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Root mean square deviation (RMSD) - YouTube
Channel: Khan Academy
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so we are interested in studying the
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relationship between the amount that
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folks study for a test and their score
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on a test where the score is between
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zero and six and so what we're going to
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do is go look at the people who took the
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tests we're going to plot for each
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person the amount that they studied and
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their score so for example this data
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point is someone who studied an hour and
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they got a one on the test and then
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we're going to fit a regression line and
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this blue regression line is the actual
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regression line for these four data
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points and here is the equation for that
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regression line now there's a couple of
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things to keep in mind normally when
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you're doing this type of analysis you
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do it with far more than four data
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points the reason why i kept this to
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four is because we're actually going to
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calculate how good a fit this regression
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line is by hand and typically you would
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not do it by hand we have computers for
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that
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now the way that we're going to measure
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how good a fit this regression line is
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to the data has several names
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one name is the standard deviation of
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the residuals another name is the root
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mean square deviation sometimes
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abbreviated rmsd sometimes it's called
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root mean square error so what we're
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going to do is is for every point we're
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going to calculate the residual and then
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we're going to square it and then we're
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going to add up the sum of those squared
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residuals
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so we're going to take the sum
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of the residuals
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residuals
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squared
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and then we're going to divide that by
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the number of data points we have minus
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two and we can talk in future videos or
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a more advanced statistics class of why
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you divide by two
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but it's related to the idea that what
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we're calculating here is a statistic
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and we're trying to estimate a true
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parameter as best as possible and n
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minus 2 actually does the trick for us
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but to calculate the root mean squared
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deviation we would then take a square
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root of this
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and some of you might recognize strong
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parallels between this and how we
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calculated sample standard deviation
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early in our statistics career and i
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encourage you to think about it but
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let's actually calculate it by hand as i
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mentioned earlier in this video to see
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how things actually play out so to do
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that i'm going to give ourselves a
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little table here so let's say that is
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our x value in that column
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let's make this our y value
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let's make this
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y hat which is going to be equal to 2.5
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x minus 2
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and then let's make this the residual
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squared which is going to be
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our y value minus our y hat value our
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actual minus our estimate for that given
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x
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squared and then we're going to sum them
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all up divide by n minus 2 and take the
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square root
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so first let's do this data point so
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that's the point 1 comma 1
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1 comma 1. now what is the estimate from
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our regression line well for that x
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value when x is equal to 1 it's going to
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be 2.5 times 1 minus 2. so it's going to
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be 2.5 times 1 minus 2 which is equal to
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0.5
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and so our residual squared is going to
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be 1 minus 0.5 1 minus 0.5
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squared which is equal to that's going
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to be 0.5 squared which is going to be
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0.25
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all right let's do the next data point
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we have this one right over here it is 2
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comma 2.
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now our estimate from the regression
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line when x is equal to when x equals 2
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is going to be equal to 2.5
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times our x value times 2
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minus 2 which is going to be equal to
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3.
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and so our residual squared is going to
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be 2
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minus 3
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to
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minus 3 squared which is negative 1
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squared which is going to be equal to 1.
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then we can go to this point so that's
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the point 2 comma 3 2 comma 3.
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now our estimate from our regression
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line is going to be 2.5
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times our x value times 2
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minus 2
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which is going to be equal to
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3.
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and so our residual here is going to be
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zero and you can see that that point
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sits on the regression line so it's
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going to be 3 minus 3 3 minus 3 squared
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which is equal to 0. and then last but
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not least we have this point right over
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here when x is 3
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our y value this person studied 3 hours
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and they got a 6 on the test
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so y is equal to 6 and so our estimate
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from the regression line you could say
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what you would have expected to get
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based on that regression line is 2.5
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times our x value times 3 minus 2 is
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equal to
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5.5
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and so our residual squared is 6 minus
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5.5 squared minus 5.5
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squared so it's 0.5 squared which is
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0.25
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so now the next step let me take the sum
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of all of these squared residuals
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so this is let me just write it this way
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actually let me just do it like this
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so the sum
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of the residuals
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residuals squared is equal to
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if i just sum all of this up it's going
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to be point it's going to be 1.5
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1.5 and then if i divide that by n minus
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2 so if i divide by n minus 2 that's
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going to be equal to i have four data
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points so i'm going to divide by 4 minus
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2 so i'm going to divide by 2
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and then i'm going to want to take the
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square root of that then we want to take
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the square root of that and so this is
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going to get us 1.5 over 2 is the same
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thing as 3 4. so it's the square root of
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three-fourths or the square root of
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three
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over two and you could use a calculator
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to figure what that is to figure out
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what that is as a decimal but this gives
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us a sense of how good a fit this
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regression line is the closer this is to
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zero the better the fit of the
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regression line the further away from
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zero the worse fit and what would be the
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units for the root mean square deviation
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well it would be in terms of whatever
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your units are for your y-axis in this
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case it would be the score on the test
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and that's one of the other values of
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this calculation of taking the square
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root of the sum of the squares of the
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residuals dividing by n minus 2.
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