PHYS 2020 Experimental Error

In conducting an experiment a person encounters one or more of three general types of errors: human error, systematic error, and random error.

Human error (a mistake) occurs when you, the experimenter, make a mistake. Examples would be when you set up your experiment incorrectly, when you misread an instrument, or when you make a mistake in a calculation. Human errors are not a source of experimental error; rather, they are ìexperimenter'sî error. Do not quote human error as a source of experimental error.

Systematic error is an error inherent in the experimental set up which causes the results to be skewed in the same direction every time, i.e., always too large or always too small. One example of systematic error would be trying to measure the fall time of a ping pong ball to determine the acceleration due to gravity. Air resistance would systematically reduce the measured acceleration, producing a systematic error. Some systematic errors can be easily corrected. For example, if a balance reads 0.25 g when there is no mass on it, this would introduce a systematic error to each mass measurementóthey would all be too large by 0.25 g. This can be corrected by zeroing the balance. Other systematic errors can only be eliminated by using a different experimental setup. Most of the simple experiments you do will have some systematic error.

All experiments have random error, which occurs because no measurement can be made with infinite precision. Random errors will cause a series of measurements to be sometimes too large and sometimes too small. An example of random error could be when making timings with a stopwatch. Sometimes you may stop the watch too soon, sometimes too late. Either case introduces random error in your measurements. (Note that when a human is involved in the actual measurement process, he/she can introduce valid experimental error that is not within the definition of human error. Your finite reaction time is not a mistake; it is a limitation of one part of the experimental process, the human making the measurement.) Random error can be reduced by averaging several measurements.

Error Analysis

One way to analyze experimental error is with a % error calculation. The % error is useful when you have a single experimental result that you wish to compare with a standard value, or when you have two experimental values obtained by different means that you wish to compare. (In the latter case it is often called % difference since there is no standard to compare to.)

The % error is calculated according to the following formula. In the formula, 

         expt# - std#

%error = ------------ x 100%

             std#
"expt. #" is your experimental value, and "std. #" is the standard or reference value. Using this formula, a positive % error tells you that your result was larger than the standard, while a negative result implies an experimental result smaller than the standard. While % error tells you the relative size of your error, it gives you no clue as to the type of that error (random error or systematic error).

In certain cases one can use a statistical quantity called the standard deviation, usually denoted by the lower case Greek letter sigma, σ, or the abbreviation "std. dev.," to tell whether the error is systematic or random. Basically, σ tells you how random or scattered your data is. Your calculator should be able to calculate the standard deviation of a set of numbers; learn how to use this feature. You can consult any elementary text on statistics for the formula for standard deviation. Note that one should use the ìsampleî standard deviation and not the ìpopulationî standard deviation. Suppose you make a series of measurements of the force on two springs vs. their displacement, which can give you the spring constant k since F = -kx. You find that k is 11.7 N/m with a s of 0.31 N/m for one spring and 25 N/m with a σ of 15 N/m for the second. What does that tell you? First, it tells you the scatter in your data. A σ that is a significant fraction of the result (in this case k) implies a large scatter in the data, and a small σ implies the data has a small scatter. In our examples we had a very small scatter in our data for the first spring since 0.31 N/m is only 2.6% of 11.7 N/m. However, for the second spring there was a lot of scatter; 15 N/m is 60% of 25 N/m. Small σ's, relative to the measured number, imply small random errors, while large σ's imply large random errors. You can calculate the relative size (percentage) of σ much like you calculate % error: 

       σ

% = ------- x 100%

     expt.#
We can also use σ to compare with a standard value. Suppose we knew that the spring constant for the first spring was actually 10.0 N/m (a 17% error) and 35 N/m (a -29% error) for the second spring. One way to present your data is with the number ± σ: 11.7 ± 0.31 N/m for the first spring and 25 ± 15 N/m for the second. If the standard value lies roughly within one or two standard deviations of the experimental result, your experimental error is mainly random error. If the standard value is more than two or three σ away from your result, the error is mainly systematic. Specifically, from the theory of statistics, if the standard value is one σ from your value, there is only a 68.3% chance the error is systematic rather than random. If it is 2σ away, there is a 95.4% chance the error is systematic, and if it is 3σ away the chance is 99.7%. You can determine how many s difference there is between the experimental and standard values using this formula: 

         expt# - std#

# of σ = ------------ 

             σ
For our example, the standard value of k for the first spring is more than 5σ away (5 times 0.31 N/m), so the error is most assuredly systematic, while for the second spring, the standard value is within one σ (25 ± 15 N/m), implying random error in k. In both cases the rather large % error tells us the overall error is large.

We can summarize:
For our two examples, there is a large, mainly systematic error in the k for the first spring; there is a larger, mainly random error in the k for the second spring.

So when you have a series of measurements and can perform error analysis including % error and σ, you can gain insight into the nature of your experimental errors. Then, when you are giving sources of error in the conclusion of your lab report, you can state sources consistent with your error analysis. For example, with the k for the first spring with its large systematic error, you would need to think of sources of systematic error that would always produce values of k that were too large. Likewise, with the k for the second spring having large, random error, you would need to think of sources of random error that could produce large discrepancies from the expected value.