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Statistics 101: Descriptive Statistics, Mean, Median, and Mode - YouTube
Channel: Brandon Foltz
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hello my name is Brandon and welcome to
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the next video in my series on basic
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statistics if you are new to the channel
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it is great to have you welcome if your
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returning viewer it is great to have you
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back if you like the video please
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subscribe give it a thumbs up and share
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it with classmates colleagues or friends
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or anyone else you think might benefit
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from watching so now that we are
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introduced let's go ahead and get
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started
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so this video is about mean median and
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mode now you may have heard of these
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before and on their own mean median and
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mode are not difficult topics however
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what I want to do in this video is three
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fold one I want to visualize them for
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you wherever possible to I want you to
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understand the relationship between the
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three and three I want you to be able to
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develop judgment as to which is
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appropriate given the data you have so
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let's go ahead and get to work now its
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mean median and mode what we are
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beginning to do is measure the center of
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our data now the center of a data set is
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absolutely foundational to everything
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else that you're going to do in
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statistics we use it in hypothesis
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testing we use it in regression and many
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many other things besides that let's go
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ahead and get these fundamental building
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blocks out of the way first a mean
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technically it's called the arithmetic
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mean because there are other types of
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means we can calculate but this is the
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simple one that we are used to and it's
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just the average of all observations in
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the data you've probably calculated the
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average of numbers before it's often
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taught in grade school so it's not
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anything that's all that unfamiliar to
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you probably the next is the medium and
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for students just starting in stats this
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can be a new concept but basically the
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median is the middle observation of a
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data set so what we do is we sort them
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from smallest to largest and if there
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are an odd number of observations it's
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the one literally in the middle after
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sorting if the number of observations is
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an even number the median is the mean or
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the average of the two numbers in the
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middle and we'll see that in a couple
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minutes
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the mode is simply the observation that
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occurs most often in the data or the
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most frequently occurring observation
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now a dataset can have one mode it can
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have multiple modes or could have no
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mode at all it simply depends on the
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data you have so first a mean so here
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are some salary data I created so we
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have 12 observations and 12 salaries so
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to calculate the mean it's very simple
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we add or sum up all the observations
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and then divide by the number of
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observations in the data so in this case
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we would add up all the salaries and
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then we would divide by 12 and the
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notation we denote that by what's called
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x-bar so it's an X with a bar over the
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top so step one we sum our observations
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when we sum all of our salary data we
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have 1 million two hundred ninety one
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thousand four hundred dollars step two
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we count our observations now notice I
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have the word length in parentheses
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there the reason I have length is
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because length is often how we describe
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what data set when we're doing
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programming applications so in
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programming when we have a data set like
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this it's basically what's called an
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array so the salary data here is an
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array with a length of 12 so as students
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get more into coding environments I like
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to describe things in a ways that you're
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going to see in code but if you're not
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into that or not going into that it's
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simply the count of the number of
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observations so in step three we just
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divide the sum by the count or the
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length of our data set so we have 1
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million two hundred ninety one thousand
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four hundred dollars divided by our
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twelve observations and we end up with a
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sample mean of one hundred and seven
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thousand six hundred and $16.67
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so that is the mean or the average
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salary of our twelve observations so
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next we have the median so here is our
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data same numbers but I put it in a
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horizontal format and you'll see why
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here in a second so observation one is
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sixty five thousand six hundred and so
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on and so forth so to find the median
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first we sort our data
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from smallest to largest so we can see
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here that we have a small salary of
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twenty nine thousand five hundred
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dollars all the way up to the maximum
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salary of five hundred thousand dollars
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so they are all in order from smallest
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to largest next we ask ourself is our
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data odd or even length or an odd or
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even count of observations if it is even
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as it is in this case we then just
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divide it in half so you can see here I
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have the first six observations shaded
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in a grey and then I have the last six
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observations shaded in sort of a light
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brown and because this data is an even
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number of count or an even length we
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just divide it in half six on one side
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six on the other so here's where we left
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off now well since it's even we find the
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mean of the two middle values so we can
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see here that observation six and
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observation seven those are the two
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middle values on each side of that
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dividing line so all we do is find the
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average or the mean of those two values
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so we have seventy three thousand six
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hundred plus seventy eight thousand
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eight hundred and then we just divide
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that by two so our median in this case
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is seventy-six thousand two hundred
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dollars so what if our data set has an
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odd number of observations or is an odd
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length well in this case we just simply
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divide our length or our count of
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observations in half and then round up
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to the next number so in this case we
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have eleven observations we divide that
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in half so we get 5.5 and let me just go
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ahead and round up to six and it's the
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sixth value that is our medium so in
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this case it's seventy three thousand
