How to combine uncertainties in different quantities
One way to answer the question
How many strides does it take to walk the Appalachian Trail
is to estimate two quantities:
the length of the trail, L
the length of one stride, d
and then divide:
L
number of strides N = -----
d
Now, each estimate will have some uncertainty;
for example, one group may guess
L = 1200 km +/- 100 km
d = 68 cm +/- 3 cm
so
N = 1,765,706 strides (but how many digits are justified?)
How can we incorporate the uncertainties in each of our two
quantities to figure out the uncertainty in the number of strides?
In this particular case (division), there is a relatively simple
rule: add the fractional uncertainty in each value to find the
fractional uncertainty in the result.
ΔL 100 km
fractional uncertainty ------ = ------- = 0.083
L 1200 km
Δd 3 cm
fractional uncertainty ----- = ------- = 0.044
d 68 cm
so ...
ΔN
fractional uncertainty ----- = 0.083 + 0.044 = 0.127
N
One can also think of this procedure in terms of percentages:
"The length of the trail is known to 8 percent, and the length
of a stride to 4 percent, so the number of strides will
be known to (8+4) = 12 percent."
Okay, now time for you to finish this example.
What is the proper uncertainty ΔN in the number of strides
needed to walk the trail, given these estimates?
Write the number of strides N with the proper
number of significant figures.
You can find a great deal more information on
combining quantities by reading
