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Lesson 4 - The Normal Distribution
For many different kinds of measurements, if the only errors which are involved are random (indeterminate) errors, those measurements obey a normal, or Gaussian, distribution. Let's take that statement apart to see what it is getting at. If the only errors are random ones, the measurements should be distributed about the mean in such a manner that more measurements lie close to the mean than lie far away from it. (Of course the question of what is meant by close and far depends on how large the random errors in a given experiment are.) If you were to plot the distribution of a large number of measurements, you would find that they look like what you may know as a "bell shaped curve". This curve, the normal or Gaussian curve, is given mathematically by the expression: 
where the variables have the same meaning as in lesson 3. Here is a graph showing two Gaussian curves, both with a mean of 11.237, but one with a standard deviation of 0.034 and the other 0.068 (which is which?). The mean is the x value corresponding to the maximum in the curve. (If you have had calculus you know how to prove that this is the case.) 
Press the button below to see an interactive demonstration of the relationship between the standard deviation and the width of the distribution (be sure to close the spreadsheet before going on).
Notice that as the curve gets wider, it also gets shorter. The Gaussian distribution is representing the whole sample, so that as a greater percentage (or fraction) of the data lie further from the mean (larger standard deviation), there is a smaller percentage (or fraction) which can lie closer to the mean. You will explore this aspect of the normal curve, along with several others, now. Pressing the next button will take you to another spreadsheet that allows you to interactively determine the fraction of data that lie within a certain range about the mean for normally distributed data.
This next button will take you to a spreadsheet which is set up to calculate a Gaussian distribution when provided with a mean and standard deviation. Sheet 2 of the spreadsheet includes some additional information, and asks several questions.
In the first lesson it was mentioned that we would use one particular form of bar graph, the histogram. The time has come. A histogram enables us to see how many data in a data set fall in various subdivisions of the total range of data. This, in turn, gives us a better idea of how the data are distributed. The last button in this lesson takes you to a spreadsheet that further introduces the idea of the histogram and shows how histograms are constructed in Excel.
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