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The standard error, Clearly Explained!!! - YouTube
Channel: StatQuest with Josh Starmer
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step quest step quest stack quest
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hello and welcome to stab quest this
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time we're gonna talk about standard
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errors and we're also gonna have a
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bootstrapping bonus we'll start by
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talking about error bars which are very
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closely related to standard errors for
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example you might collect measurements
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from three samples labeled a B and C and
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plot them on a scatter plot just like we
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see here
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you could then calculate the means for
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the three data sets and we Illustrated
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those here with three green horizontal
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bars approximately halfway up the
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clusters of data points after that we
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could calculate the standard deviations
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and add those to the graph and we've
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shown those here with red error bars in
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manuscripts and presentations people
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often don't display the original data
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but instead just show the mean in the
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standard deviation and what's called a
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dynamite plot because each column in the
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plot looks like it's the igniter for a
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stick of dynamite
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there are three common types of error
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bars the first type of standard
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deviations which we just saw and I'm
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sure you're all familiar with these tell
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you how the data are distributed around
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the mean big standard deviations tell
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you that some of the data points were
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pretty far from the mean in most cases
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you want to use standard deviations in
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your graphs since it tells us about your
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data the data points that you collected
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yourself the second type of error bar
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comes from standard errors these tell
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you how the mean is distributed not just
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the data but the means which sounds
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crazy but it'll become clear once I draw
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some pictures the third common type of
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error bar are confidence intervals and
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these are related to standard errors
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confidence intervals will be explained
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more in a future stat quest since this
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stat quest is all about standard errors
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that's what we're going to talk about
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let's start by considering a normal
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distribution in this case we can imagine
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that we weighed a lot of mice and
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plotted the distribution of differences
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from the mean the y-axis is the
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proportion of the mice that we weighed
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in the x-axis is the difference from the
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mean most of the mice had weights close
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to the average a few of the mice weighed
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much less than the average Mouse and a
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few other mice weighed much more than
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the average Mouse usually you can't
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afford to measure the weight of all the
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mice so you just take a sample in this
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example we'll just assume we took five
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measurements from the population rather
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than measuring all the mice since most
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of the mice have weights close to the
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average most of our samples are going to
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be close to zero now just like we always
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do we can calculate the mean and
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standard deviation from our sample in
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this case the mean of our sample is
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minus point two and the standard
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deviation is one point nine to three and
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we can plot the mean and standard
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deviation on our graph as the mean
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plus or minus the standard deviation
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around the mean and for all you stat
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Questers out there here's a rule of
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thumb remember that one standard
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deviation on each side of the mean is
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supposed to cover about 68% of the data
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two standard deviations on each side of
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the mean is supposed to cover about 95
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percent of the data this will come in
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handy later
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the mean is now a lighter color because
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we're going to take additional samples
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and overlay additional means and
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standard deviations on this same graph
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here we've taken another five
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measurements and from those five
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measurements we've calculated the mean
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and the standard deviation and here
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we've plotted that mean plus or minus
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one standard deviation on each side and
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now we take another five measurements
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this is the first sample or one of the
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measurements is relatively extreme
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however that one measurement doesn't
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sway the mean that far from zero that is
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to say the means are relatively close to
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each other compared to the raw data this
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is because for a mean to be far from the
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middle most if not all of the raw data
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points would have to be in a single
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cluster that is far away from the middle
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for example the sample of purple points
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all form a cluster that are far from the
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middle this could happen but very rarely
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what's much more likely is to have a
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sample where most of the points are
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close to zero and only one or two are
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far away so far we've shown that you can
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calculate the standard deviations for
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each sample but now that we have three
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means we can also calculate the standard
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deviation of those means because one
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standard deviation will cover 68% of the
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values and two will cover 95% of the
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values the standard deviation of the
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means won't be as wide as the standard
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deviations of the data here we've
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plotted the mean of the means plus or
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minus one standard deviation of the
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means notice that this standard
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deviation is much smaller than the
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standard deviations we got from the
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individual samples the standard
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deviation of the mean is called the
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standard error of the mean or more
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simply the standard error the standard
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error gives us a sense of how much
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variation we can expect in our means if
