Summary of Significant Figures and Uncertainty
Significant Figures
Note: When uncertainty is known, disregard the rules below and use uncertainty to determine the number of significant figures that should be shown:
 Round uncertainty to one significant figure.
 Round your value to the same digit (i.e. 54.3 ± 0.2 are both rounded to the tenths place).
Counting significant figures 
Number of sig figs is the number
of digits reported, not including any zeroes to the left of
the first nonzero digit. For example, 3.00 has 3 sig figs,
0.0045 has 2 sig figs and 3.0400 has 5 sig figs 
Ambiguity 
With no decimal point, the number
of significant figures in the number 100,000 is ambiguous.
For numbers without a decimal, zeros to the right of the last
nonzero digit are assumed to be not significant. However,
significance can be indicated in a number of ways:
 Using
a final decimal point. For example, "100,000." would indicate
6 sig figs.
 Underlining the last significant digit. For
example, 100,000 would indicate
1 sig fig, whereas 100,000 would indicate 4 sig figs.
 Scientific
notation.
1.00 x 10^{5} would indicate 3 sig figs.

Precision 
The precision of a number is the decimal place of the furthestright digit. For example, 3.0 is precise to the tenths place. 3.0x10 is precise to the ones place. 
Calculations 
Keep all digits in your
calculator throughout your calculation. Then
go back to determine significant figures at each stage. 
Addition and Subtraction 
Round your answer to the same decimal
place as the number with the least precision. (e.g.
4.1 + 7.00092 = 11.1) Be careful when using scientific
notation. (e.g. 4.1 + 1E3 = 4.1 + 0.001 = 4.1) 
Multiplication and Division 
Round your answer so that it has
the same number of significant figures as the number that has
the
least significant figures. (i.e. 5 * 1.000 = 5) 
Rounding 
If digit to be dropped is > 5(00…) increase last digit by one (round up).
If digit to be dropped is < 5(00…) no change in last digit (round down). 
Rounding 0.5 
If digit to be dropped is exactly 5, with no nonzero numbers afterward, round last digit to the nearest even number. (i.e. 2.5 rounds to 2; 3.5 rounds to 4) 
Click here for sig fig practice worksheet
Uncertainty
Measured value 
x ± AU = x ± δx
Example: 54 ± 2 seconds 
Absolute uncertainty 
AU = δx
Example
from above: AU = 2 seconds 
Relative uncertainty
(Fractional uncertainty) 
Example
from above: RU = 2/54 = 0.04 
Percent uncertainty =
RU x 100 
Example from above: % uncertainty = 0.04
x 100 = 4% 
Standard deviation 
When multiple measurements
are collected and averaged, AU for the average =
standard deviation of
the measured values. In Excel, find the standard deviation
using "=stdev(select cells)" 
Reporting uncertainty 
Uncertainty is always reported
as a positive value, with only one significant figure. Measured
(or calculated) value and uncertainty should be reported with
the same precision. Use the precision of whichever is least
precise.
Example 1: 564.45 ± 2.34 =
564 ± 2
Example 2: 435 ± 0.0434 = 435 ± 0 
Rules for Propagating Uncertainties
Addition and Subtraction 
AU total = SUM of AUs
First calculate AU then calculate RU = AU/q.
Example: (54 ± 2) + (83 ± 3)
= 137 ± 5 or 137 ± 4% 
q = x +....+ y  z .... u
q = 54 + 83 = 137 
δq ≤ δx + Κ δy + δz + Κ + δu
AU < 2 + 3 = 5
RU < 5/137 = 0.04 
Multiplication and Division

RU total = SUM of RUs
First calculate RU then calculate AU = RUq
Example:
(2.7 ± 0.3) x (4.5 ± 0.5)
= 12 ± 3 or 12 ± 20% 
q = 2.7 x 4.5 = 12.15 
RU < 0.3/2.7 + 0.5/4.5 = 0.222
AU < 0.222 x 12.15 = 2.7 
Click here for uncertainty practice
worksheet
Reference: Taylor, John R., An Introduction to Error Analysis 