Theory of random measurement errors: concepts, uses, and misuses

Statistics is a mathematical discipline dedicated to properties of systems in which a variable can assume different values. For instance, the height of a person can be a variable, and the corresponding statistical system is formed from measured heights within a group of people. Generally, there are two types of statistics. A priori statistics is dedicated to theoretical modelling which provides probabilities that a variable will assume given values. Mathematical formulae that describe these probabilities are called statistical distributions. They are sometimes named after those who derived them (e.g., Gaussian distribution, Poisson distribution, Bernoulli distribution) and sometimes contain mathematical terminology (e.g., gamma distribution, exponential distribution, log-normal distribution). A posteriori statistics focuses on measured occurrences of values, which results in frequency distributions. While frequency distributions always contain discrete numbers, statistical distributions can be expressed either through pairs of discrete numbers (discrete distributions) or as functions (continuous distributions). Comparisons of frequency distributions and statistical distributions are common and may lead to a deeper understanding of given experimental results.

Let us consider random variables, i.e., those which assume different values solely due to the corresponding probabilities of their appearances. Suppose that a variable x can assume values x₁,..., x_n and that the probability of x = x_i is equal to p(x_i), where 0 ≤ p(x_i) ≤ 1. The set of pairs [x_i, p(x_i)] then forms a discrete statistical distribution of the variable x, satisfying the condition ∑i=1np(xi)=1 \sum\nolimits_{i = 1}^n {p({x_i})} = 1 (which means that the probability that x will assume any value is 100 %). As explained above, the values of p(x_i) are calculated theoretically (a priori statistics). We can also carry out N measurements of x and record, for each i, the number f(x_i) (called absolute frequency) of the appearance of x_i. Using this, we can define r(x_i) = f(x_i)/N (called relative frequency), which represents the experimental probability (a posteriori statistics) of the appearance of x_i, satisfying ∑i=1nf(xi)=N \sum\nolimits_{i = 1}^n {f({x_i}) = N} and, consequently, ∑i=1nr(xi)=1 \sum\nolimits_{i = 1}^n {r({x_i}) = 1} . The set of pairs [x_i,r(x_i)] forms a frequency distribution.

Statistical properties of a system are usually expressed through descriptive statistical parameters, derived from statistical moments m_l and M_l (the lth raw moment and central moment, respectively), which are calculated as [1] ml=∑i=1nxilpxi {m_l} = \mathop \sum \nolimits_{i = 1}^n x_i^lp\left({{x_i}} \right) and, [2] Ml=∑i=1n(xi−m1)lpxi {M_l} = \mathop \sum \nolimits_{i = 1}^n {({x_i} - {m_1})^l}p\left({{x_i}} \right)

For a frequency distribution, the p(x_i) in the above equations (and henceforth), should be replaced with r(x_i).

In most cases, four parameters are used to describe a distribution. The first is x̄ = m₁, called the expectation or the mean (value) of a statistical distribution or a frequency distribution, respectively. The second is variance V = σ² = M₂, which is basically the mean squared deviation of x from x̄. The third is skewness α₃ = M₃/σ³, which accounts for the asymmetry of a distribution. The fourth is kurtosis α₄ = M₄/σ⁴, which is a measure of the tailedness of a distribution.

The probability that x ≥ x_a and x ≤ x_b is calculated as follows: [3] Pxa≤x≤xb=∑i=abpxi P\left({{x_a} \le x \le {x_b}} \right) = \mathop \sum \nolimits_{i = a}^b p\left({{x_i}} \right)

As mentioned before, there are also continuous statistical distributions besides the discrete ones. In this case, x is a continuous variable, and this changes the required mathematics. We define a continuous function g(x) ≥ 0, called probability distribution, with a property that the probability of x being in the interval (x, x + dx) is given by g(x)dx. This means that ∫ g(x)dx = 1 over the range where x is defined and that [3] changes into [4] P(xa<x<xb)=∫xaxbgxdx P({x_a} < x < {x_b}) = \int\limits_{{x_a}}^{{x_b}} {g\left(x \right)dx}

Graphically, this is the area under g(x) between x = x_a and x = x_b. Equations [1] and [2] change as well, by replacing x_i with x and ∑i=1np(xi) \sum\nolimits_{i = 1}^n {p({x_i})} with ∫ g(x)dx = 1, where the integration is carried out over the range where x is defined. The meanings of x̄, σ, a₃ and a₄ remain the same.

