The Statistics Of
    Measurement
 Types Of Errors
 Random Errors
 Normal Distribution
 Standard Errors Of
    Derived Quantities
 Propagation Of Small
    Systematic Errors

The Normal Distribution

If we recall the example before where we observed an angle with a theodolite, and then using that data we plotted a frequency count of the angles, we would get the figure shown on the next page.

If we were to take many more observations, and reduce the class intervals in size the distribution would probably tend to look like the one below.

This should be quite familiar, and is known as the normal distribution. It is a theoretical distribution with the characteristics N(m, s). The Greek letter m is the population mean, and s the population variance. The formula is as follows:

The probability of a measurement x having a value between a and b is given by the area under the curve as follows:

We estimate the mean of the normal distribution by x and the standard deviation by s. With a suitable large sample these approximate m and s.

The Standard Normal Distribution

Normally distributed data has a particular mean and variance, it is a function of the data being sampled. The standard normal distribution has been re-scaled so that it has a mean value of zero, and a variance of one.

What does this refer to?(W.S. Gossett who wrote under name 'Student', data tabulated by Fisher)

Values of ‘t’ are found in normal probability tables

The standard normal distribution will give the probability of finding a given quantity within certain limits. It gives probability by calculating the area under the curve from negative infinity to the value being tested.

When Should an Observation Be Rejected?

We can use the standard normal distribution to test whether a value is ‘probable’ or not, that is whether to accept or reject certain measurements. For example, lets look at a set of distance measurements:

xi		mean = 93.827   standard deviation = +0.024   Question, should the measurement 93.898* be rejected?
93.815	0.012
93.818	0.009
93.826	0.001
93.823	0.004
93.809	0.018
93.815	0.012
93.898*	0.071
93.827	0.000
93.821	0.006
93.823	0.004
93.822	03005

From tables of the standard normal distribution, we can determine the probability of an observation lying within three standard deviations of the mean as follows:

In other words there is a 99.74% chance of a measurement lying between ( -3s < x < 3s ) of the mean. If we are happy with odds of 99.74% then in the table above:

mean = 93.827, s = +0.024,

so the value should be between 93.899 and 93.755 which it is (but only just).

In general, around 68% of the measurements will fall within one sigma (one standard deviation), 95% of the measurements will fall within 2 standard deviations of the mean, and around 99% within three.

In the example above, if the criterion was that we would reject observations outside of two standard deviations, then by rejecting the value we get a mean of 93.820 and a standard deviation of +0.004. The standard deviation of the mean would then be +0.001, a considerable improvement.