Part 2: Compound Uncertainties.

Where a result is calculated using a formula, every measurement used in the calculation contributes to the uncertainty of the result. This is what we mean by compound uncertainties.

The method used at A Level is actually a fairly crude estimate for the uncertainty but is sufficient for the degree of precision which you’re working to.

The rules are fairly simple:

• If the formula only involves adding or subtracting, you can add the absolute uncertainties in the measurements used. (It’s hard to think of an example where you would actually do this for A Level physics!)
• If the formula involves anything other than adding or subtracting, so multiplying, dividing, indices (powers), you can only add the percentage uncertainties in the measurements.

Example:

R = V/I

Let’s say your measurement for P.d. (V) is 2.0 ± 0.1 v and your measurement for current (I) is 5.0 ± 0.1 A (I’ve kept the numbers fairly simple here to avoid confusion)

Percentage uncertainty in V = 0.1 ÷ 2.0 = 5%

Percentage uncertainty in I = 0.1 ÷ 5.0 = 2%

So R = V/I = 2.0 ÷ 5.0 = 0.40 Ω

And the percentage uncertainty in R = %U(V) + %U(I) = 5 + 2 = 7%

So R = 0.40 Ω ± 7%

If we want the absolute uncertainty in R, we can calculate it as follows:

Absolute uncertainty = ([%-uncertainty] ÷ 100%) x result

Uncertainty in R = (7 ÷ 100) x 0.40 = 0.028

R = 0.40 ± 0.03 Ω (Note that the uncertainty is typically rounded to one s.f. less than the result)

For formulae involving indices (powers) we have to multiply the percentage uncertainty by the index.

Example:

KE = ½ mv² = ½ x m x v x v

So %-uncertainty in KE = %U(m) + 2 x %U(v)