Engineering 194.06

Laboratory Exercise 2

An Introduction to Experimental Error

 

The concept of error is inherent to any experimental process.  In fact, it is a part of nature.  This brief introduction will present the concept of experimental error, explain why it is important to us as engineers, and illustrate some of the ways that we deal with error as we encounter it.

 

·        So what is error?

Most of us have grown up with the idea that the word “error” invariably means that someone made a mistake, got a wrong answer on an exam, etc. When scientists, engineers, and statisticians speak of experimental error, however, they are not referring to a mistake of any kind.  Errors in scientific measurements are simply unavoidable uncertainties.  As technically sophisticated people, we recognize that perfect measurements are simply not attainable, and we find methods of dealing with this fact.

 

There can be many sources of error.  For example, if you use a stopwatch to time a worker performing a task, you introduce an error in the time measurement because it takes a fraction of a second for you to hit the start and stop buttons.  There is no way to exactly account for that small deviation from the “true” elapsed time, and so an error is present in the reading. The same applies for the experiments we are doing in the labs.

 

·        If error is inherent to all measurements, why do we care about it?

It’s true that the presence of error is a given.  What becomes important, then, is to try to quantify that error.  That is, we need to know how large the error in our measurement is so that we can use the measurement with some degree of confidence. Another reason for estimating our “normal error” is that when a mistake does occur, we can spot it much easier, because it will lie outside the range of errors we expect to see.

 

·        How can we get a handle on error?

Simply measuring a quantity once tells us little about the error associated with that measurement.  The best way to get a handle on error is to take a series of measurements (a sample)and then calculate a mean and a standard deviation based on that sample.

 

For the population, for the mean, and s for the standard deviation.

 

The mean, represented by the Greek letter m, or mu,  is our “best guess” of the true value of the quantity we’re measuring.  It’s just the arithmetic average of all the measurements:


 

where N is the total number of measurements. 

 

The sample standard deviation, represented by the Greek letter s, or sigma,  is the actual estimate of the error.  It measures the average difference between our observed values and our best guess of the mean.  The standard deviation is calculated from the sample using the formula:

 


 

It can be shown that, in most cases, measurements subject to random errors have a normal distribution.

 

·        What is the normal distribution?

The normal, or Gaussian, distribution is represented by a bell-shaped curve as shown below.  The value of the quantity of interest is shown on the x-axis, and the probability of observing that value in a measurement situation is shown on the y-axis.  Note that the highest probability corresponds to the mean and that the probabilities are symmetric about this mean value.

 

The normal distribution occurs very often in nature – in fact, your grades in this class will likely be normally distributed! It also has some interesting characteristics. Regardless of the value of N, m, or s, about 68% of your observations will fall within one standard deviation of the mean, as shown on the graph below.  Ninety-five percent will fall within two standard deviations, and 99.7% will fall within 3 standard deviations.

 

 

 

We only have time to examine the normal distribution very briefly here. To learn more, check out the following website, courtesy of Stanford University:

 

http://www-stat.stanford.edu/~naras/jsm/NormalDensity/NormalDensity.html