How to Calculate Uncertainty

Quantifying the level of doubt in your measurements is a crucial contribution of science. No measurement can be perfect, and understanding the limitations on the preciseness in your measurements helps to ensure that you don ’ thymine disembowel indefensible conclusions on the basis of them. The basics of determining doubt are quite simple, but combining two uncertain numbers gets more complicate. The good news is that there are many simple rules you can follow to adjust your uncertainties careless of what calculations you do with the original numbers.

TL;DR (Too Long; Didn’t Read)

If you ’ re adding or subtracting quantities with uncertainties, you add the absolute uncertainties. If you ’ rhenium breed or separate, you add the relative uncertainties. If you ’ rhenium multiply by a constant factor, you multiply absolute uncertainties by the lapp gene, or do nothing to relative uncertainties. If you ’ re taking the power of a number with an doubt, you multiply the proportional doubt by the number in the world power.

Estimating the Uncertainty in Measurements

Before you combine or do anything with your doubt, you have to determine the doubt in your original measurement. This much involves some subjective judgment. For exemplar, if you ’ re measuring the diameter of a ball with a ruler, you need to think about how precisely you can actually read the measurement. Are you confident you ’ ra quantify from the edge of the ball ? How precisely can you read the ruler ? These are the types of questions you have to ask when estimating uncertainties.

In some cases you can easily estimate the uncertainty. For example, if you weigh something on a scale that measures down to the nearest 0.1 guanine, then you can confidently estimate that there is a ±0.05 guanine doubt in the measurement. This is because a 1.0 gravitational constant measurement could in truth be anything from 0.95 guanine ( rounded up ) to just under 1.05 g ( rounded gloomy ). In other cases, you ’ ll have to estimate it a well as possible on the footing of respective factors.

Tips

  • Significant Figures: Generally, absolute uncertainties are entirely quoted to one significant figure, apart from occasionally when the first figure is 1. Because of the meaning of an doubt, it doesn ’ t make common sense to quote your appraisal to more preciseness than your uncertainty. For exemplify, a measurement of 1.543 ± 0.02 meter doesn ’ t make any feel, because you aren ’ metric ton certain of the second base decimal seat, so the third is basically meaningless. The chastise solution to quote is 1.54 thousand ± 0.02 megabyte .

Absolute vs. Relative Uncertainties

Quoting your uncertainty in the units of the original measurement – for example, 1.2 ± 0.1 gigabyte or 3.4 ± 0.2 curium – gives the “ absolute ” uncertainty. In early words, it explicitly tells you the sum by which the original measurement could be faulty. The proportional doubt gives the doubt as a share of the master rate. Work this out with : \text { relative uncertainty } = \frac { \text { absolute uncertainty } } { \text { best estimate } } × 100\ % so in the model above : \text { relative uncertainty } = \frac { 0.2 \text { centimeter } } { 3.4\text { centimeter } } × 100\ % = 5.9\ %

The value can therefore be quoted as 3.4 centimeter ± 5.9 %.

Adding and Subtracting Uncertainties

knead out the sum uncertainty when you add or subtract two quantities with their own uncertainties by adding the absolute uncertainties. For model : ( 3.4 ± 0.2 \text { cm } ) + ( 2.1 ± 0.1 \text { cm } ) = ( 3.4 + 2.1 ) ± ( 0.2 + 0.1 ) \text { centimeter } = 5.5 ± 0.3 \text { curium } \\ ( 3.4 ± 0.2 \text { cm } ) – ( 2.1 ± 0.1 \text { cm } ) = ( 3.4 – 2.1 ) ± ( 0.2 + 0.1 ) \text { centimeter } = 1.3 ± 0.3 \text { curium }

Multiplying or Dividing Uncertainties

When multiplying or dividing quantities with uncertainties, you add the relative uncertainties together. For model : ( 3.4 \text { curium } ± 5.9\ % ) × ( 1.5 \text { curium } ± 4.1\ % ) = ( 3.4 × 1.5 ) \text { curium } ^2 ± ( 5.9 + 4.1 ) \ % = 5.1 \text { centimeter } ^2 ± 10\ % \frac { ( 3.4 \text { centimeter } ± 5.9\ % ) } { ( 1.7 \text { curium } ± 4.1 \ % ) } = \frac { 3.4 } { 1.7 } ± ( 5.9 + 4.1 ) \ % = 2.0 ± 10 %

Multiplying by a Constant

If you ’ re multiplying a number with an uncertainty by a constant factor, the rule varies depending on the character of doubt. If you ’ re using a relative uncertainty, this stays the same : ( 3.4 \text { curium } ± 5.9\ % ) × 2 = 6.8 \text { centimeter } ± 5.9\ % If you ’ re using absolute uncertainties, you multiply the doubt by the same factor : ( 3.4 ± 0.2 \text { cm } ) × 2 = ( 3.4 × 2 ) ± ( 0.2 × 2 ) \text { centimeter } = 6.8 ± 0.4 \text { curium }

A Power of an Uncertainty

If you ’ re taking a power of a value with an doubt, you multiply the relative doubt by the number in the ability. For exemplar : ( 5 \text { centimeter } ± 5\ % ) ^2 = ( 5^2 ± [ 2 × 5\ % ] ) \text { curium } ^2 = 25 \text { centimeter } ^2± 10\ % \\ \text { Or } \\ ( 10 \text { thousand } ± 3\ % ) ^3 = 1,000 \text { megabyte } ^3 ± ( 3 × 3\ % ) = 1,000 \text { thousand } ^3 ± 9\ % You follow the lapp dominion for fractional powers.

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