Consider a length of 29.42 m measured with a tape and correct to ±0.05 m. The range of these measurements would therefore be from 29.37 m to 29.47 m, giving 11 possibilities to 0.01 m for the answer. If the next bay was measured in the same way, there would again be 11 possibilities. Thus the correct value forthe sum of the two bays would lie between 11 × 11 = 121 possibilities, and the range of the sum would be 2×±0.05 m, i.e. between −0.10 m and 0.10 m. Now, the error of −0.10 m can occur only once, i.e. when both bays have an error of −0.05 m; similarly with 0.10. Consider an error of −0.08; this can occur in three ways: (−0.05 and −0.03), (−0.04 and −0.04) and (–0.03 and –0.05). Applying this procedure through the whole range can produce Table 2.1, the lower half of which is simply a repeat of
Probability of errors
the upper half. If the decimal probabilities are added together they equal 1.0000. If the above results are plotted as error against probability the histogram of Figure 2.2 is obtained, the errors being represented by rectangles. Then, in the limit, as the error interval gets smaller, the histogram approximates to the superimposed curve. This curve is called the normal probability curve. The area under it represents the probability that the error must lie between ±0.10 m, and is thus equal to 1.0000 (certainty) as shown in Table.
More typical bell-shaped probability curves are shown in Figure ; the tall thin curve indicates small scatter and thus high precision, whilst the flatter curve represents large scatter and low precision. Inspection of the curve reveals:
(1) Positive and negative errors are equal in size and frequency; they are equally probable.
(2) Small errors are more frequent than large; they are more probable.
(3) Very large errors seldom occur; they are less probable and may be mistakes or untreated systematic errors.
The equation of the normal probability distribution curve is:
where y = probability of the occurrence of x − μ, i.e. the probability that x the variate deviates this far from the central position of the distribution μ, σ is the spread of the distribution and e = the base of natural logarithms. If μ = 0, i.e. the centre of the distribution is at zero and σ = 1, i.e. the spread is unity, the formula for the probability simplifies to:
As already illustrated, the area under the curve represents the limit of relative frequency, i.e. probability, and is equal to unity. Thus a table of Normal Distribution curve areas (Table ) can be used to calculate
Area under the Normal Distribution curve