There are two additional parameters which are frequently used to describe the main properties of a distribution. The first is the median η of a distribution, which is defined by P(x < η) = P(x > η) = 1/2. The second is the most probable value ν of x, defined as the point at which there is a maximum of p(x_i) or P(x).

Gaussian or normal distribution is arguably the best-known statistical distribution, being applicable in numerous fields of science and technology. Since TRME relies on it, we shall briefly discuss its main properties. This is a continuous statistical distribution defined as: [5] g(x)=1σ2πe−(x−x¯)22σ2;−∞<x<∞ g(x) = {1 \over {\sigma \sqrt {2\pi}}}{e^{- {{{{(x - \bar x)}^2}} \over {2{\sigma^2}}}}}; - \infty < x < \infty

As shown in Figure 1, g(x) is a symmetrical, bell shaped curve peaked at x̄. At = x̄ ± σ, there are inflexion points, i.e., the curvature of g(x) changes from convex to concave. The shape of the curve is completely determined by x̄ and σ, whereas α₃ = 0, α₄ = 3, and η = ν = x̄.

Figure 1

Probability distribution g(x) of a random variable x for the Gaussian distribution with x̄ = 4.5 and σ = 2.5

One of the properties of Gaussian distribution is that P(x – kσ < x < x̄ + kσ) is about 0.68, 0.95, and 0.98 for k = 1, 2, and 3, respectively. Of course, one can choose any k (called coverage factor) and calculate the corresponding P, but the above integer values are most commonly used in TRME.

Measurement errors can be systematic or random. Systematic errors are specific of every measurement procedure and are usually manifested as offsets in measured results, e.g., when an instrument is showing some value in the absence of any input. These errors are not uncommon – and are solved by removing a given offset technically or analytically – but are not in the domain of TRME and are therefore not addressed in this review. TRME deals with random errors, which originate from limited accuracy of measurement instruments and procedures. Random errors do not produce offsets and are manifested as symmetric departures of measurement results from x̄, with r(x_i) following closely the Gaussian distribution.

In its approach and mathematical methods, TRME is based on statistics. Hence, formulae used in TRME and statistics are similar and sometimes even the same, but their meanings may be quite different. Suppose that a quantity x is measured repeatedly n times. In TRME, p(x_i) = 1/n, and Equation [1] gives [6] x¯=∑i=1nxin \bar x = {{\mathop \sum \nolimits_{i = 1}^n {x_i}} \over n} while the use of Equation [2] results in [7] σ2=∑i=1nxi−x¯2n {\sigma^2} = {{\mathop \sum \nolimits_{i = 1}^n {{\left({{x_i} - \bar x} \right)}^2}} \over n}

Let us first discuss x̄. We have used a general formula from statistics to calculate it and have used the same name (the mean). However, the meaning of x̄ in TRME is not the same as in statistics, where x̄ is simply the average value of x. In TRME, x̄ is the best estimate for the true value ξ of x. Since ξ cannot be determined – due to the limitations of measurement instruments and procedures – taking ξ ≈ x̄ is the best we can do. The central quantity in TRME is the uncertainty u in this approximation, so we can express a measurement result as [8] ξ=x¯±u \xi = \bar x \pm u

While different names are used for u (e.g., measurement uncertainty), its full name – uncertainty of the mean – reveals its meaning more directly. More clearly, we approximate ξ ≈ x̄, but Equation [8] actually says that ξ can be anywhere, with a given probability, in a symmetric interval around x̄. The distribution that determines this probability is Gaussian distribution peaked at x̄. This defines x̄ as the most probable value of ξ and implies that we can use a coverage factor to set the related confidence level. As already said, it is common to choose k = 1, 2, or 3, but this is just for the sake of simplicity (as any value of k can be used).

