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Standard deviation of residuals or Root-mean-square error (RMSD) - YouTube
Channel: Khan Academy
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what we're going to do in this video is
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calculate a typical measure of how well
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the actual data points agree with a
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model in this case a linear model and
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there's several names for it we could
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consider this to be the standard
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deviation of the residuals and that's
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essentially what we're going to
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calculate you could also call it the
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root mean square error and you'll see
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why it's called this because this really
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describes how we calculate it
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so we're going to do is look at the
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residuals for each of these points and
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then we're going to find the standard
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deviation of them so just as a bit of
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review the ith residual
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is going to be equal to
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the
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y-value for a given x minus the
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predicted y value for a given x now when
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i say y-hat right over here this just
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says what would the linear regression
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predict for a given x and this is the
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actual y for given x so for example and
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we've done this in other videos this is
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all review the residual here when x is
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equal to 1
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we have y is equal to 1 but what was
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predicted by the model is 2.5 times 1
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minus 2 which is 0.5 so 1
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minus 0.5 so this residual here
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this residual is equal to 1 minus 0.5
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which is equal to 0.5 and it's a
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positive 0.5 and if the actual point is
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above the model you are going to have a
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positive residual
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now the residual
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over here
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you also have the actual point being
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higher than the model
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so this is also going to be a positive
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residual and once again when x is equal
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to 3 the actual y is 6
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the predicted y is 2.5 times 3 which is
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7.5 minus 2 which is 5.5 so you have 6
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minus 5.5 so here i'll write residual is
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equal to 6 minus 5.5
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which is equal to
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0.5 so once again you have a positive
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residual
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now for
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this point that sits right on the model
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the actual is the predicted the act when
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x is two the actual is three and what
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was predicted by the model is three so
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the residual here is equal to the actual
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is three and the predicted is three so
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it's equal to zero
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and then last but not least you have
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this data point
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where the residual is going to be the
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actual when x is equal to 2
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is 2
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minus the predicted well when x is equal
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to 2 you have 2.5 times 2 which is equal
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to
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5 minus 2 is equal to 3. so 2 minus 3 is
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equal to negative 1.
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and so when your actual is below your
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regression line you're going to have a
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negative residual so this is going to be
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negative 1 right over there
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now we can calculate the standard
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deviation of the residuals we're going
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to take this first residual which is 0.5
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and we're going to square it we're going
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to add it to the second residual right
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over here i'll use this blue with this
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teal color that's zero
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i'm gonna square that
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then we have
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this third residual which is negative 1.
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so plus
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negative 1 squared and then finally we
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have that fourth residual which is 0.5
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squared 0.5
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squared
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so once again we took each of the
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residuals the which you could view as
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the distance between the points and what
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the model would predict we are squaring
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them when you take a typical standard
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deviation you're taking the distance
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between a point and the mean here we're
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taking the distance between a point and
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what the model would have predicted but
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we're squaring each of those residuals
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and adding them all up together
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and just like we do with the sample
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standard deviation we are now going to
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divide by
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one less than the number of residuals we
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just squared and added so we have four
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residuals we're gonna divide by four
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minus one
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which is equal to of course
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three
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you could view this part as a mean of
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the squared errors and now we're going
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to take the square root of it
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so let's see this is going to be equal
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to
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square root of
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this is 0.25 0.25
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this is just 0.
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this is going to be positive 1
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and then this 0.5 squared is going to be
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0.25
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0.25
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all of that over 3.
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now this numerator is going to be 1.5
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over
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3.
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so this is going to be equal to 1.5 is
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exactly half of three so we could say
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this is equal to the square root
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of one half this is one over the square
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root of two one
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divided by
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the square root of two
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which gets us 2.
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so if we round to the nearest
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thousandths it's roughly
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0.707
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so approximately 0.707
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and if you wanted to visualize that one
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standard deviation of the residuals
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below the line would look like this
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and one standard deviation above the
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line for any given x value would go one
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standard deviation of the residuals
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above it
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it would look something like that
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and this is obviously just a hand-drawn
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approximation but you do see that this
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does seem to be roughly indicative of
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the typical residual
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now it's worth noting sometimes people
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will say it's the average residual and
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it depends how you think about the word
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average because we are squaring the
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residuals so outliers things that are
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really far from the line when you square
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it are going to have disproportionate
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impact here if you didn't want to have
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that behavior we could have done
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something like find the mean of the
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absolute residuals that actually in some
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ways would have been a simpler one but
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this is a standard way of people trying
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to figure out how much a model disagrees
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with the actual data and so you can
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imagine the lower this number is
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the better the fit of the model
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