Engineering Cost Estimates Jul 2021

Estimates -- aka Adding in Quadrature

Uncertainty Analysis

To illustrate a simple but often ignored concept – what is the estimated total cost of the following?

Item Cost \pm Cost
alpha 100,000 5,000
beta 75,000 7,500
gamma 50,000 6,000
.. .. ..
total 225,000 what is the total plus-minus range?

I say the answer is:

Item Cost \pm Cost
.. .. ..
total cost 225,000 11,000 (10,828)

Notice that the plus-minus range for the total cost is less than the sum of the plus-minus ranges. This will always be true when adding in quadrature.

    \begin{align*} 5,000 + 7,500 + 6,000 = 18,500\\ 11,000 < 18,500 \end{align*}

Uncertainty Analysis Reference

In accordance with the rules of quadrature, our uncertainty is:

    \begin{align*} \delta Q = \sqrt{(\delta alpha)^2 + (\delta beta)^2 + (\delta gamma)^2}\\ \delta Q = \sqrt{(5000)^2 + (7500)^2 + (6000)^2}\\ \delta Q = 10,828 \end{align*}

Of course, no more than 2 significant digits are warranted here; therefore 10,828 \longrightarrow 11,000.

Caveat

The quantities alpha, beta, and gamma must have uncertainties which are uncorrelated and random.

Example Calculation

If say n same-cost items were considered, with a constant p percent plus-minus cost:

    \begin{align*} \delta Q = \sqrt{n{p}^2}\\ \delta Q = p\sqrt{n}\\ \frac{\delta Q}{Q} = \frac{1}{n}p\sqrt{n}\\ \frac{\delta Q}{Q} = \frac{p}{\sqrt{n}}\\ \end{align*}

Therefore, if 20 items of the same cost (but still independent variables) were considered, with a constant 10 percent (p = 0.10) plus-minus cost:

    \begin{align*} \frac{\delta Q}{Q} = 0.022 \end{align*}

Here the plus-minus range for the total cost, 2.2 percent, is much less than 10 percent.

If 20 items of a random cost (and still independent variables) were considered, with a constant 10 percent (p = 0.10) plus-minus cost, we might calculate 2.5 percent instead of 2.2 percent.

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