Total Percent Uncertainty

Total percent uncertainty = The sum of all percent uncertainties

         After calculating the percent uncertainty of all of your measurements it is important to determine the total percent uncertainty.   The purpose of the total percent uncertainty is determine how large is the "region of error" in your experiment due to the accuracy of measuring equipment.   Having a large percent uncertainty just means that given the equipment at hand this is how close to the theoretical value (or in the case of percent difference, how close to all other measured values) you can get.  

        To calculate the total percent uncertainty there are two methods.  The first method is when adding like measurements together.  An example of this type of total uncertainty if you measured the length of a track, and measured the length of a cart moving along the track.  Both are measurements of length.   The total uncertainty is 

(The uncertainty of the track measurement) + (the uncertainty of the cart measurement) / (length of the cart + length of the track)  x 100% 

It is important to realize that it's the uncertainty (the plus or minus value) used and not the percent uncertainty 

       This type of uncertainty is relatively uncommon, it requires the two measurements to added together to get the one total measurement, like one length plus another length.  This is NOT like making a measurement of volume or area where the length measurements are multiplied together, so the first method would work for total distance or displacement but not area.

The second method is a far more common method of calculating total uncertainty.   When dealing with products of measurement we add the percent uncertainty together.  This means in calculating the percent uncertainty of a volume

(percent uncertainty in the height)+(percent uncertainty in the length)+(percent uncertainty in the width)= total percent uncertainty

So as an example if the uncertainty in the measurements in length, height, and width is 1%, 3%, and 5% respectively the total uncertainty would be

1% + 3% + 5% or a total of 9% 

A better example of total uncertainty is the "falling down the incline" lab.  The lab is determine the acceleration of a cart as it moves down an incline.  The only measurements made in the experiment is are the displacement of the car and the time it takes for cart to reach the bottom of the incline.  Using the equations from kinematics to determine acceleration we get 

a= 2 d / t^2 

with d = displacement 

t = time 

a = acceleration

breaking the equation down in to parts 

a = 2  x  d  x  1/t  x  1/t 

Now consider that the percent uncertainty for time is 5% and the displacement is 3%  so the total uncertainty is 

(total percent uncertainty in acceleration)  =  

(percent uncertainty in 2)+(percent uncertainty in displacement)+(percent uncertainty in time)+(percent  uncertainty in time) 

(we don't care if it's time or 1/time it's the same percent)

Total percent uncertainty = (0%, 2 is 2 no uncertainty) + (3%) + (5%) + (5%)   or 13%