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.