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A complete measurement can only be specified by taking the uncertainty into account, it can be displayed as where where M is the average or the one measurement and u is the uncertainty. The uncertainty can also be specified as a relative value, e.g. in percent of the measured value M. A measured value that has multiple decimal digits can show more digits than the uncertainty allows, therefore rounding to the last decimal digit of the uncertainty is common.
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Example statistic error: when reading a scale with an indicator needle, the reader makes small mistakes
(Serious/major errors: errors due to the experimenter, e.g. improper instrument handling/arrangement, can be easily and generally avoided with some care and preparation. E.g. reading the ammeter as a voltage)
It can therefore be concluded, that a single measurement doesn’t have any implications as no knowledge about the statistic error exists!
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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.
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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., rounded numerical uncertainty values should be reported with two (or, if required, three) significant digits and rounded up. The measurement result should be rounded at the same decimal place as the associated uncertainty, e.g. M=123.4561V to M=123.46V if u(M)=0.7819V is rounded up to u(M)=0.79V. [1]
5. Technical Basics & Preparations
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- Gather all the necessary measurement objects and instruments:
- Caliper
- Ruler
- 10 cylinders of type A
- 2 flat objects -> Unterlegscheiben?!
- 1 O-ring
6. Experiment Procedure
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1.2 Find out, which of the two flat objects is thicker, A or B.
1.3.1 Pick one of the cylinders and measure its length and diameter.
Submit a complete measurement of the length and the diameter. Use at least 10 measurements. (repetition)
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1.3.2 Put the cylinder down. Measure the same part again, using the same caliper.
1.3.3 Now ask your neighbor to measure your wooden part with their caliper!
- Did 1.3.1, 1.3.2 and 1.3.3 yield the same result? Which of the steps was for testing reproducibility (A) and which for repeatability (B)?
1.4 Measure the volume of one of the cylinders by carrying out the following steps:
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Calculate the propagated error and supply a complete measurement of the volume (rounded to significant figures and including the uncertainty)!
1.5 Submit a complete measurement of the length and the diameter. Use at least 10 measurements.
1.6 (creative) Find a way to measure the weight of your hand (only the hand, not the arm) at home and describe your steps.???
7. Evaluation of Experiment Results
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- Matriculation number
- Value for 1.1 caliper, value for 1.1 ruler
- the thicker object (A or B)
- value for the complete measurement of one cylinder, length and diametercylinder - length and diameter: measurement (1.3), 1.3.1 and 1.3.2
- 1.3.2 (A or B), 1.3.3 (A or B)
- complete measurement of the volume of the cylinder
- complete measurement result for 1.5 - length and diameter
- (free text) explain your measurement steps and the final result of 1.6
Resources:
[1] DIN 1319-4. Grundlagen der Meßtechnik. Teil 3: Auswertung von Messungen - Meßunsicherheit (9.2). Februar 1999.