Measurement and Uncertainty

In science, we take measurements. These, however, are never exact. Measurements are only our best estimate, which is subject to some uncertainty.
The only numbers that are exact are when you are counting whole things. Ex. seven apples, 15 pencils, 48 dogs, etc.

There are two types of Uncertainty: Absolute Uncertainty and Relative Uncertainty.

Absolute Uncertainty

- This uncertainty is expressed in units of measurement, not as a ratio
- Two methods can be used

Method One:
- Take at least three measurements
- Calculate the average of these numbers
- The Absolute Uncertainty of these measurements is the difference between the average and the highest or lowest reasonable measurement
- Always remove any measurements that are unreasonable

Ex.
Trial #    Mass of an Object
   1                 27.3 g
   2                 27.5 g
   3                 27.4 g
   4                 27.9 g
   5                 27.4 g

You would remove 27.9 g, since it is unreasonable. You would then add up the other measurements and divide by four, calculating the average:
109.6 g / 4 = 27.4 g
You would then find the difference between 27.4 g and the highest or lowest measurement:
27.4 g - 27.3 g = 0.1 g
27.4 g - 27.5 g - 0.1 g
Therefore, the Absolute Uncertainty of these measurements is 0.1 g. You would record the mass as:
27.4 +- 0.1 g

Method Two:
-Determine the uncertainty of the instrument being used 
-Measure as precisely as possible
-Estimate to 10% of the smallest division made on the instrument
Ex.
On a ruler, the smallest division is 0.1 of a cm, or 1 mm. Therefore, you should measure to the nearest tenth of a mm. If you estimate a measurement to be 25.34 cm on a ruler, you would record the number as 25.34 +- 0.01 cm. If you estimate a measurement to be 6.7oC on a thermostat, you would record the number as 6.7 +- 0.1 oC.

Relative Uncertainty
-This can be expressed as in percent or by using significant figures.
-Percent is most commonly used
-Relative Uncertainty = Absolute Uncertainty / Estimated Measurement
-To get a percent, you would multiply that number by 100.
Ex. Estimated measurement = 27.23 and the Absolute Uncertainty = 0.01
(0.01 / 27.23) X 100 = 0.03672
Therefore, the Relative Uncertainty is 0.04 %