NEW DERIVATIONS

OF THE ROOT-MEAN-SQUARE ERROR OF A SINGLE MEASUREMENT

V. S. Nuvaryev

In Gauss’s derivation, based on the relation of the sum of squares of the corrections \(v_i\) to the sum of squares of the true errors \(\Delta_i\), the following formula was obtained for the root-mean-square error \(m\) of a single observation:

\[ m^2=\frac{[v^2]}{n-u}, \tag{a} \]

where \(n\) is the number of error equations and \(u\) is the number of unknowns. The derivation of the root-mean-square error in adjustment by the method of conditional equations is constructed by analogy with the method of indirect observations; in doing so, for example, Jordan includes among the equations identities of the form \(x_i=v_i\). We shall show that the formula obtained by Jordan under this construction,

\[ m=\sqrt{\frac{[vv]}{r+(n-r)-(n-r)}}=\sqrt{\frac{[vv]}{r}}, \]

where \(r\) is the number of conditional equations, can be found in another way.

In our work “The relation between indirect and conditional observations”¹ we showed in general form that the values \(l_i^0\) of the observed quantities \(l_i\), adjusted by the methods of conditional and indirect observations, are equal, i.e., the corrections \(v_i\) to the observed values are also equal in the sum \([vv]\). A system of \(n\) equations with \(u\) unknowns can be reduced to \(r=n-u\) conditional equations. The root-mean-square error of a single measurement does not depend on the method that we have chosen for the adjustment, i.e., the formula for computing \(m\) must give one and the same result. This requirement is fulfilled only under the condition

\[ m=\sqrt{\frac{[v^2]}{n-u}}=\sqrt{\frac{[v^2]}{r}}, \tag{b} \]

i.e., the formula for the method of conditional observations is obtained without additional considerations.

The question of the relation between corrections and true errors, which led to formulas \((a)\) and \((b)\), interests us not only from the point of view of obtaining an approximate value \(m\) for a single observation or for a unit of weight. These formulas, in our opinion, provide a basis for judging the merit of the results of adjustment.

In investigating derivations of \(m\) based on determining the relation of the sum \([v^2]\) to \([\Delta^2]\), one may get the idea that a derivation based on determining the dependence of the first powers \(v_i\) and \(\Delta_i\) would be more intuitive

Therefore we have developed derivations of formulas for the root-mean-square error of a single observation in which the dependence of each correction on the true errors of the measurements is first determined. In addition, we give a derivation of the root-mean-square error for the case of \(r\) equations in \(n\) unknowns \((r<n)\), based on the known relation

\[ [vv]=-[kw]. \]

The root-mean-square error of one measurement for \(n\) equations and \(u\) unknowns \((n>u)\)

For simplicity of exposition, let us take \(n\) equations containing two unknowns. The exact values of the unknowns and the quantities to be measured are connected by conditions of the form

\[ 0=-L_i+a_iX+b_iY . \]

The adjusted values must then satisfy the following equations:

\[ 0=-(l_i+v_i)+a_ix^\circ+b_iy^\circ; \]

subtracting the first equation from the second, we obtain

\[ v_i=-\Delta_i+a_i\Delta x+b_i\Delta y, \tag{a} \]

where \(\Delta_i\) is the true error of \(l_i\); \(\Delta x,\Delta y\) are the true errors of the adjusted values \(x^\circ,y^\circ\).

The adjusted values \(x^\circ,y^\circ\) are found from the normal equations by the formulas

\[ x^\circ=\frac{1}{D}\bigl([bb][al]-[ab][bl]\bigr) \]

\[ y^\circ=\frac{1}{D}\bigl([aa][bl]-[ab][al]\bigr), \tag{b} \]

where \(D\) is the determinant of the system of normal equations.

We differentiate \((b)\) with respect to the variables \(l_i\) and replace the differentials by finite changes; then

\[ \Delta x=\frac{1}{D}\left\{(a_1[bb]-b_1[ab])\Delta_1+(a_2[bb]-b_2[ab])\Delta_2+\cdots\right\} \]

\[ \Delta y=\frac{1}{D}\left\{(b_1[aa]-a_1[ab])\Delta_1+(b_2[aa]-a_2[ab])\Delta_2+\cdots\right\}. \tag{c} \]

We introduce the notation

\[ A_i=a_i[bb]-b_i[ab];\qquad B_i=b_i[aa]-a_i[ab]. \tag{d} \]

