### 7. CURVE FITTING

In many experiments, the functional relation between two or more
variables describing a physical process, *y* =
*f*(*x*_{1}, *x*_{2}, ...), is
investigated by measuring the value of *y* for various of
*x*_{1}, *x*_{2}, . . .It
is then desired to find the parameters of a theoretical curve which
best describe these points. For example, to determine the lifetime of
a certain radioactive source, measurements of the count rates,
*N*_{1}, *N*_{2}, . . ., *N*_{n},
at various times, *t*_{1}, *t*_{2}, . . . ,
*t*_{1}, could be made and the
data fitted to the expression

(69)

Since the count rate is subject to statistical fluctuations, the
values *N*_{i} will have uncertainties
_{i} =
*N*_{i} and will not
all lie along
a smooth curve. What then is the best curve or equivalently, the best
values for and
*N*_{0} and how do we determine them? The method most
useful for this is the method of *least squares*.