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Measurement Errors: What Types There Are (+ Examples)

Table of contents

Exercises with Solutions

In every measurement, there is a certain degree of uncertainty or sources of error that influence the accuracy of the measurement. It is therefore important to know the different types of measurement errors and to eliminate them as far as possible or at least to be aware of them.

Systematic Measurement Errors - these are reproducible measurement deviations, meaning they are predictable / calculable and can be eliminated.

Examples of systematic errors

Parallax errors
Environmental factors, such as heating and therefore expansion of the measurement object.
Error due to model used: for example, using formulas for mathematical pendulum instead of real pendulum (i.e. neglecting the mass of the thread).

Random measurement errors - you can neither calculate these nor somehow eliminate them with better devices. They occur due to fluctuations that cannot be influenced and due to the subjective perception of the observer.

Examples of random errors

Read-off error
Voltage fluctuations
Temperature fluctuations

All these random errors cannot be reproduced with the same measurement setup!

Exercises with Solutions

Use this formula eBook if you have problems with physics problems.

Exercise #1: Relative Error of Electric Power

You want to measure the power $ P $ across the resistor $ R ~=~ 5 \Omega ~\pm~ 2\% $. For this, you have measured the voltage $ x_{\text U} ~=~ 60 \, \text{V} $ with a voltmeter and the current $ x_{\text I} ~=~ 10 \, \text{A} $ with an ammeter.

Determine the relative error $ f_x $ for the following three approaches:

$ P ~=~ U \, I $
$ P ~=~ \frac{U^2}{R} $
$ P ~=~ I^2 \, R $

Data of the Ammeter: End of range value $ x_{\text E} ~=~ 100 \, \text{A} $. Error class: $ K_{\text E} ~=~ 1 $.

Data of the Voltmeter: End of range value $ x_{\text E} ~=~ 100 \, \text{V} $. Error class: $ K_{\text E} ~=~ 1 $.

Tip: Error class is calculated as follows: 1 $$ K_{\text E} ~=~ 100 \cdot \frac{G}{x_{\text E}} $$ where $ G $ is the magnitude of the maximum possible error.

Solution to Exercise #1.1

Rearrange 1 to find $ G $: 2 \[ G ~=~ \frac{ K_{\text E} }{ 100 } \, x_{\text E} \]

Hence, the maximum possible error of the voltmeter is: 3 \[ G ~=~ \frac{ 1 }{ 100 } \, 100 \, \text{V} ~=~ 1 \, \text{V} \]

And similarly for the ammeter: 4 \[ G ~=~ \frac{ 1 }{ 100 } \, 100 \, \text{A} ~=~ 1 \, \text{A} \]

The displayed voltage value, as well as the current value, can deviate by a maximum of $ F_{\text U} ~=~ \pm 1 \text{V} $ and $ F_{\text I} ~=~ \pm 1 \text{A} $ respectively.

The relative error of the voltage value is calculated as: 5 \[ f_{\text U} ~=~ \frac{ F_{\text U} }{ x_{\text U} ~-~ F_{\text U}} ~\cdot~ 100\% \]

The maximum relative error occurs when you set $ F_{\text U} ~=~ +1 \text{V} $: 6 \[ f_{\text U} ~=~ \frac{ 1 \text{V} }{ 60 \text{V} ~-~ 1 \text{V}} ~\cdot~ 100\% ~\approx~ 1.7 \% \]

Similarly for the relative error of the measured current (with $ F_{\text I} ~=~ + 1 \text{A} $): 7 \[ f_{\text I} ~=~ \frac{ 1 \text{A} }{ 10 \text{A} ~-~ 1 \text{A}} ~\cdot~ 100\% ~\approx~ 11.1 \% \]

The relative error of the power value (calculated using the first power formula) is thus: 8 \[ f_{\text P} ~=~ f_{\text U} ~+~ f_{\text I} ~=~ 1.7 \% ~+~ 11.1 \% ~=~ 12.8 \% \]

Solution to Exercise #1.2

In Exercise #1.1, the maximum possible error of the voltmeter ($f_{\text U} ~=~ 1.7 \%$, see Eq. 6) was calculated. Furthermore, the relative error of the measured resistance according to the exercise is $ f_{\text R} ~=~ \pm 2 \% $. Therefore, the maximum possible error of the power value, calculated with second power formula, is: 9 \[ f_{\text P} ~=~ f_{\text U} ~+~ f_{\text U} ~-~ f_{\text R} ~=~ 1.7 \% ~+~ 1.7 \% ~-~ (-2\%) ~=~ 5.4 \% \]

Solution to Exercise #1.3

In Exercise #1.1, the maximum possible error of the ammeter ($f_{\text I} ~=~ 11.1 \%$, see Eq. 7) was calculated. Furthermore, the relative error of the measured resistance is $ f_{\text R} ~=~ \pm 2 \% $. Therefore, the maximum possible error of the power value, calculated with third power formula, is: 10 \[ f_{\text P} ~=~ f_{\text I} ~+~ f_{\text I} ~+~ f_{\text R} ~=~ 11.1 \% ~+~ 11.1 \% ~+~ (+2\%) ~=~ 24.2 \% \]