Then, substituting \((c)\) into \((a)\), and taking \((d)\) into account, we obtain

\[ v_1=\frac{1}{D}\left\{(-D-a_1A_1+b_1B_1)\Delta_1+(a_1A_2+b_1B_2)\Delta_2+\cdots\right\} \]

\[ v_2=\frac{1}{D}\left\{(a_2A_1+b_2B_1)\Delta_1+(-D+a_2A_2+b_2B_2)\Delta_2+\cdots\right\} \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

Squaring, we obtain

\[ \begin{aligned} v_1^2={}&\frac{1}{D^2}\Bigl\{ \bigl(D^2+a_1^2A_1^2+b_1^2B_1^2-2Da_1A_1-2Db_1B_1+2a_1b_1A_1B_1\bigr)\Delta_1'^2 \\ &\qquad\qquad+\bigl(a_1^2A_2^2+b_1^2B_2^2+2a_1b_1A_2B_2\bigr)\Delta_2'^2+\ldots \Bigr\},\\[6pt] v_2^2={}&\frac{1}{D^2}\Bigl\{ \bigl(a_2^2A_1^2+b_2^2B_1^2+2a_2b_2A_1B_1\bigr)\Delta_1'^2 \\ &\qquad\qquad+\bigl(D^2+a_2^2A_2^2+b_2^2B_2^2-2Da_2A_2-2Db_2B_2+2a_2b_2A_2B_2\bigr)\Delta_2'^2+\ldots \Bigr\}. \end{aligned} \tag{l} \]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

To the expressions \(v_1^2, v_2^2,\ldots\) it would have been necessary to add terms containing products of errors \(\Delta_i\Delta_j\). Let us suppose that \(n\) is sufficiently large for the sum of the terms containing products of the true errors \(\Delta_i\Delta_j\) to be set equal to zero. In this case equations \((l)\) strictly determine the values of the corrections obtained by the method of least squares, if true errors \(\Delta_i\) are admitted in the given series of observations. Since the true errors \(\Delta_i\) are unknown to us, we are forced to replace them by their root-mean-square errors; then

\[ v_i^2=\frac{1}{D^2}\Bigl\{ D^2+a_i^2[A^2]+b_i^2[B^2]+2a_ib_i[AB]-2Da_iA_i-2Db_iB_i \Bigr\}m^2 \tag{f} \]

represent a certain mean value of \(v_i^2\), and \([v_i^2]\) from \((f)\) will no longer be equal to \([v_i^2]\) from \((l)\), i.e. \([v_i^2]\) from \((f)\) will not be equal to the sum of squares of the corrections belonging to the given series of observations. Summing \(v_i^2\) and denoting this sum by \([v_m^2]\), we obtain

\[ [v_m^2]=\frac{1}{D^2}\Bigl\{ nD^2+[aa][A^2]+[bb][B^2]+2[ab][AB]-2D\bigl([aA]+[bB]\bigr) \Bigr\}m^2. \]

Since

\[ D=[aa][bb]-[ab]^2; \]

\[ [A^2]=[bb]D;\qquad [B^2]=[aa]D;\qquad [AB]=-[ab]D; \]

\[ [aA]=[bB]=[aa][bb]-[ab]^2=D, \]

then

\[ [v_m^2]=\frac{1}{D^2}\Bigl\{ nD^2+2D\bigl([aa][bb]-[ab]^2\bigr)-4D^2 \Bigr\}m^2, \]

whence

\[ m^2=\frac{[v_m^2]}{n-2}. \]

We have obtained the well-known formula. In deriving it, we assumed a sufficiently large \(n\), for which we can set the sum of the terms containing products of true errors equal to zero. In addition, by \([v_m^2]\) we understand the sum of squares of the mean corrections obtained from many series of measurements. The fact that, for simplicity of exposition, we limited ourselves to two unknowns has not affected the method of derivation; hence we conclude that the generality of the derivation does not depend on the number of unknowns.

