CHAPTER 3

A general discussion of how measurement errors should be determined is given in Appendix B. There are basically two kinds of errors that enter scientific measurements: 1) Systematic errors and 2) Random errors. With careful planning of your experiment, the systematic errors can be minimized or avoided. Random errors, on the other hand, are always part of any measurement.

Taking Precautions to Avoid Systematic Errors

Did you measure the length of the spring after the experiment with no weight to be certain that the length was the same after hanging all those weights on the spring? That is, are you certain that during the measurements you did not introduce a distortion by permanently stretching the spring out of shape. Always be on the alert by taking such precautions, because systematic errors can be introduced into your measurements, if you fail to notice things like stretching the Slinky permanently out of shape. Systematic errors are usually difficult to correct for, and may mean that you need to re-design your experiment to avoid such pitfalls. What might cause your Slinky to be permanently distended?

What about the Spring Scale? Did you check its readings with no load on the Slinky both before and after you made your measurements? The spring in the spring scale is also subject to permanent distortions. You should develop a keen eye for such potential problems, and be careful and thorough in your experiment planning and procedures to avoid systematic errors.

If your Spring Scale does not read "zero" with no load, how could you remedy this? Do you see a way that you might be able to adjust the zero point or "recalibrate the zero point" of your Spring Scale? There is a plastic 'nut' at the top of your Spring Scale that can be turned to adjust the reading on your Spring Scale to read "zero" when nothing is suspended from the scale. Adjustment of the scale may be done at any time before or after an experiment, but never while the experiment is in progress. Take good care of your Spring Scale. You will be using it to make a variety of measurement during the next few weeks.

Random Errors in Measurements

When you made your three readings of the position of the lower end of the Slinky, did you get the same numbers? When you recorded three readings of the spring scale, were are the measurements the same? If you had variations in the measurements what was the cause. Because nothing was changed in the experiment, and only your reading of the meter stick or the spring scale changed as you looked away and then back again, and probably moved the position of your head. Or perhaps your lab partner made one or more of the readings. In any event chances are that not all three readings were the same because your eye (or eye + head position) led to different scale readings. In the case of the spring scale, head position matters because of a "parallax" effect. This effect gives different readings because your eye level is not exactly horizontal to the blue disk inside the spring scale. Consequently an error is introduced into your measurements. In the case of differences in the meter stick readings, these are likely caused simply by differences in your judgment about where the lower end of the Slinky lined up with the scale on the meter stick.

Errors in reading scales are unavoidable and are usually random. That is, if you made sufficient numbers of readings, 50 say, then the average of all the readings should be a very good representation of the actual reading that you would get if there were no human error in scale readings. How much each single reading differs from the average gives a good measure of the error (or the accuracy) of a single reading. The accuracy of a measurement is as important in science as the measurement itself. In fact, a measurement is essentially useless if we do not know its accuracy.

Therefore, always make more than one measurement in an experiment, and always make an estimate of the accuracy of a single measurement. In the case of the Slinky, repeated measurements of the meter stick for the second entry in column two above were, 63, 61 and 62 cm. Therefore the average value was calculated by summing the three numbers and dividing by the number of measurements (3 in this case). Alternatively, simple examination of the three numbers indicates that 62 cm must be the average. Likewise to find the error of a single measurement, we could go through a formal calculation. Or since there are only three measurements, by simply inspecting how each measurement differs from the average, 62 cm, we could come to the conclusion that our estimate of the error of a single reading of the meter stick was about ± 1cm. Note that no decimal is recorded, because no measurement of the fraction of a cm was made. In other words the measurements were made to two significant figures, where "6" and "2" are the significant figures.

For the three individual scale readings of the spring scale for the second entry in column three in the Data Table above, we found, 0.69, 0.71 and 0.70 Newtons. The average value entered in the Table above is therefore 0.70 Newtons. In this case an estimate was made of the scale readings to the second significant figure (since the zero before the decimal does not count unless there are numbers to the left). The two significant figures in the average are "7" and the "0" to the right of the seven. Note that it is important to record this zero, since it indicates that the average has two significant figures. What is the error in the spring scale reading? Again we only need to examine the three individual readings and compare each with the average to decide that our error estimate is ± 0.1 Newton.

Therefore we estimate that the error in a single measurement of the Slinky position is ± 1 cm, and for each measurement of the scale that gave the amount of pull exerted by the coins our estimate of the error is ± 0.1 Newton. The Data Table could therefore have been given with the errors included, and would convey much more information about the experiment. Why is this so?

Data Table: Sample Measurements and Errors of Slinky Jr.

No. of Pennies	Position of Lower End of Slinky Jr. (cm)	Pull of Slinky on Spring Scale (Newtons)	Displacement from Initial Position (cm)	Additional Pull on Slinky from Initial Position (Newtons)
0	48± 1	0.60± 0.10	0.0	0.0
2	55± 1	0.65± 0.10	7.0	0.05
4
6
8

Once you have analyzed your data for measurement errors, you are prepared to analyze your data to see what results you have found from the experiment. Your analysis of the data can take many forms. When data are collected, we usually wish to graph two variables to see whether they are correlated. If they are, what mathematical relationship can best describe the two variables?

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Last modified 9 Aug 1997
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