Weighted mean of three observations of differing precision
Three crews, three standard errors, one distance. Weights as the inverse of the variance, the weighted mean, the standard deviation of unit weight, and what it means when that comes out well above one.
Apply· about 22 minutes by hand· 5 steps
Given
The same distance was measured by three parties using different equipment and different procedures. Each reported a value and its own standard error.
Party 1: 1842.36 ft, σ₁ = ±0.03 ft
Party 2: 1842.29 ft, σ₂ = ±0.05 ft
Party 3: 1842.41 ft, σ₃ = ±0.02 ft
The three determinations are independent. All were reduced to the same horizontal plane and the same units, US survey feet.
No party's result is to be discarded; the task is to combine them.
Required
The weight of each observation and the weighted mean.
The standard deviation of unit weight, σ₀.
The standard deviation of the weighted mean, and a judgement on whether the reported standard errors are credible.
Work it through yourself before reading on - the solution below shows every step, so there is no way to skim it without giving the answer away.
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