## Uncertainties in Calculations

If the mass of a block is measured asand the volume asthen 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 quantityisthen 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 thenso the density is

Power relationships are a special case of this. IfthenIf a cube is measured to bealong each side then the % uncertainty in the volume isand the absolute uncertainty is 7.5% ofso the volume is

If two quantities are added, then the absolute uncertainty in the sum is the sum of the absolute uncertainties, so ifandthen

Sometimes however we can only estimate the range of a quantity. Ifthen is between 55 and 65 degrees.is betweenandThe uncertainty is the maximum value ofandi.e. 0.5 soto 2 dp.