Chapter 7: Weighted Averages

The theoretical foundation rests on the Principle of Maximum Likelihood, which seeks to maximize the probability of observing the measured values by minimizing a chi-squared expression representing the weighted sum of squared deviations from the true value. This optimization approach, known as the method of least squares, yields the mathematical framework for weighted averaging. The chapter introduces three essential formulas: the weight assigned to each measurement, calculated as the reciprocal of the uncertainty squared, which ensures that more precise measurements exert proportionally greater influence on the final result; the weighted average itself, computed by multiplying each measurement by its corresponding weight, summing these products, and dividing by the total weight; and the uncertainty of the weighted average, derived through error propagation as the reciprocal square root of the sum of all weights. A practical example involving three resistance measurements demonstrates how this weighting mechanism works in practice: a measurement with three times the uncertainty of the others receives nine times less weight, becoming essentially negligible in determining the final result. This chapter equips students with a rigorous, mathematically justified method for combining imprecise data while accounting for varying degrees of measurement reliability.