Examples of random errors

CPS Physics

Treatment of Errors and Uncertainties in Laboratory Book Submissions

Introduction

In experimental physics an error does NOT mean a mistake.

An error is the uncertainty in a measurement or it is the uncertainty in a result that is calculated from a series of measurements.

When you obtain an experimental result you should always state the range of uncertainty.

For example, suppose you have found the value of the acceleration due to gravity, g, at the Earth’s surface from a simple pendulum experiment. The result should be reported in the following way:

g = 9.78 ± 0.15 m s^-2

The ± range is due to random errors in the physical quantities you have measured and from which the result is calculated.

These notes show how to estimate these error limits. Usually the “true value” (in this example it is of course 9.81 m s^-2 to 2 decimal places) will lie within the error limits. If this is not the case it means that the experiment contains systematic errors in addition to random errors.

Random errors and uncertainties

In every experiment there are random errors. This means that the value obtained for a measurement (or for a calculated value based on a series of measurements) is not certain – not precise – and it could be either a lower value or a higher value. We can express a random error in a result, Δx, for a measured quantity x as:

i) an absolute error x ± Δx (Δx has the same dimension as x)

ii) a fractional error ± (or a percentage error ± ). is dimensionless.

From the above example, the fractional error = = 0.0153 or 1.53%

Examples of random errors

Suppose you need to use a ruler to measure a distance. Each time you take the measurement it is slightly different because of random differences in the way you place the ruler – and in the way you estimate the distance between the smallest divisions on the ruler.

Similar uncertainties/errors arise for ammeter/voltmeter readings, or for angular scales on spectrometers.

If you are timing an event with a stop watch, your human reaction time varies each time you do it and this leads to a set of results which are subject to random errors.

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