There are two basic types of error that may appear in the result:

The only way to eliminate systematic errors is to re-calibrate the apparatus or change it!

These are errors that are due to experimenter - in others words us! The size of these errors depends on how well the experimenter can use the apparatus. The better experimenter you are the smaller will these errors be. The way to reduce these errors, apart from simply being careful, is to repeat the readings and take an average.

It is important to understand the difference between the precision of a measurement and its accuracy. It is possible to measure something to a high degree of precision (for example a mass to a milligram using an electronic balance) but still to have an inaccurate answer because the instrument used had an offset zero error.

When you have made your set of
readings you must be careful when you quote your result. You will probably use a calculator
reading to eight digits to calculate and answer involving perhaps three of four quantities - do not
use the eight digits - the accuracy of the final answer cannot be better than the accuracy of any of
the single quantities.

A word of warning here: small quantities may only be ignored in comparison with large ones. For example in an answer such as 5.4000003 the 0.0000003 may be ignored in comparison to the 5.4 and the answer quoted as 5.4 but this is not the case in an answer such as 0.0000013 where the 0.0000003 is 23% of the answer. An Analogy: If you are sat on by an elephant it is not important whether or not the elephant has a fly on its back - you are still squashed flat. However to another fly the first fly is important!

There are two possibilities:

(i) Q is the sum or difference of a and b or (ii) Q is the product or quotient of a and b.

(i) Suppose that Q is the length of an object found by taking the reading of the position of its two end a and b from a ruler Q = a - b

Let the error in a be Δa and that in b be Δb. Then the error in Q is simply the error in a plus the error in b.

That is:

Then the value of Q should be written as:

Let a = 26.3 cm +/- 0.1 cm and b = 15.8 +/- 0.1 cm then:

Q = 26.3 - 15.8 = 10.5 +/- 0.2 cm or 10.5 cm +/- 1.9%

(ii) Suppose that Q is the area of a rectangle with sides a and b so that Q = ab (Figure 4).

Let the error in a be Δa an that in b be Δb.

Therefore the maximum value of Q is :

Q + ΔQ = (a + Δa)(b + Δb) = aΔb +bΔa +ab +ΔaΔb

But we can ignore the quantity ΔaΔb since it will be very small compare with the other quantities in the equation and so:

and this is true whether or not Q is a product or a quotient.

If one of the quantities, say b, is raise to a power n (i.e. Q = ab

It is probably easier to understand these two equations if they are quoted as a fractional error. In other words:

or even easier as a percentage error:

Notice that pure numbers have no errors; this can be assumed for quantities such as p and e

1. Find the maximum possible error in the measurement of the force on an object of mass 4.0 = +/- 0.1 kg travelling at 5.2 +/- 0.2 ms

The equation is F = mv

Using fractional errors ΔQ/Q = Δa/a + nΔb/b and so we have

ΔF/F = Δm/m + 2Δv/v + Δr/r = 0.1/4.0 + 2x0.2/5.2 + 0.01/0.75 = 0.025 + 0.077 + 0.013 = 0.115

But F = mv

The result should be quoted as F = 144 +/- 17 N

2. Find the maximum possible error in the density of a material in the form of a cube if the mass can be measured to +/- 5 % and the length of its sides to +/- 3.4 %.

Using percentage errors: %Q = %a + n%b and so we have %r = %m + 3%L = 5% + 3x3.4% = 15.2 %

You can see how, although the percentage error in L is smaller than the percentage error in mass, it is more important in the answer because it appears as L