How Good Is That Number?

When we calculate anything, we're always modeling - we're making assumptions and approximations, and our measurements are always inherently uncertain to some degree.  Our calculation must therefore be viewed with an understanding of its limitations and biases (not the colloquial term implying that there was an agenda, but rather any sort of consistent skew to the results due to some physical mechanism that wasn't accounted for).

One experiment that we did to determine the speed of sound a few days ago is a great case study for talking about error analysis. 

The Goal: We were trying to determine the speed of sound in air.

The Setup: Students spread out across a large field, at known distances from a student with a baseball bat and ball.  The student tossed the ball into the air and hit it.

The Measurement: Students started their watches when they saw the ball hit the bat and stopped them when they heard the impact - the travel time delay and the distance can then be used to determine the speed of sound.

Sources of Error: Again, there's a difference between everyday use of 'error' and scientific use; we don't mean 'mistakes' - if you made a mistake, then you need to fix it.  We're talking about consequences of our modeling assumptions and unavoidable measurement uncertainty.

In this experiment, two big sources jump out at us, and illustrate very nicely the two main forms of experimental error that students are likely to encounter:

• Timer error - every time you start and stop the watch, there's some uncertainty in that measured time interval.  You could have started or stopped a little too late or early, and it's impossible to tell which happened in any given measurement, just based on that one time.  This is a random error source, because it could make your measurement either too high or too low; it generally serves to scatter your data - some points are a little high and some a little low, but the trend is unaffected.
• Reaction time - this one's more subtle here, and a diagram really helps to illustrate its presence in this case:

 

          The subtle part is is the difference between the two events.  We saw the impact coming - he tossed the ball into the air, and we could anticipate the impact, so the beginning of the time interval came as absolutely no surprise to us.  The end of the interval, however, was subject to our reflexes; you can't hear the ball until the sound gets to you, so there's no way to anticipate the moment at which you should stop your watch.  This tells us that the measured time intervals will all be too long - that's a systematic error source, because it skews all of our data in the same direction, and has an effect on our average value. 

Effects of the error sources:  This is the most important part of the error analysis, because it gives us information on how we should trust our final calculated value.  In this case, we should expect our calculated speed to be lower than the true value.  We could quantify how much longer, but that's a bit deeper than I'm interested in going with this physics class.  If we can qualitatively analyze the effects of error sources, then I'll be thrilled!

• The reaction time issue will cause our times to be uniformly too long, which will cause the speeds to be too low.  This is a systematic effect, and we should expect the average value to be lower than the true speed of sound.
• The timer error will scatter our values, but not affect the average value; some will be too high (because the times were too low) and some too low (because the times were too high), but there's no effect on the average speed.