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How does an uncertainty in a measurement affect the FINAL result?
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.)
Example to illustrate rule 1
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
Example to illustrate rule 2
To find a change in temperature, T, we find an initial temperature, T1, a final temperature, T2, and then use T = T2 - T1
If T1 is found to be 20°C and if T2 is found to be 40°C then T= 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
Example to illustrate rule 3
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
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.
Relative uncertainty in final result for S is 130/4000 = 0·0325
Example to illustrate rule 4
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)r3
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
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