
Error – difference between your answer and
the ‘true’ one. Generally, all errors are of
one of three types.
◦ Systematic – problem with the method, all errors
are of the same magnitude and direction (affect
accuracy)
◦ Random – causes data to be scattered more or
less symmetrically around a mean value. (affect
precision)

No measurement is perfect

Uncertainty in data leads to uncertainty in
calculated results
◦ Our estimate of nearness to the true value is
called the uncertainty (or error)
◦ Uncertainty never decreases with calculations,
only with better measurements

Reporting uncertainty is essential
◦ The uncertainty is critical to decision-making
◦ Estimating uncertainty is your responsibility

Accuracy is the degree of closeness of a
measured or calculated quantity to its actual
(true) value. Precision, the degree to which
further measurements or calculations show
the same or similar results.
High accuracy, but low precision
High precision, but low accuracy



Absolute uncertainty expresses the margin of
uncertainty associated with a measurement.
Relative uncertainty compares the size of the
absolute uncertainty with its associated
measurement.
Percentage uncertainty =
relative uncertainty x 100%

If a measurement is written as 5.4 ± 0.2 g, then
there is a,
Absolute Uncertainty = 0.2 g
absolute uncertaint y
Relative Uncertain ty 
 0.2  5.4  0.04
magnitude of measuremen t
percentage uncertaint y  0.04  100 %  4 %


We often do math with measurements
Density = (m ± m) / (V ± V)
What is the uncertainty on the density?
“Propagation of Error” estimates the
uncertainty when we combine uncertain
values mathematically.

Rule 1:
If a measured quantity is multiplied or divided by a constant
then the absolute uncertainty is multiplied or divided by the
same constant. (In other words the relative uncertainty stays
the same.)

Example to illustrate rule 1
Suppose that you want to find the average thickness of a page of a
book. We might find that 100 pages of the book have a total
thickness of 9mm. If this measurement is made using an instrument
having a precision of 0·1mm, we can write
Thickness of 100 pages, T = 9·0mm ± 0·1mm
and, the average thickness of one page, t, is obviously given by
t = T/100
therefore our result can be stated as t = 9/100mm ± 0·1/100mm
or
t = 0·090mm ± 0·001mm

Rule 2:
If two measured quantities are added or subtracted
then their absolute uncertainties are added.

To find a change in temperature, DT, we find an initial
temperature, T1, a final temperature, T2, and then use
DT = T2 - T1
If T1 is found to be 20°C ±1°C and if T2 is found to be
40°C ±1°C then DT = 20°C ± 2°C

Rule 3:
If two (or more) measured quantities are
multiplied or divided then their relative
uncertainties are added.
Rule 4:
If a measured quantity is raised to a power
then the relative uncertainty is multiplied by
that power.
If we multiply 1.2 ± 0.5 cm and 3.0 ± 0.5 cm,
then relative error of the first one is 0.5/ 1.2 =
0.416 and for the second one is 0.5/3 = 0.166.
Sum of the relative error is = .582 or 5.82%
The product is = 3.6 cm2
Absolute error = 5.82% x 3.6 cm2 = 2.0952 cm2
The errors are expressed to one significant
figures = 2 cm2
The product = 3.6 ± 2 cm2.

250
charge ,milli coulomb
200
150
100
50
0
0
1
2
3
4
5
Time, s
6
7
8
9
Significantly different from
the accepted value.
Consistent with the
accepted value.
May be significantly different
from the accepted value.