Now we turn to the meaning and applicability of σ as given by Equation [7]. σ is called the standard deviation of the population. Again, it is useful to spell out the meaning of σ: the square root of the mean squared deviation of x from x̄ for the complete population of a studied quantity. However, seldom can one collect data for an entire population, and n measurements then refer to a sample taken from all possible values of x. It can be shown, but this is outside the scope of this review, that the denominator in Equation [7] is actually the number of independent variables (degrees of freedom), which is reduced by unity in case of a sample that is smaller than the entire population. Hence, the variance of a sample is given by [9] s2=∑i=1n(xi−x¯)2n−1=nn−1σ2 {s^2} = {{\mathop \sum \nolimits_{i = 1}^n {{({x_i} - \bar x)}^2}} \over {n - 1}} = {n \over {n - 1}}{\sigma^2}

Since σ and s represent the square roots of the mean squared deviations of x from x̄ for an entire population and a sample, respectively, which of them should be used as u that enters Equation [8]? Shall we prefer the use of s, because we normally measure restricted datasets, that is, samples? The answer to this question is: neither σ nor s can be used as u in Equation [8]. Just read their names: σ and s are measures of the mean squared deviation of x from x̄ in a set of measurements, nothing more, and u is fundamentally different – it is the uncertainty of the approximation ξ ≈ x̄. This is why it is important to keep in mind the full names of σ, s, and u, because the common short names do not reveal clearly the meaning of these quantities. Of course, short names are welcome, but one should not forget what is behind them. For the rest of this review, we shall focus on u and discuss it from the perspective of its proper meaning, but shall simply call it “uncertainty”.

Before we address u more closely, we ought to discuss one of its important properties. Let us suppose that we measure quantities q₁, q₂,…, q_z directly and obtain ξ_j = q̄_j ± u_j for all 1 ≤ j ≤z. Suppose also that there is a quantity y = y(q₁, q₂,…, q_z) that has not been measured directly but can be calculated from q₁, q₂,…, q_z by using a formula. Having ȳ_j and u_j, can we find ȳ and u_y, the mean and uncertainty of y, respectively? Yes, we can, using the following equations: [10] y¯=yq¯1,q¯2, …,q¯z \bar y = y\left({{{\bar q}_1},{{\bar q}_2},\; \ldots,{{\bar q}_z}} \right) and [11] uy2=∑j=1z∂y∂qjqj=q¯j2uj2 u_y^2 = \mathop \sum \nolimits_{j = 1}^z \left({{{\partial y} \over {\partial {q_j}}}} \right)_{{q_j} = {{\bar q}_j}}^2u_j^2

Equation [10] simply says that ȳ is obtained by inserting the values of q̄_j (for all j) in the formula that relates y and q̄_j. Regarding Equation [11], although it can be addressed rigorously from a mathematical point of view, we choose to discuss it more intuitively. One can see that uy2 u_y^2 is a sum of squared uncertainties uj2 \left({u_j^2} \right) of individual contributors to y and that each uj2 u_j^2 is multiplied by a weight factor (∂y/∂qj)qj=q¯j2 (\partial y/\partial {q_j})_{{q_j} = {{\bar q}_j}}^2 that determines how much a change of q_j affects y. Partial derivatives must be squared in order to conserve dimensional consistency. If we take a look at the expression for σ² = M₂ (see Equation [2]), we can see that σ² is also calculated as a weighted sum of squared individual deviations from x̄, the weight factors being p(x_i). The mathematical similarity between the equations for σ₂ and uy2 u_y^2 is not surprising, because both of these quantities are cumulative deviations that result from weighted individual contributions.

