WEBVTT

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- [Narrator] So we have
nine students who recently

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graduated from a small school
that has a class size of nine,

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and they wanna figure out
what is the central tendency

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for salaries one year after graduation?

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And they also wanna have a
sense of the spread around

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that central tendency one
year after graduation.

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So they all agree to put in
their salaries into a computer,

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and so these are their salaries.

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They're measured in thousands.

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So one makes 35,000, 50,000,
50,000, 50,000, 56,000,

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two make 60,000, one makes
75,000, and one makes 250,000.

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So she's doing very well for herself,

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and the computer it spits
out a bunch of parameters

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based on this data here.

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So it spits out two typical
measures of central tendency.

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The mean is roughly 76.2.

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The computer would calculate
it by adding up all of these

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numbers, these nine numbers,
and then dividing by nine,

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and the median is 56, and median
is quite easy to calculate.

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You just order the numbers and you take

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the middle number here which is 56.

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Now what I want you to
do is pause this video

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and think about for this data set,

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for this population of
salaries, which measure,

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which measure of central
tendency is a better measure?

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All right, so let's think
about this a little bit.

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I'm gonna plot it on a line here.

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I'm gonna plot my data
so we get a better sense

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and we just don't see them,
so we just don't see things

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as numbers, but we see
where those numbers sit

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relative to each other.

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So let's say this is zero.

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Let's say this is, let's see,
one, two, three, four, five.

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So this would be 250, this
is 50, 100, 150, 200, 200,

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and let's see.

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Let's say if this is 50
than this would be roughly

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40 right here, and I just wanna get rough.

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So this would be about 60,
70, 80, 90, close enough.

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I'm, I could draw this
a little bit neater,

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but, 60, 70, 80, 90.

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Actually, let me just clean
this up a little bit more too.

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This one right over here would be

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a little bit closer to this one.

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Let me just put it right around here.

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So that's 40, and then
this would be 30, 20, 10.

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Okay, that's pretty good.

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So let's plot this data.

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So, one student makes 35,000,
so that is right over there.

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Two make 50,000, or three make 50,000,

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so one, two, and three.

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I'll put it like that.

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One makes 56,000 which would
put them right over here.

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One makes 60,000, or
actually, two make 60,000,

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so it's like that.

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One makes 75,000, so
that's 60, 70, 75,000.

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So it's gonna be right around there,

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and then one makes 250,000.

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So one's salary is all
the way around there,

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and then when we
calculate the mean as 76.2

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as our measure of central tendency,

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76.2 is right over there.

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So is this a good measure
of central tendency?

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Well to me it doesn't feel that good,

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because our measure of central
tendency is higher than all

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of the data points except for
one, and the reason is is that

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you have this one that the,
that our, our data is skewed

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significantly by this
data point at $250,000.

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It is so far from the
rest of the distribution

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from the rest of the data
that it has skewed the mean,

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and this is something
that you see in general.

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If you have data that is skewed,
and especially things like

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salary data where someone might
make, most people are making

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50, 60, $70,000, but someone
might make two million dollars,

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and so that will skew the
average or skew the mean I should

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say, when you add them all
up and divide by the number

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of data points you have.

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In this case, especially when
you have data points that

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would skew the mean,
median is much more robust.

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The median at 56 sits right
over here, which seems to be

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much more indicative for central tendency.

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And think about it.

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Even if you made this instead of 250,000

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if you made this 250,000
thousand, which would be 250

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million dollars, which is
a ginormous amount of money

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to make, it wouldn't, it would
skew the mean incredibly,

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but it actually would not
even change the median,

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because the median, it doesn't matter

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how high this number gets.

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This could be a trillion dollars.

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This could be a quadrillion dollars.

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The median is going to stay the same.

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So the median is much more robust

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if you have a skewed data set.

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Mean makes a little bit more
sense if you have a symmetric

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data set or if you have things
that are, you know, where,

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where things are roughly
above and below the mean,

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or things aren't skewed
incredibly in one direction,

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especially by a handful of data

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points like we have right over here.

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So in this example, the median is a much

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better measure of central tendency.

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And so what about spread?

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Well you might say, well,
Sal you already told us

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that the mean is not so good

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and the standard deviation
is based on the mean.

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You take each of these data
points, find their distance

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from the mean, square that
number, add up those squared

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distances, divide by the
number of data points if we're

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taking the population standard
deviation, and then you,

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and then you, you take the
square root of the whole thing.

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And so since this is based on
the mean, which isn't a good

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measure of central tendency
in this situation, and this,

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this is also going to skew
that standard deviation.

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This is going to be, this is a lot larger

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than if you look at the, the actual,

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if you wanted an indication of the spread.

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Yes, you have this one data
point that's way far away

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from either the mean or
the median depending on how

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you wanna think about it, but
most of the data points seem

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much closer, and so for that situation,

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not only are we using the median,

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but the interquartile range
is once again more robust.

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How do we calculate the
interquartile range?

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Well, you take the median
and then you take the bottom

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group of numbers and
calculate the median of those.

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So that's 50 right over here
and then you take the top

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group of numbers, the
upper group of numbers,

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and the median there is
60 and 75, it's 67.5.

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If this looks unfamiliar
we have many videos

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on interquartile range and calculating

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standard deviation and median and mean.

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This is just a little bit of a review,

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and then the difference
between these two is 17.5,

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and notice, this distance
between these two, this 17.5,

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this isn't going to change,

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even if this is 250 billion dollars.

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So once again, it is both of
these measures are more robust

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when you have a skewed data set.

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So the big take away here is
mean and standard deviation,

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they're not bad if you have
a roughly symmetric data set,

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if you don't have any
significant outliers,

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things that really skew the data set,

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mean and standard deviation
can be quite solid.

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But if you're looking at
something that could get really

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skewed by a handful of data
points median might be,

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median and interquartile range,
median for central tendency,

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interquartile range for spread
around that central tendency,

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and that's why you'll see when
people talk about salaries

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they'll often talk about
median, because you can have

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some skewed salaries,
especially on the up side.

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When we talk about things
like home prices you'll see

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median often measured
more typically than mean,

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because home prices in a
neighborhood, a lot of,

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or in a city, a lot of the
houses might be in the 200,000,

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$300,000 range, but maybe
there's one ginormous mansion

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that is 100 million dollars,
and if you calculated mean

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that would skew and give a
false impression of the average

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or the central tendency
of prices in that city.