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You use the first formula you gave when you have (entirely) uncorrelated errors where the standard variances (the squares of the standard deviances) add. Gaussian distributions of errors are usually assumed. You might use the second formula if your errors are perfectly correlated, but even then only as a worst-case measure (if you know the correlation, you ...

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I think that the easiest way to understand this is in the formula for addition. If you consider your quantity $C$ which depends on $A$ and $B$ such that $C=A+B$, then the formula $\Delta C = \Delta A + \Delta B$ overestimates the error values. You can visualize this as a rectangle with $A$ on the x-axis and $B$ on the y-axis with the area enclosed being ...

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These new formulae are a form of "adding in quadrature." In my understanding it's a way of combining errors whilst acknowledging that the 'worst case' of adding two errors is not likely to happen, if those errors are independent. For example, if you measure the area of a rectangle by measuring its length with a ruler, and its width with some kind of ...

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