Percent difference

Percent Difference is the numerical interpretation of comparing two values with one another.

The general requirement for selecting two values to be compared is that the user of this technique expects the two values to be numerically equivalent. In other words, obtaining a percent difference of 0% is the optimum result as it explains that the two values are exactly the same. Not a general requirement, but common use shows that the two values usually will pertain to the same property of an object (let's say the mass of an object, a material's characteristic, or maybe the discharging time of a capacitor), but each value will be "calculated" using two different methods and/or theories. Emphasis must be made on the word "calculated", because the most important requirement for the two values that are being compared using Percent Difference is that they had needed to be calculated indirectly of measurement of the objective value. In other words, neither of the two values can be the actual or accepted value of the objective value.

The two values are determined using theories and measurements that are to be tested with respect to an accepted value by the scientific community. Example is someone measuring the length and period of a pendulum to determine the acceleration of gravity, but must relate the value to the accepted value of gravity by the scientific community.

Percent difference is similar to another comparison technique called percent error (note that a more informative result comes from a non-absolute function, this will be discussed later), which is when one determines an experimental value and is comparing it to the accepted or actual value. Percent difference is "different" in that neither of the two values are the accepted or actual value; they are both an experimental value determined by two different techniques but describing the same objective value.

Formula

The equation for determining the Percent Difference by comparing values x₁ and x₂ is:

$\%Diff\; =\; cfrac\{left(cfrac\{x\_1+x\_2\}\{2\}\; ight)\}\; imes100$

In sentence form, one is dividing the absolute difference of the two values by the average value of x₁ and x₂. Because this equation contains the absolute function, percent difference will "always" be positive and therefore it does not matter which value one assigns to the "dummy" variables (x₁ and x₂) used in these equations shown in this article. An easier form of the equation can be calculated as,

$\%Diff\; =\; cfrac\{2\; imes|x\_1-x\_2\{x\_1+x\_2\}\; imes100$

It is important to note that both values (x₁ and x₂) must contain the same units in order to be compared correctly with one another. And as mentioned before, a zero percent difference is optimum and the higher the percent value, the less precision of the two values.

Lastly, percent difference is unitless; however, in the two forms given above, the calculated value must be quoted as a percentage. One may quote the difference without using percent by not multiplying the fraction by 100,

$Diff\; =\; cfrac\{left(cfrac\{x\_1+x\_2\}\{2\}\; ight)\}$

One final note to make is that a lot of confusion lies in mistakenly assuming that percent difference is the same as percent error. The difference is that percent difference is comparing two experimental values, whereas percent error compares one experimental value with the actual/accepted value.

Percent Error

It seems the general standard of calculating the Percent Error involves using the absolute function imposed on the difference between the experimental (measured) and accepted (actual) values. However, this removes detail from the result in the form of only producing a positive percent error value. It should be suggested to ignore the absolute function and calculate the percent error as follows,

$\%Error\; =\; cfrac\{Exp-Acpt\}\{Acpt\}\; imes100$

It is "very" important to note that the numerator should be the Experimental value minus the Accepted value and "not" the other way around. By using the equation shown above, the result will be positive only when the experimental value is greater than the accepted and the result will be negative only when the experimental value is less than the accepted.

This is a very important outcome. By avoiding the absolute function when calculating for the percent error, the results will give both the reader and author more information. If the percent error is negative, the reader knows immediately that the experimental value is short of the accepted (goal) value. If the percent error is positive, the reader knows that the experimental value is above the accepted (goal) value. This technique of solving the percent error value becomes very helpful whenever an accepted value imposes a lower or upper limit for all experimental (measured) values.

A rough example would be the goal to determine the speed of light. If an experiment produced a speed that is greater than the speed of light, the reader will know immediately from a positive percent error that something is wrong. The actual value will place the "no greater than" limit on all measured values. Therefore, only negative percent errors should be expected! Using the absolute function will hide this insight and important information and could be devastating to relevant experiments.

References

* cite web
last =
first =
authorlink =
coauthors =
title = Understanding Graphing and Measurement
work =
publisher = North Carolina State University
date =
url = http://www.physics.ncsu.edu/courses/pylabs/und._meas_&_graphing.pdf
format =
doi =
accessdate = 2007-03-27
* cite web
last = Hester
first = Jerry
coauthors =
title = Physics Tutorial: %Error and %Difference
work =
publisher = Clemson University
date = 2006-01-27
url = http://phoenix.phys.clemson.edu/tutorials/error/index.html
format =
doi =
accessdate = 2007-09-24

ee also

*Relative difference