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

Random Errors

It has been found in practice that these errors are generally:

• small in nature

• positive or negative in equal proportions, that is with a large sample the algebraic sum will tend towards zero. (This is the simple case, studies in later years will show that there are other situations where this does not apply).

Methods of Survey to Minimise affect of Errors

One of the most effective methods of controlling the effect of random errors is to "work from the whole to the part". The initial control over the whole area is established carefully, and then the rest of survey work is carried out within the control network, like the field exercise involving the mapping of the football oval on campus.

Other field procedures we use to reduce the effect of errors include:

• developing observing procedures to eliminate gross and systematic errors (for example checking the difference between face left and face right angle readings;

• taking additional or redundant observations to check for mistakes;

and specifically with respect to random errors:

• take observations many times.

If we assume the simple case, that is that given a large sample the spread of values would conform to the normal distribution, we can estimate the most probable value of a certain quantity measured several times as the mean.

n = number of readings taken

The Standard Deviation 's' gives a measurement of 'dispersion' or spread about the mean, that is it will give an estimate of precision of the observations.

where the top line represents the sum of the squares of the deviations of the observations from the mean, also known as the residuals. The square of the standard variation is the variance. This can also be expressed as:

which is more convenient for calculations.

For example, consider the following table showing readings for an angle measured 12 times:

2°21' 47"	- 5	25
52"	0	0
53"	+1	1
54"	+2	4
49"	-3	9
49"	-3	9
55"	+3	9
52"	0	0
50"	-2	4
56"	+4	16
51"	-1	1
56"	+4	16

This is the standard deviation/error of one observation from the set. The standard deviation of the mean is given by another formula:

The standard deviation of the mean

In the example above, the standard deviation of the mean

We would naturally hope that the precision of the average value was better than the precision of a single pointing, which is the case.

Hence: the best estimate of the angle = 2° 21' 52".0 +0."84. It is conventional to show the significant figures of the mean to one place more than that of the observations.

How Many Times?

The formula for the precision of the mean can also show us how many observations are required to attain a given precision.

For example, suppose an angle can be measured to +5" . Typically this is measured by experience or experiment (as above), or is a function of the level of precision of the equipment being used. The specifications for the project call for a mean angle with a precision of +2". How many observations are required to achieve this precision?

Therefore 7 observations are required with this instrument to give an angle with a precision of +2 seconds.