# Percent difference formula theoretical experimental

One of the major sources of growing pains my students experience is the treatment of error and percent difference in the lab. This is meant to help you understand what error is and

One of the major sources of growing pains my students experience is the treatment of error and percent difference in the lab. This is meant to help you understand what error is and is not, and what I expect you to be able to do in your lab reports.

## What it is:

First, lets acknowledge that no experiment is done without error and that its not, in itself, a bad thing. Its just something that exists and must be considered. Often, paying attention to it will enliven your thinking about physics and how all the pieces fit together. One of my colleagues describes error as something that, through no fault of the experimenter, may cause the outcome of the experiment to be different next time. I think that about sums it up.

## Percent Difference:

You do not calculate percent error for labs in the algebra-based sequence (PHY185/186 labs for PHY111/112 lecture courses). Percent error develops a quantitative measure of how uncertainty in measurements taken propagates through to your final calculation.  You will do qualitative error analysis  describe it with words instead of numbers. You calculate a percent difference to help, which is just a comparison between your calculated or measured result and a standard. Many of you are frustrated when you ask what the formula for percent difference is and I respond that it depends. However, the (experimental  theoretical) / theoretical x 100 formula youre looking for doesnt always apply, which is why I wont tell it to you. The idea is that you take the difference between two numbers and report that difference relative to something by dividing by the something. If you have an accepted value, like when the object of the lab is to experimentally determine some constant of nature, that formula works. For a lab like Ohms Law, where you calculate one version of a number from measurements and also measure it directly, which is experimental and which is theoretical? They both rely on measured values. What matters is that you report your answer as x % difference relative to (whatever you put in the denominator). Either one works and you may choose whichever value as the comparison as it strengthens your arguments in the error and conclusion sections of the report. Since you ought to be calculating these things before writing those final two sections, Id recommend calculating the percent difference in multiple ways. It doesnt take long and motivates thinking.

## The Error Section:I insist upon you calling it percent difference because when you call it percent error you run the risk of misunderstanding what it represents  simply a gauge of the difference between two numbers. A huge misapprehension that students suffer is that a 0% difference indicates a perfect lab performance where there was no error. This is a fundamental problem. Even if you performed the lab as well as humanly possible, there is still error in play. Everything from uncertainty in a meter value to the generic old equipment many of you love so well is in play for possible error sources. In fact, I dont necessarily care if the error source(s) you propose are major contributors to error in your experiment. What I want you to show me is that you understand how an error source will enter into your results. It simply isnt good enough to say the equipment is old and probably skewed results. This is true for every experiment we do, but it clearly isnt indicative that you understand how the old equipment would impact results. Developing that the old equipment has some zero error, that this serves to push all of your measurements for that quantity higher, explaining how that quantity relates to what youre calculating (usually through a formula or plot), and predicting finally which way (up or down) that error source would serve to push your calculated value is one way to blame the equipment properly. Ideally, youd then connect this to your result. and this is what we saw in our lab, where the value derived from the meter measurement was higher than our standard ofNow, an analogy to help out.

Two competing error sources (Moe and Larry) altering the expected location of a measurement (Curly). Photo credit three stooges.net

lmagine that Moe and Larry, flanking Curly in the photo, are error sources affecting the position of Curlys head. Identifying them is great, but I want you to be able to use what you know about the physical relationships under investigation to predict how their pushing or pulling affects Curly. It looks like Larry and Moe are both pulling on Curly. In this example, if they both pull with the same strength (and Curlys ears stay attached), he may well end up exactly where he ought to be (zero % difference). It doesnt mean theyre not present and pulling. If Larry pulls harder, Curly might end up farther to one side than expected (non-zero % difference from his expected location). Same for Moe. If you discover your experimentally found value to be lower than expected, you should start looking for an error source that explains how this might have happened. Again, the focus is that you display critical thinking to construct a rational argument about how an error source affects changes to a final answer. You may assume if there is no error, youd obtain the expected result (usually a constant or theoretically calculated value).

## What it is not:

As a final note, dont ever identify human error as a source. There is no such animal. Usually these things fall into two categories: you mean to say that you just did the experiment poorly, which is wrong and not error, or you mean to say that the precision of an instrument is an error source. Having a ruler with 1 mm smallest scale division is a fine error source if you develop how drastically the uncertainty might influence results and identify it in those results. Saying your eyes arent good enough to get a good measurement is actually fine as well, but then you should estimate over what range your eyes cant distinguish a difference in measurement and proceed similarly. Strive to write your error sections so a reasonably intelligent person cant ask what does that do? which way? or how do you know?