the sample could be
anything but you care about in general
is my average phosphate level in blood
less than four point eight so why are
you so the question might arise you know
why you're distinguishing between what
the sample says and what reality is and
that's going to become clearer in a
second so so let's first go about trying
to answer this question right so the
first thing is if you took a set of
measurements and let's say you got
consistently very low values so let's
say you got two point four two point
five two point one two point seven two
point three two point nine fill up four
more numbers in the two points okay then
I guess you really don't need a
statistician right you can look at that
sample that you've got you can say look
I'm fairly certain that even if I went
on taking more and more
samples or that if I woke up another day
through all this data or took another
sample or if I took infinite many
samples in either case it looks like my
average is going to be less than 4.8
that's fine
you know intuitively that seems obvious
similarly the flipside suppose you were
to take this data and you consistently
got five point five six point one seven
point seven six point nine and so on
where every single data point is
significantly greater than the four
point eight mark and approximately in
that same region meaning it's not widely
moving around right so that's also an
intuition that you might have that if
one second it's six point five in the
next second it's two point one you know
it's wildly moving on but here I'm
giving you examples where six point one
six point seven nine seven point one
seven point three seven point four so
it's consistently significantly greater
again you don't need a statistician
somewhere your intuition you just say
look I mean based off of the sample I'm
willing to bet that my average phosphate
levels are less than four point eight mg
/ DL but then it gets a little tricky
what happens if you had you know numbers
you know some of them below some of them
above right so some of them are less
than four point eight some of them are
greater than four point eight and in
some sense what do you do that
and the instinct of the intuition there
sometimes is to say well let me take an
average of the sample right if that
average is greater than four point eight
then perhaps I should conclude that my
average is greater than four point and
this is a small problem with that kind
of an approach right
I mean assume that you got the readings
like s such as five point one 4.8 4.9
4.7 4.7 5.3 four point and you got some
set of readings and you took the average
and that average was four point eight
five okay so you think you know what the
sample showed that it's greater
and you conclude for instance that the
average phosphate level in my blood in
general is greater than 4.8 based off of
the sample that I just saw the problem
with that could be that maybe if you
just took 2 more data points let's say
you took 2 more data points you increase
your sample by 2 more and you got a 4.6
and a 4.7 and all of a sudden because of
these new data points your average you
know slides just below 4.8 right
something about doing this process where
you just take the average of the sample
and make a conclusion does not seem
correct for this reason right and you
know another way of looking at it is
also that you looked at this notion of
sampling distributions right so we know
that let's let's let's take a look at
the same graph that we looked at at the
end of last class right so you had this
thing which we described as the original
distribution okay and let us say for now
let us just say for now that truly your
this this distribution by the way
represents the amount of phosphate in mg
per DL that you will see in your blood
if you do a test at any given point of
time so you get numbers from this
distribution so sometimes you get a 5.1
that's what that's the dot there
sometimes you get a 4.4 that's the other
dot there but in the end of the day it
looks like on average right given how I
have drawn it on average it's only 4.7
mg per DL which is less than the 4.8
mark that we were interested in now you
go and take a sample right this is the
same example as the last time so you
took the sample and you and you got some
data point so you took six data points
and and this is what you got now if I
were to just take the sample average and
I'm just eyeballing it here if I just
took the average of these numbers right
I would say that average falls somewhere
here
wouldn't you say so this might be the
average of these numbers 'm or maybe it
will fall right on
maybe it'll fall right on this dot
actually so let's let's say this is
somewhere out here is average and the
truth is this sample average is greater
than the four point seven and if you
were to just go by the sample average we
might have concluded that the amount of
mg per DL is greater than four point
eight or whatever right but here's the
good part you've now had a class in
statistics that told you that the
average that you get from the sample is
not always going to fall on this four
point seven right as your number of
samples tends to infinity yes it will
converge to this point right but if
you've got a finite number of samples
and that's not right so it is not going
to always be exactly on four point seven
so what is it going to be what it is
going to be is another distribution
right so if you had n samples and this
distribution will change with more
samples if there are many many many many
samples then literally if as as the
number n tends to infinity this
distribution will pretty much lie flat
on this line but if if not you're still
getting the sample mean the mean that
you calculate from the sample is
literally a random variable that you're
getting from this distribution so it's
literally like you just pick a random
point from this distribution right so
you could get a point anywhere here you
you probably won't get a point here
right because it doesn't the probability
of getting this point from this
distribution is very low is almost zero
so you will be getting points from this
distribution and as a result just
because this number is greater than four
point seven should not make you conclude
that this mean which is what you are
trying to conclude yet
like to say something about this line
