If an unknown physical quantity x is measured n times,the measurements x1, x2,...,xn

Chapter 4, Problem 67

(choose chapter or problem)

If an unknown physical quantity \(x\) is measured \(n\) times, the measurements \(x_{1}, x_{2}, \ldots, x_{n}\) often vary because of uncontrollable factors such as temperature, atmospheric pressure, and so forth. Thus, a scientist is often faced with the problem of using \(n\) different observed measurements to obtain an estimate \(\bar{x}\) of an unknown quantity \(x\). One method for making such an estimate is based on the least squares principle, which states that the estimate \(\bar{x}\)

should be chosen to minimize

                                                     \(s=\left(x_{1}-\bar{x}\right)^{2}+\left(x_{2}-\bar{x}\right)^{2}+\cdots+\left(x_{n}-\bar{x}\right)^{2}\)

which is the sum of the squares of the deviations between the estimate \(\bar{x}\) and the measured values. Show that the estimate resulting from the least squares principle is

                                                                  \(\bar{x}=\frac{1}{n}\left(x_{1}+x_{2}+\ldots, x_{n}\right.)\)

that is, \(\bar{x}\) is the arithmetic average of the observed values.

Equation Transcription:

n

Text Transcription:

x

n

x_1, x_2, … , x_n

n

bar{x}

x

bar{x}

s = (x_{1} - bar{x})^2 + (x_{2} - bar{x})^{2} + cdot cdot cdot (x_{n} - bar{x})^2

bar{x)

bar{x} = frac{1}{n}(x_1 + x_2 + … , x_n)

bar{x}