Minimizing Random Error

In contrast to systematic error, random error can usually be estimated and minimized through statistical analysis of repeated measurements. Many books have been written on statistics and data analysis; for the labs in this class, however, a short summary should be adequate. Make repeated measurements, and find
the sample average and standard deviation (most scientific calculators will do this for you):

where N is the sample size (i.e., the number of measurements). Although there is no single accepted standard, one commonly used way of reporting error, given a sample standard deviation of , is to write . Assuming a normal distribution in the measurements and a sample size of ten or more, this implies that there is a 95% probability that the true value lies within the upper and lower error limits. For example, if you measured the height of a random sample of 25 Cornell students and found of 5'7" and of 4", you would report a student height of 5'7" 8". (In this case, this is the height we would expect if we measured another Cornell student at random, since there is no single true height.)

We can go on to estimate how close , our sample mean, is to the true or population mean. The population mean is what we would get by taking and averaging a huge number of measurements; for example, averaging the heights of all the students at Cornell. We would expect our average of a sample of 25 student heights to be closer to the true average, in general, than any single height measurement would be. However, there is still uncertainty in , and we can estimate this by calculating , the standard deviation of the mean:

For the average student height, inches, and you could report the average Cornell student height as 5'7" 2", or 5'7" 1.6", with 95 percent confidence. In other words, if we repeated our 25-student sampling procedure 20 times, we would expect our error limits to include the true mean 19 times out of the 20 samples, and miss it completely in one case out of 20.

Two pitfalls should be kept in mind when using the standard deviation of the mean: First, although you can make arbitrarily small with enough repetitions of a measurement, remember that it is the precision you are improving. The overall accuracy will be good only if the systematic error present is similarly small. Second, note that refers to the error in measuring the average; whether this is useful depends on the situation. If you are interested in finding out how tall Cornell students are, compared to Stanford students, it makes sense to compare the averages and use . If you are designing a staircase with a low{hanging beam, however, and want to ensure that most students will not bump their heads, you need to use and , which properly characterize the uncertainty in the height of the Cornell students, rather than the uncertainty in the average.

Introduction

Main Body
•Experimental Error

•Minimizing Systematic Error

•Minimizing Random Error

•Propagation of Error

•Significant Figures

Questions