Equation [6] shows that x̄ is a function of x₁,…, x_n, where each x_i has an uncertainty u_i. Therefore, we can calculate uncertainty of the mean, i.e., u = u_x̄, by using Equations [6] and [11] together with the symmetries y ↔ x̄ and q_i ↔ x_i. This results in [12] u2=u2x=∑i=1n∂x¯∂xi2ui2=∑i=1n1n2ui2 {u^2} = u{2 \over x} = \mathop \sum \nolimits_{i = 1}^n {\left({{{\partial \bar x} \over {\partial {x_i}}}} \right)^2}u_i^2 = \mathop \sum \nolimits_{i = 1}^n {1 \over {{n^2}}}u_i^2

Since x_i is the result of a single measurement, the question arises as to what should be used for its (unknown) uncertainty u_i. TRME assumes that u_i is approximately the same for every i and equal to some u₀, so that Equation [12] gives u2=ui2/n {u^2} = u_i^2/n . The next assumption is that u₀ = s, so the use of Equation [9] results in [13] u2=Σi=1n(xi−x¯)2nn−1=s2n=σ2n−1 {u^2} = {{\Sigma_{i = 1}^n{{({x_i} - \bar x)}^2}} \over {n\left({n - 1} \right)}} = {{{s^2}} \over n} = {{{\sigma^2}} \over {n - 1}}

We can see that u, s, and σ are interrelated, which is not surprising, given the statistical nature of TRME. However, while s and σ are indicators of how much individual measurement results depart from x̄, it is u which tells us how much ξ ≈ x̄ is reliable. Moreover, the approximation u_i = u₀ = s extends the meaning of s beyond simple statistics, since s in TRME is the average uncertainty of an individual measurement. With this interpretation, the relation u² = s²/n becomes more understandable: if one carries out n measurements, the squared uncertainty will be reduced by a factor of 1/n in comparison with a single measurement.

Equation [13] covers those experimental results where a direct measurement of an observable is repeated n times in order to reduce u. Equation [11] accounts for the u of a quantity which has not been measured directly but is calculated from several directly measured observables. The third frequent case is measurement results that have different statistical weights.

Suppose that a quantity x is measured using different methods or in different laboratories and that each of these measurements results in ξ_j = x̄_j ± u_j. If one wants to compile all of these results and calculate their mean value and its uncertainty, how is this done properly? Since some of the results are more precise than the others, one should approach this problem by assigning a statistical weight w_j to each ξ_j.

To calculate the weighted mean, i.e., the mean 〈x〉 of L independent contributions, is straightforward, namely [14] 〈x〉=Σj=1Lwjx¯jΣj=1Lwj \langle x\rangle = {{\Sigma_{j = 1}^L{w_j}{{\bar x}_j}} \over {\Sigma_{j = 1}^L{w_j}}}

It is easy to understand the above expression by recalling that this is how one calculates grade point average if x̄j is a grade point and w_j the number of its appearance.

We can now combine Equations [11] and [14] to derive the expression for the uncertainty 〈u〉 of 〈x〉, that is, [15] 〈u〉2=∑j=1L∂〈x〉∂x¯j2uj2=Σj=1Lwj2uj2(Σj=1Lwj)2 {\langle u\rangle^2} = \mathop \sum \nolimits_{j = 1}^L {\left({{{\partial \langle x\rangle} \over {\partial {{\bar x}_j}}}} \right)^2}u_j^2 = {{\Sigma_{j = 1}^Lw_j^2u_j^2} \over {{{(\Sigma_{j = 1}^L{w_j})}^2}}}

In principle, w_j can be defined in many different ways. On the other hand, wj=1/uj2 {w_j} = 1/u_j^2 is most commonly used, because this choice is applicable to many situations and assigns larger weights to results which are less uncertain. This definition of w_j transforms Equation [14] into [16] x=Σj=1Lx¯juj2Σj=1L1uj2 \left\langle x \right\rangle = {{\Sigma_{j = 1}^L{{{{\bar x}_j}} \over {u_j^2}}} \over {\Sigma_{j = 1}^L{1 \over {u_j^2}}}} and Equation [15] into [17] u2=1Σj=1L1uj2 {\left\langle u \right\rangle^2} = {1 \over {\Sigma_{j = 1}^L{1 \over {u_j^2}}}}

There is another parameter which is sometimes calculated in order to obtain a more complete insight into measurement results. This parameter is called maximum absolute uncertainty and represents the maximum absolute departure of measurement results from x̄.

