Errors, Theory of

the branch of mathematical statistics concerned with the analysis of measurement errors and with the refinement of the numerical values of approximately measured quantities. Repeated measurements of a constant quantity generally yield different results, since each measurement contains some error. There are three basic kinds of errors: systematic errors, blunders, and random errors. Systematic errors consistently either exaggerate or understate the results of measurements and have specific causes (the measuring instruments are improperly set up, an environmental factor) that systematically influence and alter the results of the measurements. Such errors are estimated by methods that lie beyond the purview of mathematical statistics. Errors in counting and the incorrect reading of instruments are among the causes of blunders. The results of measurements involving blunders differ markedly from other measurement results and therefore are often easily noticed. Random errors stem from various random factors that act in an unforeseen manner during each individual measurement—sometimes decreasing and sometimes increasing the results.

The theory of errors deals only with blunders and random errors. Its basic problems are the ascertainment of the laws of distribution of random errors; establishment, based on measurement results, of estimates of the unknown quantities being measured; determination of the errors in the estimates; and elimination of blunders.

Let the values x₁, x₂, …, x_n be obtained from n independent, equally accurate measurements of the quantity a. The differences

δ₁ = x₁ – a, …, δ_n = x_n – a

are called the true errors. In terms of the probabilistic theory of errors, the δ_i are interpreted as random quantities; the independence of the measurements is taken to mean the mutual independence of the random quantities δ₁,…, δ_n. The equal accuracy of the measurements is interpreted in the broad sense as identical distribution; that is, the true errors of equally accurate measurements are identically distributed random quantities. The mathematical expectation of the random errors b = Eδ₁ = … = Eδ_n is called the systematic error, and the differences δ₁ – b, …, δ_n – b are called random errors. Thus, the absence of a systematic error means that b = 0 and δ₁, …, δ_n are random errors. The quantity , where σ is the root-mean-square deviation, is called the index of precision. When a systematic error is present, the index of precision is expressed by the ratio . In the narrow sense, equal accuracy of measurements means that the index of precision is identical for all measurement results. The presence of blunders indicates that equal accuracy in both the narrow and the broad senses has been violated for some individual measurements. The arithmetic mean of the results of the measurements

is usually selected as an estimate of the unknown quantity a, and the differences Δ₁ = x₁ – x̄, …, Δ_n = x_n – x̄ are called the apparent errors. The selection of x̄ as an estimate for α is based on the law of large numbers: when the number n of equally accurate measurements lacking a systematic error is sufficiently large, the estimate x differs arbitrarily little from the unknown quantity α with a probability arbitrarily close to unity. The estimate x lacks systematic error—estimates with this property are said to be unbiased. The deviation of the estimate is

Dx̄ = E(x̄ – α)² = σ²n

Experience shows that the random errors σ_i often conform to distributions close to the normal distribution; the reasons for this are given by the limit theorems of probability theory. In this case, the quantity x has a distribution that differs little from a normal distribution with a mathematical expectation α and deviation σ²/n. If the distributions of σ_i are precisely normal, then the deviation of any other unbiased estimate for α, such as a median, is at least Dx. But this property does not hold if the distribution of σ_i is non-normal.

If the deviation σ² of the individual measurements is not known in advance, it can be estimated by the quantity

Es² = σ²; that is, s² is an unbiased estimate for σ². If the random errors δ_i have a normal distribution, the relation

obeys Student’s distribution with n – 1 degrees of freedom. This fact can be used to estimate the error of the approximate equality a ≈ x̄.

On the same assumptions, the quantity (n – 1) s²/σ² has the chi-squared distribution with n – 1 degrees of freedon. This fact permits estimation of the error of the approximate equality σ ≈ s. It can be shown that the relative error ǀs – σǀ/s will not exceed the number q with the probability

ω = F(z₂, n – 1) – F(z₁ n – 1)

where F (z, n – 1) is a function of the chi-squared distribution and