The measurements we make during an experiment are usually not the final result; they are used to calculate the final result.

When considering how an uncertainty in a measurement will affect the final result, it will be helpful to express uncertainties in a slightly different way. Remember, what really matters is that the uncertainty in a given measurement should be much smaller than the measurement itself. For example, if you write, "I measured the time to a precision of 0·01s", it sounds good: unless you then inform your reader that the time measured was 0·02s ! The uncertainty is 50% of the measured time so, in reality, the measurement is useless. We will define the quantity Relative Uncertainty as follows

(to emphasise the difference, we use the term "absolute uncertainty" where previously we simply said "uncertainty").

We will now see how to answer the question in the title.

It is always possible, in simple situations, to find the effect on the final result by straightforward calculations but the following rules can help to reduce the number of calculations needed in more complicated situations.

A few simple examples might help to illustrate the use of these rules. (Rule 2 has, in fact, already been used in the section "Using a Ruler" on page 3.)

Suppose that you want to find the average thickness of a page of a book. We might find that 100 pages of the book have a total thickness of 9mm. If this measurement is made using an instrument having a precision of 0·1mm, we can write

thickness of 100 pages, T = 9·0mm ± 0·1mm

and, the average thickness of one page, t, is obviously given by

therefore our result can be stated as t = 9/100mm ± 0·1/100mm

To find a change in temperature, DT, we find an initial temperature, T₁, a final temperature, T₂, and then use DT = T₂ - T₁

If T₁ is found to be 20°C and if T₂ is found to be 40°C then DT= 20°C.

But if the temperatures were measured to a precision of ±1°C then we must remember that

The smallest difference between the two temperatures is therefore (39 - 21) = 18°C and the biggest difference between them is (41 - 19) = 22°C

This means that

In other words

To measure a surface area, S, we measure two dimensions, say, x and y, and then use

Using a ruler marked in mm, we measure x = 50mm ± 1mm and y = 80mm ± 1mm

This means that the area could be anywhere between

that is

To state our answer we now choose the number half-way between these two extremes and for the indeterminacy we take half of the difference between them.

Therefore, we have

so ........ S = 4000mm² ± 130mm²

(well…actually 4001mm² but the "1" is irrelevant when the uncertainty is 130mm²).

Now, let’s look at the relative uncertainties.

Relative uncertainty in x is 1/50 or 0·02mm.

Relative uncertainty in y is 1/80 or 0·0125mm. So, if the theory is correct, the relative uncertainty in the final result should be (0·02 + 0·0125) = 0·0325.

Check

Relative uncertainty in final result for S is 130/4000 = 0·0325

To find the volume of a sphere, we first find its radius, r, (usually by measuring its diameter).

We then use the formula: V = (4/3)pr³

Suppose that the diameter of a sphere is measured (using an instrument having a precision of ±0×1mm) and found to be 50mm.

so, ......... r = 25·0mm ± 0·05mm

This means that V could be between

so ...... 65058mm³ < V < 65843mm³

As in the previous example we now state the final result as

V = 65451mm3 ± 393mm3

Check

Relative uncertainty in r is 0×05/25 = 0·002

Relative uncertainty in V is 393/65451 = 0·006 so, again the theory is verified

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