If we take repeated measurements of the same value, we will obtain a distribution of measurements. If we take a sufficient number of measurements, our results should approach an ideal bell-shaped curve, called a Gaussian or Normal distribution. The peak of the distribution should be centered on the mean, our best estimate of the true value of what it is we are measuring. The width of the curve is related to the standard deviation if the data, a measure of its spread. An example gaussian distribution is shown in figure 1.
The larger the data set, the closer the convergence of the system towards a normal distribution. This forms the Central Limit Theorem, which you will investigate experimentally in the Level 1 radioactivity experiment.
Fig.1: A Gaussian distribution
We can find out the probability of a measurement lying in a certain range by integrating the Gaussian Distribution function.
To integrate this directly is actually impossible between finite limits. However, it can be approximated to an extremely high degree of accuracy using numerical techniques. We will be using excel to do this for us.
As per usual, we will use an example to show how probabilities can be calculated using a normal distribution. This example was taken from page 25, example 3.2.2 in Measurements and their Uncertainties, Ifan G. Hughes and Thomas P.A. Hase.
"A box contains 100 Ω resistors which are known to have a standard deviation of 2 Ω. What is the probability of selecting a resistor with a value of 95 Ω or less? What is the probability of finding a resistor in the range 99-101 Ω?"
Excel has a function called NORMDIST(x, mean, standard deviation, 1), which integrates the Gaussian from negative infinity to x. Remembering that a Gaussian is also symmetric about it's mean, you can combine these in order to integrate over any desired range.
Fig.2 : How to answer the above question using Excel