you're trying to say something about
this line which is the mean of the
population and because you get a sample
mean which is nothing but a number from
a distribution should not make you
conclude that therefore it is greater
than 4.8 okay so so that's why you need
to do something more you need to do
something more complex than just blindly
taking the sample average okay so again
we're going to talk later about how you
do it but and and what and when you do
but right now I'm just trying to
motivate for you why you need something
else so take another example and in this
example we we take a problem of
proportions so let's say the health
department or some dentists related body
says only 5% of the toothpastes of any
given brand can be out of specification
right so outer specification might mean
that you have some ratings on amount of
fluoride toothpaste can have so let's
say you're allowed to need thousand
parts per million of fluoride and you
there are other chemical limits but the
Health Department understands that not
every toothpaste can match exactly their
eye the ideal requirement so let's say
they set out limits on the chemicals and
they say look if you're a toothpaste
manufacturer only I'm going to only
allow 5% of your toothpastes to you know
be out of specification 95% of your
toothpastes that I see in the market
need to fall need to fall within my
guidelines okay same problem comes up
again you can take a sample of ten
twenty two pieces right and it could it
could very well be that truly this
toothpaste brand is involved in a
chemical process or a manufacturing
process that creates on average only
four
and only 4% of the toothpaste that this
company makes okay are actually out of
specification but it's perfectly
possible that you went and took ten
twists from the market and your luck
right seven of those ten are out of
specification that's perfectly possible
right it might not be the most
probabilistic thing but it's perfectly
possible it's possible that this
toothpaste company is involved in a
manufacturing process in a chemical
process that creates stewed pastes and
on average four percent of all the
toothpastes they make so when I say
toothpaste think of it as a toothpaste
tube on average four percent of all the
tubes they make have a chemical
composition that is not acceptable okay
which is fine because the Health
Department says you can't go more than
five percent and this toothpaste company
is raising its hand and saying hey you
know have a it's only four percent but I
now go and randomly sample v to paste
s-- ten two pastes and I find that out
of the five toothpastes that I randomly
sampled three of them are defective all
of a sudden I'm saying three out of five
that's sixty percent you say only five
percent is allowed and I find sixty
percent and so is does that mean the
company is not creating two pastes less
than five percent rate which are in
conformance less than five percent rate
probably not again you need a little bit
more nuanced thinking and you need a
little bit more statistics to actually
answer this question based off of the
sample you can't just take the sample
average another example the third
example that we have on the one sample
case is imagine that you you you are an
insurance company okay and you find that
there is this particular mechanic shop
like a garage which does repairs and
because most people are required to have
insurance once the the the garage kind
of writes out and in
and people who file the insurance can
name attached this mechanic's invoice
and stell the company to reimburse them
for these for this rectification that's
down to the car and let's say this is a
new garage and you know the insurance
company is suspecting that these guys
are cheating that they are set up as a
place to not do any real work but just
write really high invoices so that the
insurance company so they're involved in
some kind of a fraud so one thing that
the insurance company can do is saying
okay I'm going to look at the next 10 20
or whatever repairs so let's say up to
this point this garage is not made a
single you know claim but it's just
being set up but the inside word is that
they're trying to that they're trying to
cheat the system
so once the garage gets set up this
insurance company is ready the first 10
claims or 20 claims that this garage
files or 30 claims they take those
claims and they see how that compares to
the national average in terms of average
claims right again the problem is just
because this sample is greater than the
national average can we conclude yes
these guys are cheaters and just because
the sample average is less than the
national average can we conclude these
guys are not cheaters and the answer to
both those questions is no right in some
cases like when I was talking to you
about the phosphate it might be brutally
obvious right where every single data
point is so high or so low that you're
like I don't need a statistician to tell
me that the answer to this question but
when it's not that case you need you
need a little more you need inferential
statistics to answer that question okay
okay so let's look at single sample
cases by that we essentially mean that
there was one data set and you were
essentially comparing that data set to
some benchmark number that you had in
your head let's now move to to sample
situations okay the example here is
let's say that I am running a foundry
and some guy comes in consultant comes
in and says ah you know if you change
the temperature a little bit of the
molten metal that you're pouring in or
whatever let's say let's say he says
change the temperature down by a degree
or two and I assure you that an average
number of defects that you see in your
cast a metal cast will decrease so like
a mechanical engineering application
you're like okay maybe the consultant
knows what he's talking about but how do
I test it how do I test it and the
answer to the question is it's you can
do the following which is before you do
that before you go change things you can
measure the average number of defects in
your casts and you do that for 10 20
data points okay and then you do what
the consultant says which is to change
the temperature and then you collect
another 10 20 data points now let's go
back to the question if the average of
