Something like find the mean of the absolute residuals, that actually in some ways If you didn't want to have that behavior we could have done Really far from the line, when you square it are going to have disproportionate impact here. Think about the word average because we are squaring the residuals, so outliers, things that are It's the average residual and it depends how you Now, it's worth noting, sometimes people will say Seem to be roughly indicative of the typical residual. And this is obviously justĪ hand-drawn approximation but you do see that this does Go one standard deviation of the residuals above it, it would look something like that. The residuals below the line would look like this, and one standard deviation above the line for any given X value would And if you wanted to visualize that, one standard deviation of Going to be equal to, 1.5 is exactly half of three, so we could say this is equal to the square root of one half, this one over the square root of two, one divided by the square root of two which gets us to, so if we round to the nearest thousandths, it's roughly 0.707. Now, this numerator is going to be 1.5 over three, so this is To be equal to square root of this is 0.25, 0.25, this is just zero, this is going to be positive one, and then this 0.5 squared is going to be 0.25, 0.25, all of that over three. You could view this part asĪ mean of the squared errors and now we're gonna take We just squared and added, so we have four residuals, we're going to divide by four minus one which is equal to of course three. Sample standard deviation, we are now going to divide by one less than the number of residuals Here we're taking theĭistance between a point and what the model would have predicted but we're squaring each of those residuals and adding them all up together, and just like we do with the Standard deviation, you're taking the distanceīetween a point and the mean. The model would predict, we are squaring them, when you take a typical Have that fourth residual which is 0.5 squared, 0.5 squared, so once again, we tookĮach of the residuals, which you could view as the distance between the points and what
Residual which is negative one, so plus negative one squared and then finally, we This blue or this teal color, that's zero, gonna square that. To the second residual right over here, I'll use We're going to take this first residual which is 0.5, and we're going to square it, we're going to add it And so, when your actual isīelow your regression line, you're going to have a negative residual, so this is going to be So, two minus three isĮqual to negative one. Well, when X is equal to two, you have 2.5 times two, which is equal to five Going to be the actual, when X is equal to two is two, minus the predicted. Sits right on the model, the actual is the predicted, when X is two, the actual is three and what was predictedīy the model is three, so the residual here isĮqual to the actual is three and the predicted is three, so it's equal to zero and then last but not least, you have this data point where the residual is So, once again you haveĪ positive residual. Is equal to six minus 5.5 which is equal to 0.5. So, you have six minus 5.5, so here I'll write residual Now, the residual over here you also have the actual pointīeing higher than the model, so this is also going toīe a positive residual and once again, when X is equal to three, the actual Y is six, the predicted Y is 2.5 times three, which is 7.5 minus two which is 5.5. 5, so this residual here, this residual is equal to one minus 0.5 which is equal to 0.5 and it's a positive 0.5 and if the actual point is above the model you're going to have a positive residual. So, for example, and we'veĭone this in other videos, this is all review, the residual here when X is equal to one, we have Y is equal to one but what was predicted by the model is 2.5 times one minus two which is. The linear regression predict for a given X? And this is the actual Y for a given X. Now, when I say Y hat right over here, this just says what would So, just as a bit of review, the ith residual is going toīe equal to the ith Y value for a given X minus the predicted Y value for a given X. So, what we're going to do is look at the residualsįor each of these points and then we're going to find The root-mean-square error and you'll see why it's called this because this really describes We could consider this toīe the standard deviation of the residuals and that's essentially what This case, a linear model and there's several names for it.
Going to do in this video is calculate a typical measure of how well the actual data points agree with a model, in