2. Estimating Uncertainties In
Experimental Results
All experimental scientists need to know how
well they can trust their results.
The results of any experiment are only as valid as
the degree of error in those results.
A lot of time, effort , and money has been spent
by scientists developing more “accurate”
machines to measure events more precisely.
This unit is all about making and keeping tracks
of errors during experimental measurements.
5. Estimating Uncertainties In
Experimental Results
All measured values must be accompanied by an
estimate of the error or uncertainty associated
with the measured value.
The tennis ball has a diameter of
6.4 + 0.1 cm.
Measurement value Estimated error value
6. Estimating Uncertainties In
Experimental Results
Let’s look at some other possible ways of trying
to report this value:
6.4 + 0.15 cm What is inconsistent here?
6 + 0.1 cm What is inconsistent here?
6.42 + 1 cm What is inconsistent here?
7. Estimating Uncertainties In
Experimental Results
So what does 6.4 + 0.1 cm really mean?
The real or actual diameter of the tennis ball lies
between a maximum and a minimum value.
The actual value lies
Maximum value: 6.5 cm somewhere in between
these two values!
Minimum value: 6.3 cm We can not be any
more precise than
this!
8. Estimating Uncertainties In
Experimental Results
Types of Errors:
Measurement errors fall into two main types:
Systematic errors:
These errors consistently influence a set of
measurements in a particular direction , either too
high or too low.
These errors are associated with the precision of the
measuring device (eg. not calibrated correctly), or
errors in experimental procedures.
9. Estimating Uncertainties In
Experimental Results
Random errors:
These errors arise due to fluctuations in the
experimental conditions or in the judgment of the
experimenter.
These errors are random, some being too high while
others being too low and tend to average out if the
experimenter repeats the experiment often enough.
After you have identified the factors that may influence your
results in the collection of experimental results, it is important to
design strategies to minimize both of these two types of errors.
10. Estimating Uncertainties In
Experimental Results
Think:
Drop a tennis ball from some height allowing it
to hit the ground and measure the height to
which it rebounds to.
1) Think and discuss all of the factors that could
affect the outcome.
2) Think and discuss all of the possible error sources
including both Systematic and Random.
11. Estimating Uncertainties In
Experimental Results
Dealing with errors:
Adding and Subtracting Measured Values:
A student measures the mass of a 123.4 + 0.1 g
beaker + copper to be :
A student measures the mass of a
beaker to be : 113.8 + 0.1 g
Mass of Copper is: 9.6 + ? g
But what about the uncertainty? What happens to it? Does it
stay at 0.1? Or does change to a higher or lower number?
12. Estimating Uncertainties In
Experimental Results
The rule is:
When adding or subtracting numbers the
numerical uncertainty is simply added!
In order to determine the mass of copper the student
subtracted two measured values: therefore simply
add the numerical error!
Mass of Copper is: 9.6 + 0.2 g
Numerical error
13. Estimating Uncertainties In
Experimental Results
Now try these:
4.5 + 0.2 m + 2.3 + 0.1m + 6.3 + 0.3 m = 13.1 + 0.6 m
67.9 + 0.2 g - 45.7 + 0.2 g = 22.2 + 0.4 g
(34.5 + 0.2 cm) + (12.3 + 0.3 cm) - (14.3 + 0.2 cm) =
32.5 + 0.7 cm
(1.5 + 0.5 m) - (4.3 + 0.5 m) + (8.8 + 0.3 m) = 6+1m
14. Estimating Uncertainties In
Experimental Results
Multiplying or Dividing Measured Values:
This becomes a little more complicated.
The rule is:
When measured values are multiplied or divided the percentage
errors are added.
What is a percentage error?
Answer: a numerical error changed to be represented as a
percentage of the measured value
15. Estimating Uncertainties In
Experimental Results
How is this done?
Easily:
Remember the copper:
Mass of Copper is: 9.6 + 0.2 g
0.2
Percent error = X 100 = 2%
9.6
Mass of Copper is: 9.6 + 2 % g
17. Estimating Uncertainties In
Experimental Results
Now try These:
Change numerical to percentage error:
13.1 + 0.6 m 13.1 + 5 % m
22.2 + 0.4 g 22.2 + 2 % g
32.5 + 0.7 cm 32.5 + 2 % cm
6+1m 6 + 17 % m
18. Estimating Uncertainties In
Experimental Results
Now try these:
Remember when measured values are multiplied
or divided, add the percentage errors!
1) 22.2 cm + 2 % x 45.2 cm + 5% = 1000 cm2 + 7 %
2) 2.31 g + 2 % ÷ 0.76 mL + 3% = 3.0 g/mL + 5 %
3) 45 + 1 m x 342 + 3 m = 15400 m2 + 3 %
4) {(2.2 cm + 2 % x 5.4 cm + 5%) + 14 + 0.3 cm2} =
Careful on this last one!
26 + 1 cm2
19. Estimating Uncertainties In
Experimental Results
How to determine the numerical error?
1) Reading a scale:
• Use ½ of the smallest division
2) Fluctuating scale:
• Look at the range of fluctuations and
divide by 2
• 1/2(maximum value – minimum error)
