Errors during the measurement process

3.5.1 Statistical analysis of measurements subject to random errors

Mean and median values

The average value of a set of measurements of a constant quantity can be expressed as either the mean value or the median value. As the number of measurements increases, the difference between the mean value and median values becomes very small. However, for any set of n measurements x₁, x₂ … x_n of a constant quantity, the most likely true value is the mean given by:

This is valid for all data sets where the measurement errors are distributed equally about the zero error value, i.e. where the positive errors are balanced in quantity and magnitude by the negative errors.

 The median is an approximation to the mean that can be written down without having to sum the measurements. The median is the middle value when the measurements in the data set are written down in ascending order of magnitude. For a set of n measurements x1, x2 ÐÐÐ xn of a constant quantity, written down in ascending order of magnitude, the median value is given by:

Thus, for a set of 9 measurements x₁, x₂ … x₉ arranged in order of magnitude, the median value is x₅. For an even number of measurements, the median value is midway between the two centre values, i.e. for 10 measurements x₁ … x₁₀, the median value is given by: (x₅ + x₆)/2.

Suppose that the length of a steel bar is measured by a number of different observers and the following set of 11 measurements are recorded (units mm). We will call this measurement set A.

                          398 420 394 416 404 408 400 420 396 413 430                       (Measurement set A)

sing (3.4) and (3.5), mean = 409.0 and median = 408. Suppose now that the measure[1]ments are taken again using a better measuring rule, and with the observers taking more care, to produce the following measurement set B:

                                409 406 402 407 405 404 407 404 407 407 408                 (Measurement set B)

or these measurements, mean D 406.0 and median D 407. Which of the two measure[1]ment sets A and B, and the corresponding mean and median values, should we have most confidence in? Intuitively, we can regard measurement set B as being more reli[1]able since the measurements are much closer together. In set A, the spread between the smallest (396) and largest (430) value is 34, whilst in set B, the spread is only 6.

 Thus, the smaller the spread of the measurements, the more confidence we have in the mean or median value calculated.

Let us now see what happens if we increase the number of measurements by extending measurement set B to 23 measurements. We will call this measurement set C.

                      409 406 402 407 405 404 407 404 407 407 408 406 410 406 405 408

                                   406 409 406 405 409 406 407                                             (Measurement set C)

Now, mean = 406.5 and median = 406.

This confirms our earlier statement that the median value tends towards the mean value as the number of measurements increases.

Standard deviation and variance

Expressing the spread of measurements simply as the range between the largest and smallest value is not in fact a very good way of examining how the measurement values are distributed about the mean value. A much better way of expressing the distribution is to calculate the variance or standard deviation of the measurements. The starting point for calculating these parameters is to calculate the deviation (error) d_i of each measurement x_i from the mean value x_mean:

Solution

First, draw a table of measurements and deviations for set A (mean D 409 as calculated earlier):

Note that the smaller values of V and  for measurement set B compared with A correspond with the respective size of the spread in the range between maximum and minimum values for the two sets.

Thus, as V and  decrease for a measurement set, we are able to express greater confidence that the calculated mean or median value is close to the true value, i.e. that the averaging process has reduced the random error value close to zero.

Comparing V and  for measurement sets B and C, V and  get smaller as the number of measurements increases, confirming that confidence in the mean value increases as the number of measurements increases.

We have observed so far that random errors can be reduced by taking the average (mean or median) of a number of measurements. However, although the mean or median value is close to the true value, it would only become exactly equal to the true value if we could average an infinite number of measurements. As we can only make a finite number of measurements in a practical situation, the average value will still have some error. This error can be quantified as the standard error of the mean, which will be discussed in detail a little later. However, before that, the subject of graphical analysis of random measurement errors needs to be covered.