Mean-square error of a single measurement with \(n\) unknowns connected by \(r\) \((r<n)\) conditions

First derivation

Suppose that the corrections are connected by \(r\) conditional equations of the form

\[ \begin{aligned} a_1v_1+a_2v_2+\ldots+a_nv_n+w_1&=0\\ b_1v_1+b_2v_2+\ldots+b_nv_n+w_2&=0\\ &\cdots \end{aligned} \tag{a} \]

Then, solving the normal equations

\[ \begin{aligned} [aa]k_1+[ab]k_2+\ldots+[ar]k_r+w_1&=0\\ [ab]k_1+[bb]k_2+\ldots+[br]k_r+w_2&=0\\ &\cdots\\ [ar]k_1+[br]k_2+\ldots+[rr]k_r+w_r&=0, \end{aligned} \tag{b} \]

we obtain the values of the correlates in the form

\[ k_1=-\frac{1}{D} \begin{vmatrix} w_1 & [ab] & \ldots & [ar]\\ w_2 & [bb] & \ldots & [br]\\ \cdot & \cdot & \cdot & \cdot\\ w_r & [br] & \ldots & [rr] \end{vmatrix} = -\frac{1}{D}\left(D'_1w_1-D'_2w_2+D'_3w_3\ldots\right) \tag{c} \]

\[ k_2=-\frac{1}{D} \begin{vmatrix} [aa] & w_1 & \ldots & [ar]\\ [ab] & w_2 & \ldots & [br]\\ \cdot & \cdot & \cdot & \cdot\\ [ar] & w_r & \ldots & [rr] \end{vmatrix} = -\frac{1}{D}\left(-D''_1w_1+D''_2w_2-D''_3w_3\ldots\right) \]

where by \(D\) is meant the determinant of the system of equations \((b)\), and by \(D_k^i\) is meant the minor of the element of this determinant standing in column \(i\) and in row \(k\).

Multiplying the left- and right-hand sides of \((c)\) respectively by \(w_1,w_2,\ldots\), we then have

\[ -w_1k_1=\frac{1}{D}\left(D'_1w_1^2-D'_2w_1w_2+D'_3w_1w_3\ldots\right) \]

\[ -w_2k_2=\frac{1}{D}\left(-D''_1w_1w_2+D''_2w_2^2-D''_3w_2w_3\ldots\right) \tag{d} \]

\[ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot, \]

where

\[ w_1=[a_i\Delta_i]\qquad w_2=[b_i\Delta_i] \tag{e} \]

Suppose that for sufficiently large \(n\) the sum of the terms containing the products \(\Delta_i\Delta_j\) vanishes. Then, replacing the squares of the true errors by the squares of the mean errors, we obtain

\[ -k_1w_1=\frac{1}{D}\left(D'_1[aa]-D'_2[ab]+D'_3[ac]-\ldots\right)m^2; \]

\[ -k_2w_2=\frac{1}{D}\left(-D''_1[ab]+D''_2[bb]-D''_3[bc]+\ldots\right)m^2; \tag{f} \]

\[ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \]

We note that the brackets in expression \((f)\) constitute the expansion of the determinant \(D\) by the elements of the first, second, etc. columns. Then it follows directly that

\[ -[kw]=[v^{2}m]=rm^{2}, \]

\[ m^{2}=-\frac{[v^{2}m]}{r}. \]

Second derivation

All operations with mean quantities become most transparent in the case where the relation between the first powers of the corrections and the true errors has first been found. Here it must be borne in mind that all necessary reductions must be made before replacing the true errors by mean square ones. Otherwise the result of the derivation may be incorrect. These considerations prompted us to propose yet another derivation of the mean square error of a single measurement in adjustment by the method of conditional observations.

It is known that

\[ v_i=a_i k_1+b_i k_2+\ldots+r_i k_r \tag{g} \]

Then, taking into account expressions \((c)\),

\[ v_i=-\frac{1}{D}\{a_i(D_1'w_1-D_2'w_2+D_3'w_3-\ldots)+b_i(-D_1''w_1+D_2''w_2-D_3''w_3+\ldots)+ \]

\[ +c_i(D_1'''w_1-D_2'''w_2+D_3'''w_3-\ldots)+\ldots\}; \]

on the basis of \((e)\) we obtain

\[ v_i=-\frac{1}{D}\{[a_i(a_1D_1'-b_1D_2'+c_1D_3'-\ldots)+b_i(-a_1D_1''+b_1D_2''-c_1D_3''+\ldots)+ \]

\[ +c_i(a_1D_1'''-b_1D_2'''+c_1D_3'''-\ldots)+\ldots]\Delta_1+ \]

\[ +[a_i(a_2D_1'-b_2D_2'+c_2D_3'-\ldots)+b_i(-a_2D_1''+b_2D_2''-c_2D_3''+\ldots)+ \tag{h} \]

\[ +c_i(a_2D_1'''-b_2D_2'''+c_2D_3'''-\ldots)+\ldots]\Delta_2+\ldots\}.\!^{1)} \]