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we took a bunch of independent five
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measurement samples so to review this is
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how we calculate this
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an error of the mean first you take a
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bunch of samples each with the same
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number of measurements or in in this
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case in equals five the second step is
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to calculate the mean for each sample
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here we calculated the mean and standard
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deviation for each sample but for the
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standard error all we need to do is
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calculate the mean once we've calculated
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the means for each sample we can
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calculate the standard deviation of the
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means in this case the standard error
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equals zero point eight six here we
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notice that the standard error is much
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less than the standard deviations
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because the means aren't as widely
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dispersed as the raw data we've shown
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how to calculate the standard error of
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the mean but there are other standard
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errors for example we can also take the
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standard deviation of the standard
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deviations this is called the standard
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error of the standard deviations which I
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guess is to avoid a tongue-twister it
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tells us how the standard deviations of
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multiple samples are dispersed you can
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calculate the standard deviation of any
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statistic for example the median the
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mode percentiles are anything anything
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that you can calculate for multiple
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samples you just calculate the standard
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deviation and then you have the standard
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error of that so if we calculated many
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mediums we could calculate the standard
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deviation of those mediums and we have
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the standard error of those mediums to
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summarize everything we've talked about
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so far know that the standard error is
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just the standard deviation of multiple
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means taken from the same population so
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if there's a population and we can take
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a bunch of different samples from it all
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we have to do to get the standard error
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is to calculate the standard deviation
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of the means of each sample well at this
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point you might be wondering if we can
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calculate standard errors without
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spending a lot of time and money on
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doing the same experiment a bunch of
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times the good news is the answer is yes
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in rare cases there's a formula you can
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use to estimate
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the standard error for the mean is 1 the
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formula for that is very simple it's
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just the standard deviation divided by
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the square root of the sample size
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however there aren't many other cases
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the good news again is that we can use
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something called bootstrapping for
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everything else
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every time we don't have a simple
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formula we can bootstrap it the nice
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thing about bootstrapping is it's very
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simple conceptually and it's easy to
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make a computer do this work here's a
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bootstrapping example just like before
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we have an experiment where we took 5
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measurements as an aside usually for
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bootstrapping it's good to have 10 or
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more measurements in a single experiment
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now we bootstrap our data with the
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following steps first we pick a random
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measurement from the sample that we just
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took this random measurement isn't a new
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measurement that we haven't taken before
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it's not a new number that we haven't
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seen it's part of the sample that we
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already have now we just write that
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value down in this case it's one point
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four three in step three we just go back
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to step one and pick a new random
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measurement and write that value down
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and we do that five times our second
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measurement is minus one point three
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eight the third measurement is minus
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three point one one our fourth
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measurement is one point four three
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we've already picked that measurement
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before but that's okay when you're
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bootstrapping you just pick five
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measurements from your sample and it
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doesn't matter if you've picked the same
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one before our last measurement is minus
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zero point one zero step four in
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bootstrapping is to calculate the mean
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median mode or whatever the statistic it
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is we're interested in understanding the
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standard error of and we calculate that
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with our sample in this case we're
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interested in the standard error of the
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mean so all we do is calculate the mean
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from our new bootstrap sample the fifth
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step is to go all the way back to the
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beginning step one and repeat that until
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you have a lot of means or medians or
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whatever you're interested in
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calculating the standard error of the
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sixth and final step in the
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bootstrapping procedure is to simply
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calculate the standard deviation of all
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the means that we generated in steps one
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through five that's all there is to it
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in this case we calculated the standard
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error of the mean and we've plotted it
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as a black line in the graph so if
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there's no fancy formula to help us
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calculate the standard error we can do
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it ourselves from scratch we can just
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use bootstrapping and get the job done
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and that's it in this static quest we
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learned that the standard error is a
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measure of how we might expect the means
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from many different samples to vary from
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one sample to another we also learned
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that if we don't have a fancy formula
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for calculating the standard error we
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can do it ourselves using bootstrapping
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okay so tune in next time and we'll talk
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about how to use bootstrapping to
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calculate confidence intervals and
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that's when things get really cool
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