Its definition is [18] Δx=|xi−x¯| max \Delta x = |{x_i} - \bar x{|_{\;\max \;}}

Since Δx is linear in the departure of x_i from x̄, the maximum absolute error Δy of y = y = y(q₁, q₂,…, q_z) contains linear terms, and Equation [11] changes into [19] Δy=∑j=1z∂y∂qjqj=q¯jΔqj \Delta y = \mathop \sum \nolimits_{j = 1}^z {\left| {{{\partial y} \over {\partial {q_j}}}} \right|_{{q_j} = {{\bar q}_j}}}\Delta {q_j} where the absolute values of ∂y/∂q_j ensure that every contribution to the sum is positive.

Equations [18] and [19] are also useful when repeated measurements always give the same result, for instance, when the length of an object is measured with a ruler of a limited precision. Does the fact that we always obtain the same number mean that there is no uncertainty? Obviously not. In this case, we should estimate the uncertainty in the sense of Δx.

An analysis of experimental results provides numerical values of x̄ (or ȳ or 〈x〉) and u (or Δx or Δy or 〈u〉). Usually, these values are used to calculate relative uncertainty [20] R=ux¯ or uyy¯ or Δxx¯ or Δyy¯ or ux×100% R = \left({{u \over {\bar x}}\,or\,{{{u_y}} \over {\bar y}}\,or\,{{\Delta x} \over {\bar x}}\,or\,{{\Delta y} \over {\bar y}}\,or\,{{\left\langle u \right\rangle} \over {\left\langle x \right\rangle}}} \right) \times 100\% which expresses uncertainty as a percentage of the mean. R is often included in the presentation of a result. For instance, if we obtain x̄ = 4.7823 and u = 0.876, this is usually presented as [21] ξ=4.8±0.9 \xi = 4.8 \pm 0.9

The number of digits of x̄ and u is smaller than the ones we have started from. The reason for that becomes clear if we recall the meaning of u: it sets a Gaussian interval around x̄ where ξ lies with a probability given by the coverage factor. Hence, u is always rounded to one digit, as the second digit would represent the uncertainty of an uncertainty, and this would not make much sense. The only exception is when the second digit can be rounded to 5, and in this case, u is expressed with two digits (e.g., u = 0.014827 → u = 0.015). Once u is rounded properly, the number of digits of x̄ (which are called significant digits) is set accordingly, to match the value of u. For example, ξ = 4.78±0.9 is not correct, because 0.08 (the third digit of the mean) is more than ten times smaller than the uncertainty 0.9. Similarly, ξ = 5±0.9 is also wrong, since u should not be more precise than x̄. Using too many digits for u and setting the number of significant digits of x̄ wrongly are the most frequent mistakes in the presentation of a measurement result. There are no strict rules for the number of digits of R, because R is just an indicator and not a fundamental quantity. The number of digits should, however, be kept low in this case as well (e.g., 0.4 %, 7 %, 14 %, 112 %).

There are cases when the result contains very large or very small numbers, which can be solved by using the powers of ten and/or unit prefixes. For example, it is more elegant to write ξ = (4.8±0.9) μm or ξ = (4.8±0.9) × 10⁻⁶ m than ξ = (0.0000048±0.0000009) m.

Suppose that a quantity is measured many times by the same instrument and the same method, with a goal to reduce u. For instance, one weighs an object repeatedly, every time obtaining a slightly different result for the mass. Provided that there are no systematic errors, these differences are the result of the imperfections of the scale and are distributed evenly around the mean value of the mass. In order to calculate the mean and its uncertainty, one uses Equation [6] and Equation [13], respectively. Then one selects the coverage factor to define u in the spirit of Equation [8], properly rounds u and then the mean, calculates R, and presents the result as demonstrated in Equation [21]. This is simplest use of TRME.

Even though σ and s can be calculated from the data, their roles are marginal. In fact, to use σ or s instead of u in Equation [8] is the most common mistake, i.e., a typical misuse of TRME.

