In view of the fact that this question is being asked again and again on this forum, and to save me time posting the same answer again and again, this is a summary of how you combine uncertainties
at A Level.*
You have two values, each with an absolute ± uncertainty.1. If you add or subtract the two (or more) values to get a final value
The absolute uncertainty in the final value is the sum of the uncertainties.
eg.
5.0 ± 0.1 mm + 2.0 ± 0.1 mm = 7.0 ± 0.2 mm
5.0 ± 0.1 mm - 2.0 ± 0.1 mm = 3.0 ± 0.2 mm
2. If you multiply one value with absolute uncertainty by a constantThe absolute uncertainty is also multiplied by the same constant.
eg.
2 x (5.0 ± 0.1 mm ) = 10.0 ± 0.2 mm
The constant can be any number. eg Pi
3. If you multiply or divide two (or more) values, each with an uncertainty
You add the % uncertainties in the two values to get the % uncertainty in the final value.
eg
5.0 ± 0.1 mm
x 2.0 ± 0.1 mm
This is
5.0 ±
2% x 2.0 ±
5%
Result
10.0 ±
7%
This is 10.0 ±
0.7 mm
2(0.7 is 7% of 10.0)
4. If you square a valueYou multiply the % uncertainty by 2
If you cube a value you multiply the % uncertainty by 3
etc
If you need the
square root of a value, you
divide the % uncertainty by 2.
This is because square root in index form is to the power ½
√x = x
½The general rule isMultiply the
% uncertainty by the
index.
What happens to % uncertainty when I multiply by a constant?The
% uncertainty doesn't change. The
absolute uncertainty is multiplied by the constant. (see 2 above)
In the example given above we multiplied 5.0 ± 0.1 by a constant, 2
2 x (5.0 ± 0.1 mm ) = 10.0 ± 0.2 mm
The absolute uncertainty is multiplied by 2.
The original % uncertainty was 5.0 ± 2%
In the final value of 10.0 ± 0.2 mm
the % uncertainty is
still 2%
Note: This is consistent with 3. above.
When you multiply a value by a constant, it is assumed the constant has no uncertainty. We do not associate an uncertainty with the value of Pi or the number 2, for example. So they have a % uncertainty of
zero.
So when you multiply the value by the constant and add the % uncertainties, there is only the % uncertainty in the value itself and zero in the constant. Result: no change in % uncertainty.
What if the formula I use to calculate my final value has both adding and multiplication/division?Let's take an example. Assume you have all the uncertainties in the values in the formula and we want the uncertainty in
ss = ut + ½at²
Step 1.
Add the % uncertainties in
u and
t to find the % uncertainty in
utStep 2.
Multiply the % uncertainty in
t by 2 (Rule 4 above) and add it to the % uncertainty in
a to find the % uncertainty in
½at² (The constant ½ has no uncertainty)
Step 3.
Convert those % uncertainties to absolute uncertainties in
ut and in
½at²Step 4.
Add the absolute ± uncertainties in
ut and
½at² found in 3. above to get the absolute uncertainty in the final value of
s*If you would like a more advanced treatment of this topic I recommend the following.
http://www.rit.edu/cos/uphysics/uncertainties/Uncertaintiespart1.html#systematic.