even if your ukulele is out of tune
don't sweat it
you can still watch stat quest all day
long hooray hooray stat quest hello I'm
Josh stormer and welcome to stat quest
today we're going to talk about some
statistics fundamentals specifically
we're gonna talk about population
parameters note this stat quest assumes
you already know about histograms
statistical distributions and
specifically the normal distribution if
not check out the quests the links are
in the description below now imagine we
counted the number of mRNA transcripts
from gene X in five different liver
cells
note if mRNA transcripts in liver cells
doesn't mean anything to you instead
imagine we counted the number of green
apples in five different grocery stores
or you could imagine counting green
t-shirts in five different clothing
stores
or imagine whatever you want to measure
in five different units since I work in
a genetics lab I'll stick with mRNA
transcripts and liver cells this green
dot represents a liver cell that had
three mRNA transcripts for gene X and
this green dot represents a liver cell
that had 13 mRNA transcripts 19
transcripts 24 and 29 now if we had a
lot of time and money on our hands we
could count the number of mRNA
transcripts for gene X in every single
liver cell
however for the sake of this example
you'll just have to imagine 240 billion
green dots on this line representing the
240 billion cells in a human liver
because I don't have time to draw them
all wah wah
now we can draw a histogram of the
measurements
the histogram tells us that most of the
cells had between 20 and 30 mRNA
transcripts
and relatively few cells had less than
10 transcripts and relatively few cells
had more than 30 transcripts
we can use the histogram to calculate
probabilities and statistics for example
if we wanted to know the probability of
observing a liver cell with 30 or more
mRNA transcripts for gene X then we
would figure out how many liver cells
had 30 or more mRNA transcripts for gene
X and divided by the total number of
liver cells
in this case there are 38 billion cells
with 30 or more transcripts and we
divide that by 240 billion
do the math and the probability of
observing a cell with 30 or more
transcripts is 0.16 BAM note this
histogram made from mRNA counts in all
240 billion liver cells corresponds to a
normal distribution with mean equals 20
and standard deviation equals 10 the
mean 20 is right in the middle and the
standard deviation 10 corresponds to how
wide the curve is around the mean in
other words the standard deviation tells
us how the data are spread around the
mean
just like with the histogram we can use
the distribution to calculate
probabilities and statistics for example
if we wanted to know the probability of
observing a liver cell with 30 or more
mRNA transcripts for gene X then we
would calculate the area under the curve
for all values equal to or greater than
30 and divide by the total area under
the curve
in this case the area under the curve
greater than 30 equals 0.16 and the
total area is 1
now we do the math
and that tells us that the probability
of observing a cell with 30 or more
transcripts is 0.16 since we got the
same value with the histogram it means
the normal curve is a good approximation
of the real data BAM note if we had
counted green apples in a single chain
of grocery stores then the distribution
would represent the number of green
apples in every single store in that
chain and that means we could use the
distribution to calculate statistics
about apples in that grocery store chain
BAM oh no it's a terminology alert watch
out
because this histogram represents every
liver cell or all the grocery stores in
a specific chain a statistician would
say that it represents a population
thus the mean and standard deviation of
the normal curve which represents the
population are called population
parameters and we call the mean the
population mean and we call the standard
deviation the population standard
deviation or the population SD for short
note if the histogram had looked like
this then we could fit an exponential
distribution to the data
the shape of an exponential distribution
is determined by the rate which in this
case equals 0.1 and even though the
exponential distribution looks different
from the normal distribution it would
still represent the population of liver
cells and that makes the rate the
population rate and we could use the
exponential distribution to calculate
probabilities and statistics just like
when we had a normal distribution
alternatively if the shape of the
histogram had looked like this
then we would fit a gamma distribution
to the data and since the shape of the
gamma distribution is determined by two
parameters shape and rate then shape and
rate or population parameters
note the concepts that we discuss in the
rest of the stack quest apply to almost
every statistical distribution however
we'll just focus on the normal
distribution in these examples so
returning to the original normal curve
since we rarely if ever have enough time
and money to measure every single thing
in a population we almost always
estimate the population parameters using
a relatively small sample in this case
we have measurements from only five of
the 240 billion cells so we will use
these five measurements to estimate the
population parameters
the reason why we want to know the
population parameters is to ensure that
the results drawn from our experiment
are reproducible
in other words if someone else measured
gene X in five different liver cells
then they would get five different
measurements
however the new measurements will come
from the same population and insights
derived from the population like the
probability of observing more than 30
mRNA transcripts in a single cell will
apply to both experiments and future
experiments so instead of just
describing the five measurements that we
made we want to estimate the population
parameters and use those as the basis
for the results double bam
note if you're coming from a
machine-learning background it might be
helpful to think of these five
measurements as the training data set in
the curve that represents the population
is what we want to predict with our
machine learning method BAM
going back to our five measurements I
can tell you that the estimated
population mean is seventeen point six
and the estimated population standard
deviation is ten point one note we'll
talk about how to estimate the
population mean and standard deviation
in a follow-up stat quest for now just
know that it's not hard now
when we repeat the experiment the
estimated population mean is nineteen
point two and the estimated population
standard deviation is twelve point seven
thus each time we do the experiment we
get different estimates of the
population parameters
and both sets of estimates are different
from the true population values now if
you've been paying attention what I just
said should be a little disturbing
earlier we said the whole idea behind
population parameters was to give us
reproducible results so how does getting
different estimates each time give us
reproducible results
to answer this question let's start by
assuming we only have two measurements
when we just have these two measurements
the estimated population mean equals 11
and the estimated population standard
deviation equals eleven point three
compared to the actual values the
estimated mean 11 is way off from the
true population mean 20 and the
estimated standard deviation eleven
point three is a little larger than the
actual standard deviation 10 however if
we had three measurements then the
estimated mean equals fifteen point
three
which is closer to the true value than
before
and the estimated standard deviation
equals 11 which is slightly closer to
the true value than before
however like we saw earlier when we have
all five measurements then the estimated
mean equals seventeen point six which is
even closer to the true value and the
estimated standard deviation equals ten
point one which is even closer to the
true value than before and if we had ten
measurements then our estimates would be
even better that means that the more
data that we have the more confidence we
can have in the accuracy of the
estimates one of the main goals and
statistics is quantifying how much
confidence we can have in population
estimates specifically statisticians
often calculate p-values and confidence
intervals to quantify the confidence in
estimated parameters
and like we just saw generally speaking
the more data the more confidence we
have in the estimates going back to the
to replicate experiments even though
these experiments resulted in different
estimates for the population mean and
standard deviation we can use statistics
to quantify our confidence in how
different they are in this case a
p-value or alternatively a confidence
interval would tell us that while the
estimates are different they are not
significantly different and that means
the results generated from the first
experiment should not be significantly
different from the results generated
from the second experiment and that
means we should be able to replicate the
results
triple bam in summary a population
represents every single liver cell or
every grocery store in a chain of
grocery stores or whatever unit it is
you are measuring something awesome and
the parameters that determine how a
distribution fits the population data
are called population parameters we
rarely if ever have population data so
we always estimate population parameters
along with that we also calculate how
much confidence we should have in those
estimates
generally speaking the more data we have
the more confidence we have in the
estimates
by estimating the population parameters
and quantifying our confidence in them
we can generate results that are
reproducible in future experiments
PS if you'd like to learn more about how
we can quantify our confidence in
estimated population parameters
check out the quest on confidence
intervals the link is in the description
below hooray we've made it to the end of
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