Errors and Uncertainty

Here we will consider what to do with all that data that you just carefully collected from your experiment

[ Errors and Uncertainty | Estimating uncertainty | Average values ]
[ Combining uncertainty | Error bars on graphs | What to plot? | Examples ]


Error and Uncertainty

All readings, data, results or other numerical quantities taken from the real world by direct measurement or otherwise are subject to uncertainty. This is a consequence of not being able to measure anything exactly. Uncertainty cannot be avoided but it can be reduced by using 'better' apparatus. The uncertainty on a measurement has to do with the precision or resolution of the measuring instrument. When results are analysed it is important to consider the affects of uncertainty in subsequent calculations involving the measured quantities.

If you are unlucky (or careless) then your results will also be subject to errors. Errors are mistakes in the readings that, had the experiment been done differently, been avoided. It is perfectly possible to take a measurement accurately and erroneously! Unfortunately it is not always possible to know when you are making an error (otherwise you wouldn't make it!) and so godd experimental technique has to able to guard against the affect of errors

Types of Error:


Estimating uncertainty

Estimating the uncertainty on a reading is an art that develops with experience. There are two rules of thumb:

Firstly, take repeat readings. If there is a spread of readings then the uncertainty can be derived from the size of the spread of values. What you are doing in effect is seeing how repeatable the results are and this will give an order of magnitude idea of the uncertainty likely on any given reading. (See the section on dealing with averages below). For example, if three readings of time are 42s, 47s and 38s then the average is just over 42s with the other two readings being about 4s away from the average ... so use 42s ± 4s. The uncertainty is taken as 4s

Secondly, if the results are repeatable to the precision of the measuring apparatus then the uncertainty is taken as half of the smallest reading possible. For example, when measuring something with a ruler marked off in mm, the uncertainty is ± 0.5mm. When using a normal protractor the uncertainty on the angle is ± 0.5 degrees etc


Average values

If the experiment generates many repeat readings (as any really good experiment should) then there is a way to analyse the results and obtain a good value for the associated uncertainty:
  1. Take an average of the results
  2. Work out the deviation of each result from the average
  3. Average the deviations (ignore any minus signs) - this is the uncertainty

For example:

Voltage (V)Deviation from average
2.00.1
2.20.1
1.80.3
1.90.2
2.60.5
2.30.2
1.70.4
2.40.3
2.20.1
1.90.2
Av = 2.1Av = 0.24

Thus we use V = 2.1 ± 0.2 Volts


Combining uncertainty

In many equations two or more values are combined mathematically so it is important to know what happens to the uncertainties. The uncertainty on a value can be expressed in two ways, either as an 'absolute' uncertainty or as a 'percentage' uncertainty. The absolute uncertainty is the actual numerical uncertainty, the percentage uncertainty is the absolute uncertainty as a fraction of the value itself. Consider our previous example:

Voltage = 2.1 ± 0.2
The quantity = 2.1 V
Absolute uncertainty = 0.2 V (it has units)
Percentage uncertainty = 0.2 / 2.1 = 0.095 = 9.5% (no units as its a ratio)

In all of the following examples we consider combing 2 values:

and

Addition: The uncertainty on the sum of the two values is the sum of the absolute uncertainties

Subtraction: The uncertainty on the difference of the two values is the sum of the absolute uncertainties

Multiplication: The uncertainty on the product of the two values is the sum of the percentage uncertainties

Division: The uncertainty on the division of the two values is the sum of the percentage uncertainties




There are two important points to note here:

1. If two values that are very similar are subtracted then the uncertainty becomes very large ... this can render the results of an experiment meaningless. For example, consider 4.0 ± 0.1 - 3.5 ± 0.1 = 0.5 ± 0.2. The percentage uncertainty on the individual values is about ± 2.5% whereas the percentage uncertainty on the result is ± 40%

2. Be very careful to convert percentage uncertainties back to absolute values after combining the various values. Only the absolute uncertainty has any real physical meaning


Error bars on graphs

Having taken measurements and calculated the associated uncertainties, it is often necessary to plot these values graphically. Uncertainties are represented as 'error bars' on graphs - although this is a misleading title, it would be better to call them 'uncertainty bars'. Error bars are simply a line used to represent the possible range of values, the line or curve drawn through the points can pass through any part of the error bar. The graph below shows how the error bars are drawn. The values on the x-axis are shown with a constant absolute uncertainty, the values on the y-axis are shown with a percentage uncertainty (and so the error bars gets bigger)



What to plot?

The art of analysing experimental data is knowing what to plot, in most experiments it is not enough to simply plot the recorded values directly, instead some more appropriate graph is needed.

It is always the case that a linear graph gives the most useful analysis and so the data is manipulated to give the required linear relationship

The mathematical relationship for a linear relationship is y = mx + c

In a Physical situation each of these quantities has physical meaning and appropriate units - this includes the gradient and the y-intercept. Don't forget to include units when calculating values from a 'Physics' graph!

Formulaplot
y-axis
plot
x-axis
Notes
y = mx + cyxGradient = m, y-intercept = c
y = kx2yx2Gradient = k
y = k / xy1 / xGradient = k
y = k / x2y1 / x2Gradient = k
y = ekxln(y)xGradient = k
y = k sqrt(x)y2xGradient = k2


Examples

For the dynamics equation s= ½at2 (u=0) used to determine the value of g by free fall
plot s (y-axis) vs t2 (x-axis)
which will be a linear graph with a gradient of ½a

For the nuclear physics equation for gamma ray intensity R = k / (x + x0)2 where R = rate, x = distance, k & x0 are constants
plot x (y-axis) vs 1 / sqrt(R) (x-axis)
which gives a linear relationship with Gradient = sqrt(k) and y-intercept = -x0. To see why, re-arrange the equation to make x the subject (i.e. x = ....)

For the dynamics equation v2 = u2 + 2as
plot v2 (y-axis) vs s (x-axis)
which gives a linear relationship with gradient = 2a and y-intercept = u2