Measurement and Uncertainty

Every measurement is not exact, they are approximate estimations which we try to get as close as possible to the "exact value".
The two types of uncertainties are "Absolute Uncertainty" and "Relative Uncertainty".


Absolute Uncertainty
Absolute uncertainty is the inaccuracy in a measurement quantity.

Method 1 to express uncertainty:

1. First, collect at least three pieces of data from your measurements
2. Then, we will cancel out unreasonable pieces of data (if present)

3. Add all values together, then divide by the trials number to get an average
4. Subtract the average from the highest and lowest values
5. Record answer

Example:
Lengths of buffalo hair

Trial              Length
1                    22.3cm
2                    10.4cm
3                    22.4cm
4                    22.1cm (Lowest measurement)
5                    22.6cm (Highest measurement)

Since the second trial is clearly erroneous data, we will ignore out this measurement.
Average of lengths: (22.3+22.4+22.1+22.5) ÷ 4 = 22.3cm (Do not forget the UNITS!!!)
Difference between highest measurement: 22.6-22.3 = 0.3cm
Difference between lowest measurement: 22.3-22.1 = 0.2cm
(We use 0.3 as the uncertainty because it is the greater number)

Answer: 22.3 ± 0.3cm is the average length of buffalo hair.


Method 2 to express uncertainty:

We can figure out the uncertainty scale of each measuring tool, and then when we are measuring, we'll make the estimation as precise as possible. This way, we can use the uncertainty scale of the instrument to determine the answer.



Relative Uncertainty

The relative uncertainty is the percentage ratio between the uncertainty of an estimate to the real value of a measurement.

The formula for calculating relative uncertainty is the following:

Absolute    uncertainty 
estimated measurement

We can express relative uncertainty in two ways:
~ in percentage (%)
~ in significant figures

In percentage

1. First, we use the formula above to calculate an answer
2. Then, we multiply the answer by 100%

Example:Measuring the length of part of a rotten hot dog
The absolute uncertainty is ±0.5cm
The estimated measurement is 10cm
Thus the relative uncertainty is 0.5cm /10cm * 100%=5%


In significant figures
The last digit of a measurement is always uncertain, where the number of sig. figs. represents relative uncertainty.


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