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Measurements are supposed to give a certain information about a measurand, but they can never be exact due to influencing factors as the measuring procedure, environmental influences, the skills of the person measuring etc. Therefore, when measuring physical quantities, the measured values always scatter around the “true value”. Each measurement yields a different result, although the measured quantity doesn’t change. This inaccuracy is called “error” or more precise: uncertainty. To determine this uncertainty, multiple measurements can be conducted, while all other influences should be kept the same.
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- Puente León, Messtechnik. Berlin, Heidelberg: Springer Berlin Heidelberg, 2019.
- Fornasini, The Uncertainty in Physical Measurements. New York, NY: Springer New York, 2008.
- Kameke, Messwerte und Messunsicherheit. Hamburg, TUHH, 2019.
- Lectures: MTMotivationMT-Motivation, Measurement Theory
Necessary further reading: how to use a caliper!, error propagation, rounding to significant figures!
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Errors that can influence a measurement can be either systemic or statistic. Systemic errors are dependent e.g. on the instrument and always have the same amount and sign and are reproducible, so they appear every time the measurement is conducted. Some kind of offset can be used to compensate for systemic errors, but there could also be unknown systemic errors that therefore can’t be compensated. Statistic errors cannot be corrected because they are random by definition, but they can be described by average values and standard deviation, if multiple measurements were conducted. The standard deviation then is a measure for of the scattering of the values around the average, for a big standard deviation the values scatter very far from the average.
Example systemic error: a scale always shows a few grams, although nothing is on the scale.
Example statistic error: when reading a scale with an indicator needle, the reader makes small mistakes
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A complete measurement result for a statistically determined quantity x is therefore
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4.3 vernier caliper gauge
There are four types of measurements that can be made with a vernier caliper gauge: Inside measurements, outside measurements, depth and steps.
When the vernier caliper is closed, the scale should read exactly 0. To take a measurement of an inside dimension, the jaws of the caliper are now opened until they exactly cover the area to be measured. To measure an outside dimension, the caliper jaws are opened and then closed again until they touch the object. For measuring depths (e.g. drilled holes), the depth gauge at the end of the caliper can be used - the main element of the caliper is pressed against the higher surface and the depth gauge is positioned at the deepest point of the hole.
For reading, the main scale in mm is now used first to read the digit before the decimal point. This can be read at the position in front of the dash of zero on the slider. In order to read the decimal place, the position in the vernier scale (german: Nonius-Skala) must be found where the line of the large scale exactly coincides with one of the smaller scale.
4.4 significant figures
Significant figures are figures in a number, that carry meaning. To avoid misunderstandings, only significant ending zeroes are written, non significant zeros are omitted by using the scientific notation with power of ten.
Some examples:
| number | significant figures |
|---|---|
| 34.22 | 4 |
| 0.0043 | 2 |
| 0.120 | 3, due to the last zero being written, we define it as a significant zero |
| 2,345 · 106 | 4 |
| 2 345 000 | 7, due to the last zero being written, we define it as a significant zero |
| 1.003 | 4 |
| 50 | 2 |
| 50.0 | 3 |
The result of an addition/subtraction has as many decimal places as the number with the fewest decimal places.
Examples: 11.234+0.0007=11.235 11+3.432=14 11.00+3.432=14.43
The result of a multiplication/division has as many significant figures as the number with the fewest significant figures.
Examples: 3.123*4.54=14.18 3.123*0.1=0.3 3.123*0.0012=0.0037
A result can never be given with more accuracy than the biggest decimal number of the uncertainty, therefore, 2.3456±0.123 should be written as 2.346±0.123.
5. Technical Basics & Preparations
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- What is supposed to be measured? What values?
- Which accuracy? Is it even possible to measure this with this accuracy?
- What measuring device can be used for this? Does it provide the necessary accuracy?
- What are the possible (systemic) errors and external limitations that exist in the setup?
- Then: find out about the complete measurement
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General instructions
- Always try to pick the measurement instrument that alters the measurement the least and is the most accurate – there might be a trade-off! Use the instrument you deem most useful!
- Try to minimize systemic errors where possible!
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- Gather all the necessary measurement objects and instruments:
- Caliper
- Ruler
- 10 cylinders of type A
- 2 flat objects -> Muttern oder Unterlegscheiben?
- 1 O-ring
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