How this instrument works
When you calculate something from two or more measured quantities — a sum, a product, a ratio — the uncertainty in each measurement carries through into the final result, but the rules for combining those uncertainties depend on which arithmetic operation is being performed. This instrument applies the standard propagation-of-uncertainty formulas from the GUM (JCGM 100:2008, 'Guide to the Expression of Uncertainty in Measurement'), covering three operations at once: addition or subtraction, multiplication, and division.
For addition and subtraction, absolute uncertainties combine in quadrature — the square root of the sum of their squares — rather than simply adding directly, because independent random errors partially cancel rather than always stacking in the same direction. For multiplication and division, it's the relative (fractional) uncertainties that combine in quadrature instead, since a percentage error in one factor translates into roughly the same percentage error in the product or quotient, regardless of the two values' absolute sizes.
These formulas assume x and y are independent, uncorrelated measurements with no shared source of error between them — the standard simplifying assumption behind this quadrature approach, and the one used throughout the GUM's basic propagation rules. If two measurements share a systematic error source (say, both were made with the same miscalibrated instrument), their errors are correlated rather than independent, and a fuller treatment would need an additional covariance term this instrument doesn't include. For most everyday lab and engineering uncertainty estimates, though, the independent-error assumption is the standard, widely used starting point.
- Enter your first measured value into x, and its uncertainty into Uncertainty in x (dx).
- Enter your second measured value into y, and its uncertainty into Uncertainty in y (dy).
- Both x and y must be nonzero, since the product and quotient formulas need their relative uncertainties.
- Read Uncertainty of z = x + y or x − y for the combined uncertainty on a sum or difference.
- Read Uncertainty of z = x × y and Uncertainty of z = x / y for the combined uncertainty on a product or quotient.
Worked example — x = 10 ± 0.5, y = 20 ± 1.0
Two measured quantities are x = 10 ± 0.5 and y = 20 ± 1.0. Entering x=10, dx=0.5, y=20, dy=1.0, Uncertainty of z = x + y or x − y reads 1.118034, computed as √(0.5² + 1.0²) = √1.25 — notice this is smaller than simply adding 0.5+1.0=1.5 directly, because the quadrature combination accounts for the two independent errors partially offsetting each other on average.
For the product and quotient, the instrument first finds the relative uncertainty: √((0.5/10)² + (1/20)²) = √(0.0025+0.0025) = √0.005 ≈ 0.070711. Uncertainty of z = x × y reads 14.142136, computed as |10×20| × 0.070711 = 200 × 0.070711. Uncertainty of z = x / y reads 0.035355, computed as |10/20| × 0.070711 = 0.5 × 0.070711 — the same relative uncertainty applied to each operation's own resulting magnitude.
Questions
Why don't absolute uncertainties just add directly for a sum?
Because that would assume the two errors always point in the same, worst-case direction — but independent random errors are just as likely to partially cancel as to stack. Combining them in quadrature (square root of the sum of squares) reflects the more realistic, statistically expected behavior of independent errors, and it always gives a smaller, less pessimistic result than simply adding dx and dy directly, since √(dx²+dy²) is always less than or equal to dx+dy.
Why do multiplication and division use relative uncertainty instead of absolute?
Because a fixed absolute error means very different things depending on the size of the number it's attached to — a ±1 error on a measurement of 1,000,000 is trivial, while the same ±1 error on a measurement of 2 is enormous. Relative (fractional) uncertainty, dx/x, normalizes for that scale difference, which is exactly what's needed when combining errors through multiplication or division, where each factor's percentage error carries through roughly proportionally into the final result's percentage error.
Does this formula assume x and y are independent?
Yes — the quadrature approach used throughout this calculator specifically assumes x and y are independent, uncorrelated measurements with no shared source of error, which is the GUM's standard baseline assumption. If the two measurements share a common error source — the same miscalibrated instrument used for both, for instance — their errors are correlated, and a complete treatment would need an additional covariance term between dx and dy that this instrument doesn't include.
Why do x and y both need to be nonzero?
Because the product and quotient formulas depend on the relative uncertainties dx/x and dy/y, and dividing by zero is undefined — a measured value of exactly zero would make its relative uncertainty term blow up rather than produce a meaningful number. The sum/difference formula, dz=√(dx²+dy²), doesn't actually need x or y at all, only their uncertainties, but this instrument computes all three results together, so both values need to be nonzero for the full set to be valid.
Can I use this for subtraction as well as addition?
Yes — the propagated uncertainty formula for x−y is identical to the one for x+y, since squaring an uncertainty removes any effect of the sign of the operation itself. Only the central value (x+y versus x−y) differs between addition and subtraction; the uncertainty attached to either result is exactly the same quadrature combination, √(dx²+dy²), which is why this calculator reports a single dz(sum) figure covering both cases.