Uncertainties in calculations
Addition and subtraction
Multiplication and division
Exponents
(if the power is negative, discard the negative sign for the uncertainty calculation)
Exponents are simply another way of writing multiplication and division.
For example v^3 = v × v × v and v^3 = ((1 ÷ v) ÷ v) ÷ v
Raising a quantity to the power of 3 will triple the percentage uncertainty.
Square rooting a quantity is equivalent to raising it to the power of 0.5, the percentage uncertainty therefore halves.
Zero uncertainty
Worked example
A rock falls from a bridge into the water. It is initially at rest, it accelerates downwards at 9.80 ± 0.02 ms^{2}, and takes 1.8 ± 0.1 s to hit the water. How far does it fall?
The relevant equation is d = v_{i}t + ½at^{2}
Using the values already given in the question, an incomplete table can be drawn up:
Quantity 
Value 
Absolute Uncertainty 
Percentage Uncertainty 
v_{i} 
0 ms^{1} 
0 ms^{1} 
0 % 
a 
9.80 ms^{2} 
0.02 ms^{2} 

t 
1.8 s 
0.1 s 

t^{2} 

½at^{2} 

v_{i}t 

d 
Quantity 
Value 
Absolute Uncertainty 
Percentage Uncertainty 
v_{i} 
0 ms^{1} 
0 ms^{1} 
0 % 
a 
9.80 ms^{2} 
0.02 ms^{2} 
0.204 % 
t 
1.8 s 
0.1 s 
5.55 % 
t^{2} 
3.2 s^{2} 
0.36 s^{2} 
11.11 % 
½at^{2} 
16 m 
1.8 m 
11.31 % 
v_{i}t 
0 m 
0 m 
5.55 % 
d = v_{i}t + ½at^{2} 
16 m 
2 m 
11 % 
The percentage uncertainties in a and t are respectively 0.204% and 5.55%.
The percentage uncertainty in t^{2} is 2 × the percentage uncertainty in t: 2 × 5.55 = 11.11%. So t^{2} = 3.24 ± 0.36
t^{2} = 3.2 ± 0.4 s^{2}
The percentage uncertainty in ½at^{2} is 0 + 0.204 + 11.11 = 11.31%.
So ½at^{2} = 15.876 ± 1.8
½at^{2} = 16 ± 2
The percentage uncertainty in v_{i}t is the sum of the percentage uncertainties of v_{i} and t: 0 + 5.55 = 5.55%
So v_{i}t = 0 ± 5.55%
v_{i}t = 0 ± 0 m
The absolute uncertainty in d is the sum of the absolute uncertainties of v_{i}t and ½at^{2}: 0 + 1.8 = 1.8
So d = 15.876 ± 1.8
d = 16 ± 2 m
The length of an object is found on an ordinary 30 cm ruler.
The ruler is marked at every millimetre (0.1cm), the uncertainty in each of our measurements is therefore half a millimetre (0.05cm).
The length of the object is found by making two measurements and finding the difference between them:
 The first measurement is made when we align one end of the object to 0.00cm on the ruler.
 The second measurement is made where the other end of the object aligns to on the ruler. In this case it is 23.40cm.
To find the length of the object with uncertainty, we must calculate the difference between these two measurements.
Length = (23.40 ± 0.05) cm  (0.00 ± 0.05) cm
= (23.40  0.00 ± 0.05 + 0.05) cm
= (23.40 ± 0.1) cm
= (23.4 ± 0.1) cm