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six hundred if you notice that on either
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side of the sixth value we have five
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below it and five above it and the six
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is situated right in the middle
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now mode is very straightforward it is
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the observation that occurs the most so
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in this case we have two salaries of
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$54,000 and quite simply that is the
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mode now again datasets can have more
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than one mode so if we had two salaries
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that were seventy eight thousand eight
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hundred dollars we would have two modes
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we'd
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of a $54,000 mode in the 70 $8,800 mode
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and some datasets don't have a mode at
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all if all the values in our dataset are
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unique then none of them has more than
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one observation and therefore there is
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no mode and this is actually a warning
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the mean can be influenced by extreme
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observations either low and/or high that
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are different from the rest of the
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observations so here's our data set from
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before notice we have someone in our
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data set that has a salary of $500,000
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now look at the other 11 observations
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they all tend to be around you like
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$75,000 give or take so our sample mean
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was one hundred and seven thousand six
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hundred and sixteen dollars and 67 cents
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however our median was seventy-six
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thousand two hundred dollars that is a
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massive difference a huge difference and
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the question is which measure is more
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accurately representing the center of
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our data set so an analyst could
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accurately represent the center of the
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salary data this mean over here is much
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higher than the median due to the
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presence of someone making five hundred
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thousand dollars when everyone else is
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around seventy-five thousand dollars now
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also a good analyst would double-check
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that value that would double check the
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data to make sure that extreme
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observation of five hundred thousand
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dollars is not a data entry or recording
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error in this case it could be really
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really easy to type in five hundred
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thousand dollars when it's supposed to
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be fifty thousand dollars so if you do
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have an extreme observation and your
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data
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don't just say to yourself oh that's
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just how it is no go back look at your
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data look at the records you have look
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at everything you have at your disposal
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to make sure that data is valid in the
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first place before beginning with your
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analysis so our final concept is called
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the trimmed mean so here's our data set
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again then what we do is we sort them
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smallest to largest like we did for the
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median
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and then we remove the same number of
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observations from each end of our data
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set now sometimes this is expressed as a
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percentage like a five percent trimmed
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mean or a 10 percent trimmed mean so in
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the end it doesn't really matter so long
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as you are removing the same number of
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observations from each end of the data
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set so in this case we will remove the
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smallest of $29,500 and we will remove
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the largest of $500,000 so from there we
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just calculate the mean again so in this
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case we have a sample mean or original
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sample mean of one hundred and seven
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thousand six hundred and $16.67
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our median was 76 thousand two hundred
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dollars and in this case our trimmed
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mean is 76192 median and the trimmed
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mean are they're almost identical so by
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removing the same number of extreme
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values on both ends in this case we get
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a value where the trimmed mean and the
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median are almost identical so what we
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can do is conclude that that's probably
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the best representation of the center of
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our data and our original sample mean is
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heavily influenced by that person who
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had a five hundred thousand dollar
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salary among eleven other individuals
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who had salaries around seventy-five
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thousand dollars now here's a note
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trimmed means are better for a single
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variable what we call univariate so you
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know meaning one variate variable once
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you get other variables involved
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relationships between variables are
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created and things become much more
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complicated now however there are
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methods for trimming or removing
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multivariate extremes but that is beyond
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the scope of this video and I do
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actually talk about that in more
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advanced videos in my playlists so when
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you are doing a univariate analysis what
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I would recommend is reporting all three
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of these values report your original
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sample mean the median and then a
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trimmed mean either 5 percent or 10
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percent or something like that so as
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long as you include the original sample
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mean with the trimmed mean that's
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perfectly fine however never leave out
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the original
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I mean you got to have both so you can
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compare the two and of course have the
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median in there okay so a quick
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conclusion and then we are done so the
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mean median and mode can provide
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information about the center of your
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data the mean is the most often used
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however the median can sometimes be a
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better measure the mean can easily be
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influenced by extreme values so be
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careful extreme values may be a
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recording or data entry error so double
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check the original data look for a large
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difference between the mean and the
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median it could be a warning sign that
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you have an extreme observation that is
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pulling the mean in one direction or
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another and finally a trimmed mean can
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also be calculated and reported which
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chops off the same number or percentage
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of observations on both ends of the data
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and again always report that with the
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original sample mean okay so that wraps
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up this video on mean median and mode
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again on their own very simple concepts
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however we want to visualize them we
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want to understand the relationship
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between the three of them and then
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choose which one is the best depending
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on the data we have and again that's
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very important so thank you very much
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for watching I appreciate your time and
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I will see you again in the next video
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take care
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