The Arithmetic Mean and Its Standard Error

Although the true value of a qurmtity can never be determined, the error of the measurement can be reduced if the experiment is repeated many times. 

Consider an experiment that has been repeated N times, where iV is a large 

The result of the measurement is reported as the arithmetic average defined by 

This equation is the same as Eq. 2.31. As N increases, a better estimate of the true value of n is obtained—i.e., the error of the measurement becomes smaller. The true value of n, which is also called the true mean, can only be obtained with an infinite number of measurements. Since it is impossible to perform an infinite number of trials, n is always calculated from Eq. 2.71. 

The error of n depends on the way the individual measurements are distributed around n—i.e., it depends on the width of the Gaussian of Fig. 2.10. As the width becomes smaller, the error gets smaller, and therefore the measurement is better. The standard error of n is defined in terms of the standard deviation of the distribution. Using Eq. 2.34 and setting /(x,) = 1 /TV, the standard deviation of the distribution becomes 

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