But \(v_i\) is a linear function of individual deviations that obey the normal law. Consequently, we may use the formula for the square of the mean deviation; therefore, squaring \(v_i\), replacing \(\Delta_i\) by \(m^{2}\), summing, and equating to zero the terms containing products \(\Delta_i\Delta_j\), we obtain:

\[ v_{im}^{2}=\frac{1}{D^{2}}\{a_i^{2}([aa]D_1'^{\,2}+[bb]D_1''^{\,2}+\ldots-2[ab]D_1'D_2'+ \]

\[ 2[ac]D_1'D_3'-2[bc]D_3'D_3'+\ldots)+b_i^{2}([aa]D_1''^{\,2}+[bb]D_2''^{\,2}+ \]

\(^1\) Formula \((h)\) rigorously determines the dependence of the corrections \(v_i\), obtained by the method of least squares, on the true errors \(\Delta_i\) pertaining to the given series of observations. If we had at our disposal a large number of series of observations, \(v_i\) would, within known limits, take all possible values. In passing from \(\Delta_i^{2}\) to \(m^{2}\), \(v_i^{2}\) must be replaced by \(v_{im}^{2}\), i.e. by the square of a correction that is the mean from many series of measurements. This circumstance is very important to note, since one might think that in the formula \(rm^{2}=[v^{2}]\), by \([v^{2}]\) are meant the squares of corrections pertaining to the given series of measurements.

\[ \begin{aligned} &+[cc]D_3^{\prime\prime 2}+\ldots-2[ab]D_1''D_2''+2[ac]D_1''D_3''-2[bc]D_2''D_3''+\ldots)+\\ &+c_i^2\bigl([aa]D_1^{\prime\prime\prime 2}+[bb]D_2^{\prime\prime\prime 2}+[cc]D_3^{\prime\prime\prime 2}+\ldots-2[ab]D_1'''D_2'''+\\ &\qquad\qquad +2[ac]D_1'''D_3'''-2[bc]D_2'''D_3'''+\ldots\bigr)+\\ &\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\ &+2a_i b_i\bigl[D_1'\bigl(-[aa]D_1''+[ab]D_2''-[ac]D_3''+\ldots\bigr)+\\ &\qquad +D_2'\bigl([ab]D_1''-[bb]D_2''+[bc]D_3''-\ldots\bigr)+\\ &\qquad +D_3'\bigl(-[ac]D_1''+[bc]D_2''-[cc]D_3''+\ldots\bigr)+\ldots\bigr]+\\ &+2a_i c_i\bigl[D_1'\bigl([aa]D_1'''-[ab]D_2'''+[ac]D_3'''-\ldots\bigr)+\\ &\qquad +D_2'\bigl(-[ab]D_1'''+[bb]D_2'''-[bc]D_3'''+\ldots\bigr)+\\ &\qquad +D_3'\bigl(+[ac]D_1'''-[bc]D_2'''+[cc]D_3'''+\ldots\bigr]+\ldots\bigr]+\\ &+2b_i c_i\bigl[D_1''\bigl(-[aa]D_1'''+[ab]D_2'''-[ac]D_3'''+\ldots\bigr)+\\ &\qquad +D_2''\bigl([ab]D_1'''-[bb]D_2'''+[bc]D_3'''+\ldots\bigr)+\\ &\qquad +D_3''\bigl(-[ac]D_1'''+[bc]D_2''-[cc]D_3'''+\ldots\bigr)+\ldots\bigr]+\ldots\Bigr\}m^2. \end{aligned} \]

Let us represent the coefficient of \(a_i^2\) in the following form:

\[ D_1'\bigl([aa]D_1'-[ab]D_2'+[ac]D_3'-\ldots\bigr) +D_2'\bigl(-[ab]D_1'+[bb]D_2'-[bc]D_3'+\ldots\bigr)+ \]

\[ +D_3'\bigl([ac]D_1'-[bc]D_2'+[cc]D_3'-\ldots\bigr)+\ldots . \]

