Uncertainty in a single measurement
Bob weighs himself on his bathroom scale. The smallest divisions on the scale are 1pound marks, so the least count of the instrument is 1 pound.
Bob reads his weight as closest to the 142-pound mark. He knows his weight must be
larger than 141.5 pounds (or else it would be closer to the 141-pound mark), but
smaller than 142.5 pounds (or else it would be closer to the 143-pound mark). So
Bob's weight must be
weight = 142 +/- 0.5 pounds
In general, the uncertainty in a single measurement from a single instrument is half
the least count of the instrument.
Fractional and percentage uncertainty
What is the fractional uncertainty in Bob's weight?
uncertainty in weight
fractional uncertainty = -----------------------value for weight
0.5 pounds
= ------------142 pounds
=
0.0035
What is the uncertainty in Bob's weight, expressed as a percentage of his weight?
percentage uncertainty =
=
uncertainty in weight
----------------------- * 100%
value for weight
0.5 pounds
------------ * 100%
142 pounds
=
0.35%
Combining uncertainties in several quantities: adding or subtracting
When one adds or subtracts several measurements together, one simply adds together
the uncertainties to find the uncertainty in the sum.
Dick and Jane are acrobats. Dick is 186 +/- 2 cm tall, and Jane is 147 +/- 3 cm tall. If
Jane stands on top of Dick's head, how far is her head above the ground?
combined height
=
186 cm
=
333 cm
+
147 cm
uncertainty in combined height
combined height
=
2 cm
=
5 cm
=
333 cm
+
3 cm
+/-
5 cm
Now, if all the quantities have roughly the same magnitude and uncertainty -- as in the
example above -- the result makes perfect sense. But if one tries to add together very
different quantities, one ends up with a funny-looking uncertainty. For example,
suppose that Dick balances on his head a flea (ick!) instead of Jane. Using a pair of
calipers, Dick measures the flea to have a height of 0.020 cm +/- 0.003 cm. If we
follow the rules, we find
combined height
uncertainty in combined height
???
combined height
=
186 cm
+
0.020 cm
=
186.020 cm
=
2 cm
+
0.003 cm
=
2.003 cm
=
186.020 cm
+/-
2.003 cm
???
But wait a minute! This doesn't make any sense! If we can't tell exactly where the top
of Dick's head is to within a couple of cm, what difference does it make if the flea is
0.020 cm or 0.021 cm tall? In technical terms, the number of significant figures
required to express the sum of the two heights is far more than either measurement
justifies. In plain English, the uncertainty in Dick's height swamps the uncertainty in
the flea's height; in fact, it swamps the flea's own height completely. A good scientist
would say
combined height
=
186 cm
+/-
2 cm
because anything else is unjustified.
Combining uncertainties in several quantities: multiplying and dividing
When one combines several measurements together, one can often determine the
fractional (or percentage) uncertainty in the final result simply by combining the
uncertainties in the several quantities.
Jane needs to calculate the volume of her pool, so that she knows how much water
she'll need to fill it. She measures the length, width, and height:
length
L
=
=
5.56
5.56 m
+/+/-
0.14 meters
2.5%
width
W
=
=
3.12
3.12 m
+/+/-
0.08 meters
2.6%
depth
D
=
=
2.94
2.94 m
+/+/-
0.11 meters
3.7%
To calculate the volume, she multiplies together the length, width and depth:
volume =
=
L * W * D =
(5.56 m) * (3.12 m) * (2.94 m)
51.00 m3
In this situation, since each measurement enters the calculation as a multiple to the
first power (not squared or cubed), one can find the percentage uncertainty in the
result by adding together the percentage uncertainties in each individual
measurement:
percentage uncertainty in volume =
(percentage uncertainty in L) +
(percentage uncertainty in W) +
(percentage uncertainty in D)
=
2.5 % + 2.6% + 3.7%
=
8.8%
Therefore, the uncertainty in the volume (expressed in cubic meters, rather than a
percentage) is
uncertainty in volume
= (volume) * (percentage uncertainty in volume)
= (51.00 m3) * (8.8%)
= 4.49 m3
Therefore,
volume
=
=
51.00
51.00 m
+/+/-
4.49
8.8%
m3
Is one result consistent with another?
Jane's measurements of her pool's volume yield the result
volume
=
51.00
+/-
4.49
m3
When she asks her neighbor to guess the volume, he replies "52 cubic meters." Are
the two estimates consistent with each other?
In order for two values to be consistent within the uncertainties, one should lie within
the range of the other. Jane's measurements yield a range
51.00 - 4.49
46.51
m3
<
volume
<
m3
<
volume
<
51.00 + 4.49
55.49
m3
m3
The neighbor's value of 52 cubic meters lies within this range, so Jane's estimate and
her neighbor's are consistent within the estimated uncertainty.