Suppose that we have to determine the density ϱ of a cylinder by measuring its mass μ, diameter d, and height h. Our measurement instruments are a calliper (with a precision of 0.05 mm) and a scale (with a precision of 1 mg). We measure μ, d, and h about ten times, observe small variations due to the imperfections of the instruments, and use Equations [6] and [13] to obtain μ = 28.903±0.004 g, d = 23.04±0.07 mm, and h = 12.42±0.04 mm.

We first calculate ρ¯=4μ¯/d¯2 πh¯ \bar \rho = 4\bar \mu /{\bar d^2}\,\pi \bar h . Since ρ is a function of d, h, and μ, which are directly measured quantities with their own uncertainties, Equation [11] applies and gives [22] uρ=ρ¯uμμ¯2+2udd2+uhh2 {u_\rho} = \bar \rho \sqrt {{{\left({{{{u_\mu}} \over {\bar \mu}}} \right)}^2} + {{\left( {2{{{u_d}} \over d}} \right)}^2} + {{\left({{{{u_h}} \over h}} \right)}^2}}

By using the coverage factor k = 2 (which means that we multiply the result of Equation [22] by two and thus obtain a confidence level of ~95 %) and after rounding the result properly, we obtain ϱ = 5.58±0.08 g/cm³ (R = 1.4 %).

This exemplifies the general method of calculating the mean and uncertainty of a quantity that has not been measured directly but can be calculated from directly measured quantities.

Sometimes it occurs that repeated measurements of a quantity always give the same result. This however, does not mean that there is no uncertainty, that the measurements are prefect. The reason for always obtaining the same result is that the measurement instrument is not precise enough to record small differences. In such cases, one ought to estimate uncertainty, which then has the meaning of a maximum absolute uncertainty.

Imagine that we have the same cylinder as in the previous example. This time, we only have a scale with a precision of 0.1 g and a ruler with 1 mm spaced ticks, which is too crude to obtain anything else than μ = 28.9 g, d = 23 mm, and h = 12.5 mm, even if the measurements are repeated many times. Regarding the ruler, we can distinguish values with a precision of 0.5 mm, not better than that. Hence, when we say d = 23 mm, this actually means that d can be anywhere between 22.75 mm and 23.25 mm. This sets the maximum absolute uncertainty of d to Δd = 0.25 mm, and the same applies to Δh because we use the same ruler. By the same reasoning, Δμ = 0.05 g. This is the usual approach to the estimation of uncertainties in cases when there is no other option.

Hence, we start with μ = 28.90±0.05 g, d = 23.00±0.25 mm, and h = 12.50±0.25 mm. Since d, h, and μ are directly measured quantities expressed through their means and maximum absolute uncertainties, we use Equation [19] to obtain [23] Δϱ=ρ¯Δμμ¯+2Δdd¯+Δhh¯ \Delta \varrho = \bar \rho \left({{{\Delta \mu} \over {\bar \mu}} + 2{{\Delta d} \over {\bar d}} + {{\Delta h} \over {\bar h}}} \right)

This gives ϱ=ϱ¯±Δϱ=5.6±0.2 g/cm3 \varrho = \bar \varrho \pm \Delta \varrho = 5.6 \pm 0.2\,{\rm{g}}/{\rm{c}}{{\rm{m}}^3} (R = 3.6 %). Note that we have not used any coverage factor, because the maximum absolute uncertainties are not related to Gaussian distribution.

It is worthwhile to mention that uncertainties are also often estimated not because a measurement instrument is not precise enough but because a detailed analysis of measurement errors is less important than keeping the presentation free of defocusing details. In any case, estimation of uncertainties always depends on the nature of an experiment, but once uncertainties have been estimated, the above procedure applies.