the sample the first sample so now you
have sample a sample B sample a
corresponds to before the temperature
was change sample B corresponds to after
the temperature was changed now
obviously I mean in all likelihood
there's going to be some average to
sample and there'll be some average to
sample B in all likelihood these two are
not going to be the same one is going to
be higher or lower than the other right
just like in the single sample case if
they're so dramatically different these
samples and when I say dramatically
different I mean dramatically different
with respect to some amount of
variability there is in the two samples
as well then you don't need a
statistician to tell you it's like so
obvious the changing the temperature
dramatically reduced the number of
defects right but in many cases you
don't know and in those cases it's not
obvious to say the average of sample a
was different from the average of sample
B right I mean go back to I'm going to
erase this but it is the red ink but go
back to this example let's say that
let's say that this is the original
distribution of the number of defects
okay so this is the original
distribution and let's say this is three
defects and this is like nine defects
now that's too high let's say this is
five defects and this is one defect so
this original distribution is what I
care about that goes from here to here
these are the number of defects you will
see in a cast okay
now obviously if I take a sample of six
or so I get a random point let's say
this is the random point I get okay and
it is the other one okay done it is so I
basically I I took the original
distribution I took a sample of six
casts and when I take a sample of six
casts like we discussed the the average
is not going to always be exactly three
it's going to be some number that falls
into this distribution this the
distribution of sample means and I've
drawn a random number that I got out
here from that distribution okay so good
now let's say that this consultant who
was telling you to go increase or
decrease the temperature was completely
wrong let's say he had no clue what he
was saying he was just lying but
fortunately or unfortunately whatever he
said does not make a difference I mean
he lied in that you know he said it's
going to improve the process it did not
improve the process luckily he didn't
make the process worse okay now you go
take and because the process has not
changed the original distribution has
not changed now after the temperature is
change the number of defects you're
going to receive are also exactly in
conformance to this distribution it's in
conformance to this outside distribution
the original distribution right so that
hasn't changed
essentially this mean hasn't changed
okay this mean hasn't changed now you go
take a sample of six you get another
sample average and the sample average
again is going to belong to this
distribution and let's say this sample
average was this value okay
so this is the new sample average so
this is new okay so that's new now you
can't say hey this new sample average is
higher than the old sample average
therefore the population mean is
different it's not I mean I just gave
you an example where the population mean
truly did not change but how you could
have seen two different sample averages
and concluded that one is greater than
the other so this is why again you need
more than looking at the sample average
of a and sample average of B and saying
hey one is higher than the other so we
should believe that what the consultant
said was correct or the other way around
you know if the if you conclude that
what the consultant said is definitely
made things worse
that's also not correct it's perfectly
possible that there was truly a change
but because of luck you know you saw
things the other way around another
example you can think of and this the
next example is one where I wanted to
emphasize variation rather than just the
mean let's say you have two different
manufacturing processes and and and you
want to compare their variance of the
finished product in each batch so you
have manufacturing process a
manufacturing process B and they make
batches of you know finished material or
finished product and within each batch
there's some amount of variance of each
product that's some amount of variance
right variance is the inherent variation
between one part to the next and let's
say I care about that variance let's say
I tune in a particular batch I don't
want one product to look very different
from the next product I don't care about
the mean but I want them to all be
consistent again you would use the same
concept you would take the first batch
which is made from machine a calculate
the variance of it calculate the
variance of that sample take the second
batch made from machine B calculate the
variance of that sample and again the
overarching concept exists you can't
just say sample of sample variance of
machine a is lower than sample variance
of machine B therefore I conclude that
machine a is better than machine B it's
perfectly possible that
Shiina is actually worse than machinery
but because you ultimately only have a
sample that machine a got lucky
ultimately it goes back to this idea
that you see in this distribution that
sometimes you get a number on this side
of the distribution sometimes you get a
number on this side of the distribution
the more data points you take the
overall variance of this distribution
reduces which is why if you had infinite
number of points right none of this
problems exists but you don't and if
you're dealing with that then the right
thing to do is to use inferential
statistics to look closely at the data
another example that is often quoted is
things like our a 10 standard girl's
taller than ten standard boys in India
for instance we all know that in terms
of average heights men have larger
average heights in women but we've also
heard that girls start growing taller
earlier so I don't know a stand standard
a break-even point at least some
statistics textbooks seem to think so so
you can have a simple question like the
population here is ten standard girls in
India ten seven boys in India you're
ultimately taking a sample and based off
of that sample what can you say about
the population and you know sometimes
