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# Measurements in Physics

### 1. Interpolation

Interpolation is the process to obtain values from an established graph or table that contain given points. these values to find or located between these given points.
Extrapolation is the same process, but used to extend the graph, or find values beyond the given points.

Here is the method for an interpolation: For any functionf :
x -> f(x)
If we know the values of a and b and their results f(a) and f(b), we can get the value of any X between a and b:

F(X) = {[f(a) – f(b)] X + [a f(b) – b f(a)]}/(a - b)

### 2. The Method of Least Squares

The Method of Least Squares cames from Statistics. It’s the most used method to find the best fit line which is :
y = m xi + b.
The purpose of this method, is to minimize the difference between the observed (yi) and predicted values y and to determine the coefficients : m and b.

The squared quantity: (y-yi)2 may be the "least", that is the minimum. We express this by:

dSi[(y-yi)2] /dm=0 (1) and
dSi[(y-yi)2] /db=0 (2)
(i runs from 1 to n; where n is a chosen number)

We have:

and

#### Example:

Here is a prgram in PHP and AJAX that calculates the slope and the y-intercept for given data. Enter the values of xi and their corresponding values yi:

### Least squares

 x y Results:

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