One can often see a technical report or a manual where there are expressions containing numbers and the ± sign. For instance, “the appropriate voltage for this device is 16.1±0.77 V”. Is this just another misuse of TRME, because of the inconsistent rounding of the mean and uncertainty? The answer is no, because this case is not in the domain of TRME at all. The meaning of 16.1±0.77 V is not that the applied voltage is uncertain for 0.77 V, but that the device can work properly with voltages between 15.33 V and 16.87 V. It would be equivalent to saying “you can use any voltage in the range 15.33–16.87 V, and your device will work properly”.

A statistical population consists of many individual entities that differ from one another but may share a common trait that can be investigated. Investigations of statistical populations are numerous in different fields of science, such as epidemiological studies in medicine, zoological or botanical studies in biology, or geographical or geophysical studies. Related analyses of collected data may include descriptive statistics, appropriate statistical tests, comparative tests, and sometimes also TRME.

TRME is often misused in studies of statistical populations, and one can find all sorts of doubtful analyses and presentations, many containing confusing uses of σ, s, and u. In the Tang et al. study (7), for example, the mean height is 130.9±6.2 cm (referring to a sample of 3,194 eight-year-old boys). The meaning of this result is unclear. If 6.2. refers to u calculated using Equation [13], there are several problems with that. First, Equation [13] is applicable to situations where a measurement of a same sample is repeated using the same method and not to a set of measurements of different samples even if the same method is used. Second, if 6.2. is meant to represent u, the above result is not rounded properly, and one would also expect to see a comment on the used coverage factor. It may be, however, that 6.2 refers to σ or s (it should be s, because the result is related to a sample and not the entire population of eight-years-old boys in the area). In that study, the difference between σ and s is irrelevant (because n >> 1), but in some cases it might be necessary to specify it clearly. Regardless of whether it is σ or s, the notation 130.9±6.2 is wrong. Namely, 130.9 cm and 6.2 cm should be written separately, since the mean is independent of σ and s. When presenting σ or s, any reasonable number of digits can be used (there is no rounding), but the number corresponding to last digit should not be smaller than the maximum absolute error of a given measurement method.

In fact, TRME is of limited importance in studies of statistical populations, since it may account for the precision of an individual measurement but does not say much about the studied property of the whole sample/population. More information on that can be obtained in the analyses that either use a graphical presentation (8) or give the values of x̄, σ (or s), α₃, α₄, ν, and η (9).

Suppose that one has to analyse a set of independent results, originating from different measurements of the same quantity, and calculate the overall mean and uncertainty from these results. A good example of this situation is a calculation of the diameter D of a star from the measurements obtained from different observatories. The quality of these measurement may vary from observatory to observatory, since some are in a more favourable position than others or use better equipment. Each observatory provides the analyst with its result expressed as D_i = D̅ _i±u_i.

A relatively common misuse of TRME is to calculate the mean of D using Equation [6] with x_i → D̅ _i, disregard individual uncertainties u_i, and then calculate u using Equation [13] (or even Equation [7] or Equation [9]), again with x_i → D̅ _i. This is wrong for two reasons: 1) D̅ _i is not an individual measurement of D in a series of the same measurements, and 2) u_i carries information on the precision of each D_i. This problem can be solved by using Equation [16] to calculate 〈D〉 and Equation [17] to calculate u = 〈u_D〉, which have the meaning of the mean and uncertainty of D as calculated from the results from all of the observatories.

One may argue that the use of Equations [6] and [13] is not always completely wrong, since there is a situation when they give approximately the same result as Equations [16] and [17]. Namely, if u_i ≈ u₀ for every i, Equation [16] becomes 〈x〉≈ Σ_ix̄_i/n, where n is the number of independent results. However, the similarity is only numerical, and the use of Equation [6] cannot be justified conceptually. Under the same conditions, Equation [13] gives 〈u〉2≈u02/n {\langle u\rangle^2} \approx u_0^2/n , which resembles Equation [17] if we assume u₀ ≈ s with x̄ → 〈x〉 and x_i → x̄_i. Again, the similarity is only numerical and cannot be justified, especially because Equations [16] and [17] require no approximation. In addition, the statistical weight wi=1/ui2 {w_i} = 1/u_i^2 takes care that more precise results contribute more to the final result.