measurements, the best estimate of the experimental value would be found by taking the arithmetic mean. I will represent the arithmetic mean by placing a small line over the symbol: For the measurement values given above, the arithmetic mean is =1+2+33=195 mm+196 mm+197 mm3=196 mm.So, our best estimate of the length of the line segment is =196 mmA general mathematical statement for calculating the arithmetic mean of Nmeasurement values is given by X=x1+x2+x3+xi⋅⋅⋅+xNN=1Nxii=1N∑.(1)Estimating the Experimental Error:Once we have made a determination about the best measured value, we turn our attention to the experimental error. First, note that we can not determine the error exactly!There is always as much uncertainty in our “knowledge”about the error of a measurement as there is about our knowledge of the best value of the measurement itself.Next, we need a process by which we can arrive at a reasonable estimate of the experimental error. There are several such processes. However, in this introduction, I am going to limit the discussion to the method used most often by experimental physicists. Physicists most often use a number called the standard deviationto estimate the experimental error.

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The Standard Deviation:We begin with what is called the deviation. We can define the deviation of the imeasurement bydi=xi−X.(2)Using the measurement values from the example given above, we can write:d1=x1−X=195 mm−196 mm=−1 mm,d2=x2−X=196 mm−196 mm=0 mm,d3=x3−X=197 mm−196 mm=1 mm.At first blush, one might think that simply taking an average of the deviations would give us a reasonable estimate of the experimental error. However, when we do generate such an average, we finddave=d1+d2+d3++dNN=x1−X()+x2−X()+x3−X()++xN−X()N=x1+x2+x3+⋅⋅⋅+xNN−NXN=X−X=0.Of course zero is the worst estimate of the experimental error as it suggests that we are certainof our measurement.The standard deviationis defined in terms of the square of the deviation. The standard deviation is signified by σ(a lower case Greek letter named sigma), and is given byσ=1Ndi2i=1N∑,(3)This formula assumes that a very large number of measurements were made. In this lab, however, we will never be measuring a specific quantity more than twelve times. So, for small sets of data, a better formula for the standard deviation is σs=1N−1di2i=1N∑.(4)

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If we return to the data given in our example, we haveσs,=13−1⎡⎣⎢⎤⎦⎥0 mm()2+−1 mm()2+1 mm()2⎡⎣⎤⎦=1.0 mm.(5)(The standard deviation is almost always rounded to one significant digit. One important exception is if the single significant digit is the number one, then it is customary to use two significant digits.) In our example, the standard deviation turns out to be one millimeter. This wasalso the smallest standard unit of the measuring devise. We call the smallest standard unit used in a measurement the least count. The least count is always a quick, useful way to reasonably estimate error.One final complicating point.