# rules for uncertainties

```In general, the result of any measurement of a quantity x is stated as
(measured value of x) = Xbest
- x
Rules for Reporting
Measurements should be reported using one of the following two formats:
Standard notation:
Scientific notation:
345.5 +/- 0.5 m/s
(3.455 +/- 0.005) x 102 m/s
It is good practice to leave the ‘0’ in front of any decimal point when writing a number
that is less than 1. For example, in the Standard notation example above, the error of 0.5
m/s is written as:
0.5 m/s
not .5 m/s
Rule for Stating Uncertainties
Experimental uncertainties should almost always be rounded to one significant figure.
For example: 56.78 +/- 0.04834 cm should be rewritten as 56.78 +/- 0.05 cm
456.8 +/- 0.789 km should be rewritten as 456.8 +/- 0.8 km
20,078 +/- 4.32 kg should be rewritten as 20,078 +/- 4 kg
Exception to the above rule: If the leading digit in the uncertainty x is a ‘1’, then
keeping two significant figures in x may be more appropriate. For example, suppose that
some calculation gave the uncertainty x = 0.13. Rounding this number to x = 0.1 would
be a substantial percentage reduction, so we could argue that retaining two figures might
be less misleading, and quote x = 0.13. The same argument could perhaps be applied if
the leading digit is a 2, but not if it is any larger.
For example: 34.567 +/- 0.001432 should be rewritten as 34.5670 +/- 0.0014
However, 34.567 +/- 0.0025474 should be rewritten as 34.567 +/- 0.003
Rule for Stating Answers
The last significant figure in any stated answer should usually be of the same order of
magnitude (in the same decimal position) as the uncertainty.
For example: 345.87 +/- 30 m/s should be rounded to 350 +/- 30 m/s
92.81 +/- 0.3 m/s should be rounded to 92.8 +/- 0.3 m/s
92.81 +/- 3 m/s should be rounded to 93 +/- 3 m/s
92.81 +/- 30 m/s should be rounded to 90 +/- 30 m/s
Qualification to both the above rules: Any numbers that are used in intermediate
calculations should not be rounded until the final answer is achieved. This will reduce
inaccuracies caused by the compounding of rounding errors.
Sample test questions:
Rewrite each of the following measurements in its most appropriate form:
1)
X = 8.6711345 +/- 0.999864 m/s
2)
m = 44278 +/- 2 m
3)
k = 7.7899 x 10-7 +/- 3 x 10-9 kg
4)
g = 987.9004 +/- 52 m/s2
5)
n = 3000 +/- 20 meters
6)
56.9 +/- 0.6 liters
7)
7.18 +/- 0.0143 m
8)
45.78934 x 10-2 +/- 5 x 10-3 cm
1)
2)
3)
4)
5)
6)
7)
8)
X = 8.7 +/- 1.0 m/s
m = (4.4278 +/- 0.0002) x 104 m
k = (7.79 +/- 0.03) x 10-7 kg
g = 990 +/- 50 m/s2
no change necessary. This is already in the most appropriate form.
56.9 +/- 0.6 liters
7.180 +/- 0.014 m - due to the Exception to the Rule for Stating Uncertainties
(4.58 +/- 0.05) x 10-1 cm
```