Project Issue: Measurement and Uncertainty

Measurement

In this project, you will be dealing with issues and ideas that relate to measurement and the precision involved with making measurements. You will have a variety of measurement devices that you may choose and find that the different devices measure to different precisions. The measuring device you choose when making a measurement is determined somewhat by practicality. For example, you wouldn't use a meterstick to measure the thickness of a hair anymore than you would use a micrometer to measure the distance between Chicago and New York. In general, a measuring device (eg. meterstick, balance, stopwatch, etc.) can be read to an accuracy of 1/2 of the smallest increment. So, a meterstick can generally be used to measure lengths to an accuracy of 0.5 mm. If you want to measure a length that is about 0.5 mm, a meterstick would not be a good choice of measurement device. Always choose the right tool for the job.

In addition to practicality, there may be other concerns when choosing a measurement device. You might need to measure a quantity (such as a length or time interval) to a certain degree of precision. Always keep in mind that any result calculated from measured quantities can be no more precise than the least precise measurement along the way. For example, if you want to know the volume of a block of wood and you measure the lengths of the sides to 1 significant figure accuracy, then the volume of the block you compute can be no more accurate than one significant figure (even though your calculator might give you eight or more digits).

Lastly, just because a measurement device measures a quantity to a greater precision does not necessarily mean that you can measure something to a greater accuracy with it. Suppose you were to measure the thickness of a page in your book. In general, a micrometer will measure to the nearest 0.01 mm while a vernier caliper will measure to the nearest 0.1 mm. This makes the micrometer the more precise of the two measuring devices. You might understandably think that it would be more accurate to use the micrometer than the calipers. If you measure the thickness of a page in your book with a micrometer, you can measure to 1 place accuracy. If you measure the thickness of the book (excluding the covers) and divide by the number of pages in the book, you can get the thickness of a page to three significant figures.

When in an industrial environment, particularly a production line type atmosphere, one is extremely interested in issues related to quality control. Before tolerances can be set on measurement of a part for determining the quality of the part, one must know the uncertainty in the measurement that is introduced by the measuring instrument. That is, you must know the limits of your measurement device and the person(s) using that device. A statistical analysis called gage capability is used widely in industry today in order to measure the ability of a piece of equipment to measure a particular quantity. This analysis takes into consideration the repeatability (part to part variation) and reproducibility (operator to operator variation) of measurements made with a specific device. Basic to this type of statistical process control analysis is the idea of error (or uncertainty).


Types of Error

In physics lab this term, you are going to spend some time in nearly every lab focusing on the amount of error introduced with each measurement, and how those errors combine to effect your final results in the lab. With this in mind, it is instructive to discuss the various types of error in experiment and how to deal with each.

Illegitimate Error - Mistakes and blunders, such as misreading a ruler or copying down a wrong number.

Systematic Error - A reproducible error that biases the data in a given direction, such as using a timer that is slow, so that all the times measured are slow by the same factor.

Random Error - Fluctuation in the results of a measurement when the measurement is repeated, such as measuring the distance which you moved a mirror three times, obtaining the values 2.145 cm, 2.143 cm, and 2.142 cm. Random error is not the fault of you or the equipment.

Dealing with error calls for integrity. If you have performed an experiment, then you are the only person (or group of persons) that knows what happened during the given experiment. It is your responsibility to report it accurately so as not to confuse yourself, and so that you do not mislead others. You should never report or use a measurement that you know or suspect has illegitimate error. Instead, the measurement or measurements should be repeated in order to obtain more reliable data before any subsequent measurement or analysis takes place.

In cases of systematic error, where a faulty piece of equipment was the source of the error each time it was used, the data can be corrected. In order to do this honestly, the error must be reproducible. This does not give you free license to use fudge factors. (such as "Gee, all my data are off from what I should be getting, I will just correct for it here".) Any adjustment to your data must be detailed and well explained. The faulty piece of equipment must be found and a reliable piece of equipment used to measure accurately any error introduced.

Even when all of the other types of error have been eliminated, there will still be fluctuations in our data. We assume that these errors are random in that they are not correlated to one another, any faulty piece of equipment, or a faulty operator. We will discuss in detail the ways of estimating random error and the ways these errors combine in a later section of this lab.

We need to quantify the concept of observed value and error. We call D x the absolute error in the measurement of a quantity x. There are three ways of reporting the error; as an absolute error, a relative error, and a percent error.

The absolute error places bounds telling you that the actual value is expected to fall between x - D x and x + D x. The absolute error is an easy way to report the error and makes it easy to check that you are not reporting too many significant figures. For example, suppose you had a measurement that was reported to be 10.02 ± 0.3 m. Because the reported value is uncertain in the tenths place, it is wrong to report it to the hundredths place. It should be reported to the same decimal place as the absolute error: 10.0 ± 0.3 m. Another point is that the absolute error and the reported value must have the same units.

While it is hard to beat the simplicity of the absolute error, it is lacking in that it does not give you an indication of how large an error is. If you had an absolute error of 0.5 m in measuring a 2.0 m bookcase, that would probably not be tolerable. The same absolute error of 0.5 m might be very acceptable if you were measuring the distance to the moon. Relative error and percent error report a sense of how "good" a measurement is.