How to Analyze a "Measure the Value of ..." Experiment

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Many physics experiments have the goal of determining a single, constant value - free-fall acceleration, g, for instance - and then comparing the experimental result with some "accepted value". In a given run of such an experiment, we would reasonably expect to make several, if not many, trials. Our best estimate of the value in question (for this run) would be the mean value of all the trials, and a reasonable estimate of the uncertainty of our best estimate would be the standard deviation of the mean.

So, does our experimentally-obtained value match the expected value, or not? This is a difficult and controversial question to answer definitely (and I'm sure that what I am going to tell you will get me a few snotty emails) but here is the idea:

We will assume that our data corresponds to a Gaussian distribution, which is not generally an unreasonable assumption. Certainly, then, if the expected value is within 1 standard deviation (of the mean) of our estimate we have good, if not excellent, agreement. However, about a third of all data values will reasonably be outside this range, so if the expected value is not within one standard deviation of our estimate it is still reasonably possible that they represent the same quantity. About 95% of all data values will fall within 2 standard deviations of the mean, so if the expected value is within 2 standard deviations of our best estimate, it is certainly possible that they are the same value. On the other hand, if the expected value is more than three standard deviations from our best estimate, it is highly unlikely that they are from the same data set.

The number of standard deviations from the mean is called a "t score", and it is calculated as follows:

The t-score is, then, the number of standard deviations separating your experimental value and the "expected" (or "accepted") value. Having calculated a t-score, you can use the following criteria to form a conclusion:

t = !x_bar - x_expected|/sigma


My results are probably consistent with theory.*
My results are possibly consistent with theory.*
My results are inconclusive.
more than 3
My results are not consistent with theory.
*Of course this isn't your entire (verbatim) conclusion - you need to summarize your results, discuss possible sources of uncertainty, and report on anything you think can improve the experiment, among other things, but it is your basic conclusion.

last update June 4, 2004 by JL Stanbrough