Significant Figures,
Uncertainties,
and Simple Statistics
Significant Figures
• It is important to be honest when reporting
a measurement, so that it does not appear
to be more accurate than the equipment
used to make the measurement allows.
• We can achieve this by controlling the
number of digits, or significant figures,
used to report the measurement.
Rules for Significant Figures
• All non-zero numbers are significant
– 24 has 2 sig figs
– 462 has 3 sig figs
• Zeros within a number are always
significant
– 1024 has 4 sig figs
– 100 000 005 has 9 sig figs
• Zeros that do nothing but set the decimal
point are not significant
– 1200 has 2 sig figs
– 0.003 has 1 sig fig
• Trailing zeros that are not needed to hold
the decimal point are significant
– 2.10 has 3 sig figs
– 1.000 has 4 sig figs
Examples
• 120.030
– 6 sig figs
• 0.00230
– 3 sig figs
• 350
– 2 sig figs
• 6.02x1023
– 3 sig figs
Uncertainties
• Whenever a measurement is made there
is an error or uncertainty associated with it
• This is because we can never measure
anything perfectly
• The uncertainty is a combination of the
limitations of the equipment and the
person using the equipment
• This uncertainty is expressed as a plus or minus
(±) value
• For example, a measurement of the side
of a block may be 12.35 ± 0.05 cm
• This means that the “real” value is
anywhere from 12.3 to 12.4 cm
• A simple estimate of the uncertainty is half
of the smallest value we can measure
– A ruler with marking every 1 mm would have
an uncertainty of 0.5 mm.
Propagating Uncertainties
• When number with uncertainties are
combined, the uncertainty increases
• Addition and Subtraction
– Uncertainties add
• Multiplication and Division
– Percent uncertainties add
Example
• A 25.0 ± 0.3 g block of wood has the
following dimensions:
– Length: 5.00 ± 0.05 cm
– Width: 3.00 ± 0.05 cm
– Height: 3.00 ± 0.05 cm
• Calculate the density of the block of wood
Density 
mass
volume
• Volume:
V  5.00  3.00  3.00  45.00 cm 3
• Uncertainty in Volume:
– Calculate percent uncertainties
0.05
 .01
5.00
0.05
 .0167
3.00
– Add percent uncertainties
0.01  0.0167  0.0167  0.0434  4.34%
• Density
Density 
25.0
 0.556 gcm 3
45.00
• Uncertainty in Density
0.3
 0.012
25
0.0434  0.012  0.0554  5.54%
• Convert final percentage uncertainty to
absolute uncertainty (and round to 1 sig
fig)
0.0554  0.556  0.03
• Round answer to same place value
0.56  0.03 gcm 3
Simple Statistics
• Mean
– Measure of central tendency for normally
distributed data
– Sum of the data divided by the number of
data
• Standard deviation
– Measure of how the individual observations of
a data set are dispersed or spread out around
the mean
• Variance
– measures how far a set of numbers is spread
out
• T-test
– Used to determine if two sets of data are
significantly different from each other
– studentsttest.com