Suppose there were altogether \(r\) conditional equations; then the number of terms in the coefficient of \(a_i^2\) will be \(r\). The first of these terms is the expansion of the determinant of the system of normal equations \((b)\) with respect to the elements of the first column, i.e. it is equal to \(D\). The remaining terms are equal to zero, since the expressions standing in parentheses represent the sum of the products of the minors of the elements of one column of the determinant of the system \((b)\) by the elements of another column. On this basis the coefficients of \(a_i^2\), and likewise of \(b_i^2, c_i^2,\ldots\), will respectively be equal to

\[ DD_1',\quad DD_2'',\quad DD_3'''\ldots DD_i \qquad (i=1,2,\ldots,r), \]

and the coefficients of \(2a_i b_i, 2a_i c_i,\ldots, 2b_i c_i\ldots\), respectively, are equal to

\[ -DD_2',\quad DD_3',\quad \ldots,\quad -DD_3''\ldots, \]

or

\[ v_{im}^{2}=\frac{1}{D}\left\{ a_i^2D_1'+b_i^2D_2''+c_i^2D_3'''+\ldots-2abD_2' +\right. \]

\[ \left. +2acD_3'-\ldots-2bcD_3''+\ldots \right\}m^2 . \]

Summing, we obtain

\[ [v_n^2]=\frac{1}{D}\left\{ ([aa]D_1'-[ab]D_2'+[ac]D_3'+\ldots)+ \right. \tag{i} \]

\[ +(-[ab]D_2'+[bb]D_2''-[bc]D_3''+\ldots)+ \]

\[ \left. +([ac]D_3'-[bc]D_3''+[cc]D_3'''-\ldots)+\ldots \right\}m^2 . \]

The minor of element \(i\) in column \(k\) is equal to the minor of element \(k\) in column \(i\), i.e. \(D_i^k=D_k^i\), since for the normal equations the minor \(D_i^k\) is obtained from the minor \(D_k^i\) by replacing rows by columns. On this basis, each expression standing in parentheses is equal to the determinant of system \((b)\), i.e.

\[ [v_m^2]=rm^2. \]

Thus, having posed the problem in a general form, we have arrived at the result

\[ m^2=\frac{[v_m^2]}{r}, \]

i.e. the root-mean-square error of a single measurement can be strictly determined under two conditions:

1. The number of observed quantities must be sufficient for the sum of the terms containing products of first powers of the true errors to be set equal to zero.

2. If, for \(m\), one uses certain mean corrections obtained from a large number of series of observations, and not from the given series.

If only one series of observations is used to compute \(m\), then the formula

\[ m^2=\frac{[v^2]}{r} \]

gives an approximate value of the root-mean-square error of one observation; the agreement of the result determined by this formula with the root-mean-square error determined, for the given conditions of measurement, from the reduction of a larger number of observations indicates a distribution of errors in the given series of observations close to the “mean” distribution. The equality

\[ [v_m^2]=rm^2=\frac{r}{n}[\Delta^2] \]

gives us the possibility of regarding the series of corrections \(v_i\) as a representation of the series of true errors \(\Delta_i\), reduced by a factor of

\[ \sqrt{\frac{r}{n}}. \]

\[ * \quad * \]
\[ * \]

Here we must touch upon the foundations of the derivation of the formulas

\[ m(\bar m)=\sqrt{\frac{1}{2(n-u)}} \quad \text{and} \quad m(m)=\sqrt{\frac{1}{2r}}. \tag{k} \]

The formulas

\[ m=\sqrt{\frac{[v^2]}{n-u}} \quad \text{and} \quad m=\sqrt{\frac{[v^2]}{r}} \tag{l} \]

are obtained from the assumption that the number \(n\) of measured quantities \(l_i\) is sufficiently large for the sum of the terms containing products of first powers of the true errors \(\Delta_i\) to be set equal to zero.

Let us consider from this point of view, for example, the derivation of the formula \(m(m)\) for \(u=1\).

Then\(^1\)

\[ m^2=\frac{[v^2]}{n-1}=\frac{[\Delta^2]-\dfrac{[\Delta]^2}{n}}{n-1}. \]

The true value \(m_0^2\), which is obtained for an infinitely large number of quantities \(r\) (i.e., for \(n=\infty\)), is taken, and the difference is formed

\[ m^2-m_0^2=\frac{[\Delta^2]-\dfrac{[\Delta]^2}{n}}{n-1}-m_0^2. \tag{m} \]

In the further derivation a number of assumptions are made, namely:

1. \([\Delta]^2\) is set equal to \([\Delta^2]\), i.e., \(n\) is assumed sufficiently large;

2. \(m\) is replaced by \(m_0\), where

\[ m_0^2=\frac{[\Delta^2]}{n}, \]

where \(n=\infty\).

If in formula \((m)\) \([\Delta]^2=[\Delta^2]\), then the question arises as to what the difference is between the definition of a “sufficiently large number” \(n\) and \(n=\infty\), since true errors acquire no new properties when \(n=\infty\), and assumption 1 is also valid for this case.

