When conducting an experiment, often many 
variables are recorded to calculate a resultant.
For example, to calculate the speed of a car, 
the distance it travels 
and the time it took to travel that distance 
are recorded.
Consider that the time was measured 
using a stopwatch that is accurate to 
1/10th of a second, and that the distance 
was measured using a steel tape.
The velocity can be written as a function of 
the time and distance. 
More specifically, the velocity will be equal to 
the distance divided by the time. 
These measurements, like any measurement, 
have an amount of uncertainty. 
The stopwatch may record up to the 
1/10th of a second, 
but it's uncertainty could be plus or minus 
2/10ths of a second. 
The recorded time was 10.7 seconds, 
but the actual time could have been anywhere 
from 10.5 to 10.9 seconds. 
This is not due to avoidable error, such as the 
reaction time of pressing the button, 
but rather from physical equipment limtations, 
such as the quality of manufacturing the clock. 
Uncertainty can be given within a word problem, 
or more commonly as a value that is added to or 
subtracted from the measurement, as shown. 
All experimental measurements have 
uncertainty, though it is not always shown.
The velocity, calculated from these 
measurements,
will also have an amount of uncertainty. 
The uncertainty of a resultant is a function of the 
uncertainties of all the measurements 
used to calculate it. Uncertainty is represented 
with the lowercase greek letter sigma.
Let's look at the uncertainty of the velocity.
It is equal to the root sum of the squares, 
or RSS, of a contribution from
the uncertainty of the distance, and a 
contribution from the uncertainty of the time.
Let's take a closer look at one of these 
"contributions."
Sigma sub d is the uncertainty of our distance 
measurement.
This other term indicates to take the partial 
derivative of the velocity, V,
with respect to the distance, d. Recall the 
equation for the velocity.
To take the partial derivative, treat all variables 
other than the variable of interest, d, 
as if they were constants, and take the 
derivative with d as the variable.
The partial derivative with respect to the time, t, 
is also needed.
Because the time is in the denominator, its 
partial derivative is a bit more complex.
Now the uncertainty of the velocity can be 
calculated.
Recall the partial derivatives and substitute them 
in. 
Recall the time, distance, and their 
uncertainties; substitute these values in as well.
Based on the two contributing factors seen here 
in the RSS, which of the two measurements
has a larger effect on the uncertainty of the 
velocity? If more accuracy is needed
for the velocity, which measurement needs to be 
more accurate?
Note that the uncertainty of the velocity can be 
calculated independently of the velocity itself;
calculate the velocity and report the solution in 
the proper format.
Many of these uncertainty calculation equations  
have been shown specifically for this example. 
Let's expand these equations to include any 
general case.
The resultant, R, is a function of variables 
x sub 1 through x sub n. 
This may be any function, with any variables.
The uncertainty of R is a function of the 
uncertainties of all of those variables.
This function is always: the uncertainty 
of R is equal to
the RSS of the partial derivative of each variable, 
multiplied by the variables' uncertainty.
If one of the variables used to calculate the 
resultant is itself calculated from 
another measurement or measurements
(y sub i), then a "local" uncertainty analysis 
must be done on that variable, x sub i 
(which can be any of the variables x sub 1 
through x sub n)
to find its uncertainty before solving for R.
Lastly, there is another way to report 
uncertainty; 
relative uncertainty, or U, is a percentage. It is 
equal to the uncertainty of R divided by R. 
This indicates the accuracy of the observation 
(experimental measurements), 
scaled to the resultant. 
For example, an uncertainty of 0.5 m over a 
distance of 60 m indicates far more accuracy 
than an uncertainty 0.2 seconds over a time of 
10.7 seconds.
Let's rearrange the equation for the velocity 
uncertainty.
The partial derivative of V with respect to d, 
or 1 over t, can be multiplied by "1" (d over d)
and then simplified by substituting V in for d/t. 
The partial derivative for t can also be simplified.
With these substitutions, 
the relative uncertainties of the distance and 
time become a part of the equation, 
which makes it easier to visualize which 
measurements affect the resultant the "most." 
Manipulating a partial derivative to include the 
resultant can also simplify 
the equation significantly, especially if the 
resultant is calculated 
from many measurements or variables.
