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# Residuals

Given a set of *N* data points
(*x*_{0}, y_{0}),
(*x*_{1}, y_{1}),
...,
(*x*_{N}, y_{N}),
and a function *F* defined on the interval
[*x*_{0}, x_{N}],
we can get an idea of how "far away" *F*
is from the data set by computing the
**residuals**
r_{i} = y_{i} - F(x_{i}).

### Residuals (lengths of the blue line segments) between a function *F*
and a set of data points.

The residuals indicate the vertical distance from our function at
*x*_{i} to our data point. If residuals are
large, then our function is not very close to the data points.
To get a global view of how close *F* is to the set of **all**
data points, we might try to add up the residuals. Unfortunately, however,
some residuals that are positive might cancel with others that are negative, and so
we might find that the sum of the residuals is nearly zero, even though our
function does not pass close to any of the data points.

A more useful estimate is to compute the
**sum of the squares of the residuals.**
The square of a residual is always positive, so if the
sum of the squares of the residuals is a small number, then
our function passes close to many of the data points. This is the method used
by the well-known statistical method of linear regression, also
known as the method of "least-squares fit".

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