## Uncertainties in Calculations

If the mass of a block is measured as and the volume as then we can work out the density: The largest possible value of the density is The smallest possible value of the density is We can round these to give If the uncertainty in a quantity is then the fractional uncertainty is and the percentage uncertainty is The % uncertainty in the density above is Rules for Multiplication, Division or Powers of Uncertain Values

When two or more quantities are multiplied or divided we take the resulting % error as the sum of the %percentage uncertainties.

For the example above, and The total % uncertainty in the density is then so the density is Power relationships are a special case of this. If then If a cube is measured to be along each side then the % uncertainty in the volume is and the absolute uncertainty is 7.5% of so the volume is If two quantities are added, then the absolute uncertainty in the sum is the sum of the absolute uncertainties, so if and then Sometimes however we can only estimate the range of a quantity. If then is between 55 and 65 degrees. is between and The uncertainty is the maximum value of and i.e. 0.5 so to 2 dp. 