THE RANDOM ERRORS AND ESTIMATION OF THEIR INFLUENCE ON THE MEASUREMENT RESULT

While the systematic errors theoretically (and in the most cases – to a coinsiderable extent also practically) can be excluded from the measurement results, but it is impossible to avoid the random errors experimentally.

But it is necessary, using a statistical material of the multifold observations being researched PQ, to know how to estimate the random-error influence on the measurements results in each case. For this purpose the mathematical methods of probability theory are applied.

In the simplest case we have the following problem on the random errors influence estimation.

Having made n equivalent observations of the same physical quantitty, we have a series of results:

                           X₁, X_2, X_3,…,X_n,

    Where X_i is a result of i-th observation; n is an amountof observations of a series.

The systematic errors have already excluded, that is this measurement is correct.Estimate the random errors influence on the measurement  result.

 The most authentic value from a given series is arithmetic mean :

                                            = ,

The random error  of eachi-th observation:

_i =X_i – X_true,

     Where X_true – is the true value of PQ under research .

So far as the X_true is inaccessible, also the random error concept has only theoretical, but not practical sense. In practice, instead of it, the residual error of the i-th measurement n_і is used:

At unlimitedly large observations amount the arithmetic mean tends to the true value, and the residual error tends to the random one.

For checking out the calculations correctness it is necessarytoremember also, that the algebraic amount of the residual errors approaches to zero :

What criterion must be used to characterize and to compare accuracy of different series of observations?

The most widespread one is the (experimental) standard deviation σ. It is a parameter of the observation results distribution function, which characterizes their dispersion. It  is   determined as a positive value of a square root from a variance of the observation results and it is expressed in the same units, as a measurand. The variance is a less convenient parameter, because its unit is square  of  the   measurand unit.

Let us compare two targets (Fig. 12) with the results of fire from two guns (analogy to two series of measurements of the same PQ by the different instruments; and, analogy of a measurement error is a deflection of the hit point from the target center).

 Fig. 12. Comparison of the results of fire from two guns.

Concentration of the hits is characteristic for the left target, though some noticeable deviations seldom occur. The right target is characterized by more even "dissipation" of the hits.

Comparing these targets, an experienced shooter will say at once, that a gun, from which one shot at the left target, is better.

It is more difficult to make such estimation, having only two series of numbers without a visual image. But having computed the experimental standard deviations for each of the series, we shall see, that for the left target it is much less, than for the right one:

                       s_L < s_R .

           For the experimental standard deviation (ESD) calculation it is necessary to know the random error distribution law. This law relates value of the random error to probability of its appearance.

One of the most widespread laws in measurement practice is normal distribution law of random errors, the so-called the Gauss law(Fig.13).                    Fig. 13. The normal distribution law of random errors

The probability density of the random error is plotted on the ordinate axis, the random error itself is plotted on the abscissa axis.

      In Fig.13 two curves are plotted. They characterize two series of observations with the different ESD - s₁ and s₂. The curve with ESD s₁ can characterize the results of fire in the left target (Fig. 12), and s₂ - in the right one, and both the curves characterize the Gauss law, which is based on two axioms.

1. Axiom of contingency: at a very large amount of observations the random errors of the same, but with inverse sign, occur equally often.

2. Axiom of distribution: most often the small errors arise, and the large ones occur the less often, the larger they are.

The graphs are strong proof of veracity of these axioms.

Really, the axiom of contingency is confirmed by a symmetrical disposition of curves with respect to the ordinate axis. On the right side there are the positive random errors, and on the left side there are the negative ones.

 The axiom of distribution is confirmed by that the most probability of occurrence (the highest ordinate) is inherent for the zero and close to it small errors. At the same time the large errors both as positive, and negative, large errors have the small ordinates of curves, that is a small probability of occurrence.

The graph s₁ is characterized by that the large part of the area restricted by it is concentrated in a zone of small errors, and the graph with the s₂ is differed by more even distribution of the error dissipation: there is no big difference between occurrence probability of both the small and large errors .

When the random errors are distributed under the normal law, 95% of them are entered into the values limits ±2σ, 99,73 % - into the values limits ±3σ.

These limits, between which with the known probability the random errors are concentrated , are called confidence interval, and this probability is called confidence. For example, at a normal distribution law of errors the confidence interval from +3σ to -3σ corresponds to the confidence coefficient P=0,9973.

The errors by the modulus larger than 3σ occur very seldom (not more often, than 27 times by 10 000 measurements), therefore they are called the gross errors, and, as a rule, rejected from a series of observations results.

The experimental standard deviation σ can be approximately calculated by the Bessel formula:

For a measurement accuracy improvement, that is, a reduction of the σ, it is necessary to augment an amount of observations. According to the theory, if each value of the above given series is considered as arithmetic mean of “n” observations, the estimation of its ESD will be diminished in   times:

A reliable, though labour-consuming way of the measurement accuracy raise is as follows: to increase themeasurement accuracy in “n” times, it is necessary to increase the amount of observations in n² times.

8.1.The random errors distribution laws approximations .