the story is obvious from the sample
itself sometimes you need inferential
statistics to come in and tell you
whether you can say something concrete
or perhaps you cannot say anything
concrete and that's also something
inferential statistics will tell you but
ultimately it's not as simple as just
saying the average of sample a is
greater than average of sample B and
therefore I'm going to conclude one way
or the other okay so let's go to what it
is that we will be trying to do with
inferential statistics I'm just going to
go through the overarching principle and
keep this in mind we'll revisit the
slide a couple of times but I think the
ultimate test in some sense of you
understanding what it is and how it is
and why it is will really become
you're once we do the actual math with
each test and and ultimately this these
will come in the form of tests I mean
some of you might have heard of these
tests like t-test Z test chi-square test
F test ANOVA and so on and and we'll go
through each of these tests but here is
the overarching principle with respect
to hypothesis testing it's to have us
have a null hypothesis and an alternate
hypothesis so for instance in the
fluoride case the null hypothesis could
very well be let me erase this the null
hypothesis could very well be that you
have less than 4.8 mg per DL so the null
hypothesis is that the actual phosphate
levels the average phosphate levels in
blood for person X is less than or equal
to four point eight and so the alternate
hypothesis would be that it's greater
than four point eight right the
important thing is the null hypothesis
in the alternate hypothesis in some
sense together should be mutually
exclusive right which means that if it's
greater than four point eight it can't
be less than or equal to four point
eight right and vice versa and and you
know collectively exhaustive they should
essentially cover the entire space that
you're interested in talking about
so I mean meaning that the average
phosphate level is either less than or
equal to four point eight or greater
than equal to four point eight it can't
be neither
so it's collectively exhaustive okay so
then you what we will be doing and
you've not been taught this yet but you
will be doing some basic calculations or
arithmetic on the data to create a
single number called the test statistic
so you don't know what that is yet but
what you'll do is you would so the
reason I'm explaining this to you is to
give you an idea that you are going to
be working with the sample so it's not
like magic in that you you're not going
to say something about the population
without dealing with the sample so you
will be doing some math you know and and
it some of that might involve taking
things like the sample mean the sample
standard deviation but you'll be doing
some math on that and when you finish
that map you'll be getting something
called the test statistic some of you
might have heard of these test
statistics then we call the Z statistic
or the T statistic and so on the crux of
null hypothesis the crux of hypothesis
testing is that if we assume the null
hypothesis to be true and make some
assumptions about distributions of
various variables and those that we
won't go into that much but if we assume
the null hypothesis to be true then
technically the test statistic should be
no different then drawing a random
number from a specific probability
distribution right
so in some sense right what we're saying
is if the null hypothesis is true if the
if the true the true mean right is equal
to 4.8 4.8 this time because the null
hypothesis is true the null hypothesis
was is it less than or equal to four
point eight so here the null hypothesis
is true so let's take the extreme case
the true mean is four point eight and
let's say there are some assumptions
like that this distribution is normal
maybe that these were the samples that
you are taking are independent if we
said there's some set of assumptions
that you take if all that of it is true
then we want to do certain things such
that you will get another new
distribution you'll be doing some some
math with these data points see these
data points that you got from the sample
you'll be doing some math with those
data points you will do that math such
that the test statistic would be no
different than a single random number
that you draw from a very specific
probability distribution if that is the
case then you test the probability that
the test statistic you calculated
belongs to this theoretical distribution
right you basically say hey it looks to
me like if the null hypothesis is true
then whatever I've calculated out here
with the sample should be like drawing a
random number from this distribution so
let me calculate the actual
statistic and see how likely it is that
this number came from that probability
distribution and that is what we call as
a p-value now once you have done this
process you might say look if the null
hypothesis is true then the test
statistic that I should have calculated
should have come from this distribution
but look at the number that I got in my
hand it's so unlikely that I could have
gotten this test statistic from this
distribution therefore the null
hypothesis perhaps was not true and I'm
going to reject the null hypothesis or
in some cases the test statistic that
you get looks like it does belong to
this distribution the specific
distribution and therefore you can only
you cannot really see anything it's like
you just have no grounds for rejecting
the null hypothesis you can just say I
fail to reject the null hypothesis the
important thing for you you might want
to take a look at this procedure a
couple of times for the important thing
that you might want to digest from this
is that the p-value itself is associated
with the probability of seeing this data
if the null hypothesis were true and not
really the probability of seeing of this
hypothesis being true given the data so
it's probability of data given
hypothesis not probability of hypothesis
given data so I hope that kind of
clarifies inferential statistics for you
and in the next class we will look at
some specific tests and go over how you
actually do these tests and even go
deeper and talk about why we do some of
the mathematical operations that we do
thank you