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The Central Limit Theorem, Clearly Explained!!! - YouTube
Channel: StatQuest with Josh Starmer
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even if you're not normal
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the average
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is normal
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hello I'm Josh starmer and welcome to
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stat Quest today we're going to talk
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about the central limit theorem and it's
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going to be clearly explained
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note for this stat quest to make any
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sense at all you should be familiar with
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the normal distribution if not check out
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the normal distribution clearly
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explained
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it would also be helpful if you were
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familiar with the concept of sampling
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from a statistical distribution if not
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check out sampling from a statistical
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distribution clearly explained
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the central limit theorem is the basis
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for a lot of statistics and the good
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news is that it's a pretty simple
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concept
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in this stat Quest I'll explain what the
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central limit theorem is and why it's
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important
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like most things in statistics I think
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the central limit theorem is easiest to
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understand if we look at some examples
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so let's start with a uniform
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distribution
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this one goes from zero to one
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it's called The Uniform distribution
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because there is an equal probability of
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selecting values between 0 and 1.
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the probabilities are all equal and thus
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are uniform
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we can collect 20 random samples from
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this uniform distribution
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and then calculate the mean of the
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samples
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and on the right we can draw a histogram
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of the mean value
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since we only have one mean value the
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histogram isn't very interesting
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but after we collect 10 more samples and
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collect 10 more means
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the histogram starts to look a little
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more interesting
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here's the histogram after collecting 20
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samples and calculating 20 means
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30 mains
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40 means
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50 means
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60 means
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70 means 80 means 90 means and 100 means
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after adding 100 means to the histogram
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it's pretty easy to see that these means
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are normally distributed
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however to make it easy to see that the
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means are normally distributed we can
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overlay a normal distribution
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you might have noticed that in the last
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two slides I put means are normally
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distributed in bold I did this because
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this is what the central limit theorem
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is all about
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even though these means were calculated
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using data from a uniform distribution
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the means themselves are not uniformly
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distributed instead the means are
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normally distributed
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bam
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here's another example
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this time we'll start with an
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exponential distribution
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just like before we can collect 20
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random samples from this exponential
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distribution
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and just like before we can calculate
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the mean of the 20 samples
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and lastly we can draw a histogram of
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that mean over here on the right
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after we collect 10 samples and
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Calculate 10 means the histogram starts
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to look a little more interesting
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here's the histogram after 20 means
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30 means 40 means 50 means 60 means 70
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means 80 means 90 means and 100 means
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after adding 100 means to the histogram
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we can see that they are normally
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distributed
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even though these means were calculated
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using data from an exponential
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distribution
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the means themselves are not
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exponentially distributed instead the
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means are normally distributed
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[Music]
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so far we have seen that the means
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calculated from samples taken from a
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uniform distribution
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are normally distributed
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and means calculated from samples taken
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from an exponential distribution
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are also normally distributed
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well it turns out that it doesn't matter
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what distribution you start with
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if you collect samples from those
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distributions
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the means will be normally distributed
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yes there's a little asterisk here that
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means there's some fine print that will
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come later for now just know it's really
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fine print and not worth spending too
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much time worrying about
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double bam
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cool but what are the practical
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implications of knowing that the means
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are normally distributed
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when we do an experiment we don't always
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know what distribution our data comes
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from
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to this the central limit theorem says
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who cares
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the sample means will be normally
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distributed
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because we know that the sample means
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are normally distributed
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we don't need to worry too much about
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the distribution that the samples came
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from
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we can use the means normal distribution
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to make confidence intervals
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do t-tests where we ask if there's a
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difference between the means from two
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samples
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and Anova where we ask if there is a
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difference among the means from three or
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more samples
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in pretty much any statistical test that
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uses the sample mean
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triple bam
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note out there in the wild some folks
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say that in order for the central limit
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theorem to be true the sample size must
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be at least 30.
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this is just a rule of thumb and
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generally considered safe however as you
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can see in the examples here where I use
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a sample size of 20 the rule was meant
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to be broken
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here's the fine print
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in order for the central limit theorem
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to work at all you have to be able to
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calculate a mean from your sample
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off the top of my head I can think of
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only one distribution the Koshi
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distribution that doesn't have a sample
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mean and after doing biostatistics for
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20 years I've never come across it in
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practice
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that said if you know of distributions
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that don't have means put them in the
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comments below and tell us what they're
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used for I'm curious about how common
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this occurs
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hooray we've made it to the end of
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another exciting stat Quest if you like
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this stat Quest and want to see more of
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them please subscribe and if you want to
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support stack Quest well consider buying
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one or two of my original songs alright
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until next time Quest on
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