But then

\[ m^2-m_0^2=\frac{[\Delta^2]-\dfrac{[\Delta^2]}{n}}{n-1}-\frac{[\Delta^2]}{n}=0. \]

The same considerations can be brought forward with respect to the derivation for the general cases as well. Hence we conclude that the formulas \(m(m)\) require further investigation.

The root-mean-square error of a single measurement and of unit weight

Let a quantity be observed \(n\) times with equal accuracy, its true value being

\[ X=\frac{[L]}{n}; \]

then we may write:

\[ 0=L_i-\frac{[L]}{n}. \tag{a} \]

If, instead of the true values \(L_i\), in \((a)\) we substitute the measured values \(l_i\), then we obtain the deviation from zero \(\delta_i\), the square of whose mean value will be equal to:

\[ m^2(0)=\delta_m^2=\left\{\left(\frac{n-1}{n}\right)^2-\frac{n-1}{n}\right\}m_l^2, \]

or

\[ \delta_m^2=\frac{n-1}{n}m_l^2. \]

Summing \(\delta_i^2\), we have

\[ [\delta^2]=(n-1)m_l^2, \]

\(^1\) See, for example, Iordans, Handb. d. Vermess., Band I, 1920, § 146; moreover, in our notation the true errors, instead of \(\varepsilon_i\), are denoted by \(\Delta\); Russian translation, 1939, pp. 635–639.

whence the root-mean-square error of a single measurement is

\[ m=\sqrt{\frac{[\delta^2]}{n-1}}. \]

For the root-mean-square error of a unit of weight there exist the formulas

\[ \mu=\sqrt{\frac{[p\Delta^2]}{n}} \tag{b} \]

\[ \mu=\sqrt{\frac{[p\delta^2]}{n-1}}, \tag{c} \]

where \(\Delta_i\) are the true errors, and \(\delta_i\) are the probable errors of the arithmetic mean.

Let us attempt to determine \(\mu\) on the basis of other considerations; for this purpose we shall use the formula for the mean error of a linear function of direct observations.

Suppose that it is proposed to carry out \(P\) equal-accuracy measurements of one and the same quantity, and that the measurements are divided into \(n\) groups with numbers of measurements \(p_1, p_2,\ldots p_n\).

Let the true value of the measured quantity be \(L\), and the measured values \(l_i\).

Then, denoting the arithmetic means by \(x_1^0, x_2^0,\ldots x_n^0\), we have:

\[ x_1^0=\frac{[l']}{p_1}, \qquad x_2^0=\frac{[l']}{p_2}\ldots \tag{d} \]

If, instead of \(l_i\), the expressions (d) are understood to contain the true value \(L\), then

\[ \Delta_1=x_1^0-L=0; \qquad \Delta_2=x_2^0-L=0\ldots \tag{e} \]

If, however, instead of \(x_i^0\) in expressions (e), we substitute the corresponding values from (d), then we obtain the true errors, which are a linear function of the direct observations,

\[ \Delta_1=\frac{[l']}{p_1}-L, \qquad \Delta_2=\frac{[l']}{p_2}-L\ldots, \tag{f} \]

and therefore

\[ m^2(0)=\Delta_1^2=\frac{\mu^2}{p_1}, \qquad m^2(0)=\frac{\mu^2}{p_2}\ldots, \tag{g} \]

whence\(^1\)

\[ [\Delta^2]=\mu^2\left[\frac{1}{p}\right], \qquad \mu=\sqrt{\frac{[\Delta^2]}{\left[\frac{1}{p}\right]}}. \tag{1} \]

Let us turn to the case when the true value \(L\) is unknown, i.e., let us now consider formula (c).

It is known that between the mean error of one measurement, or of a unit of weight, and the mean error of the arithmetic mean there exists a relation determined by the formula

\[ \mu^2=pM^2, \quad \text{where } \mu^2=\frac{[\delta^2]}{n-1}. \]

\(^1\) Formula (1) was indicated by Prof. A. S. Chebotarev as a check formula (St. n. kv. M., 1936, p. 342). It can be shown that the error in \(\mu\) according to this formula is smaller than according to the formula (b) that is used.

It is assumed that there are individual measurements from which their deviations from the arithmetic mean are determined.

However, in determining the root-mean-square error of a unit of weight from the deviations of the values of the arithmetic means \(x^0_1, x^0_2 \ldots\) from the most probable result \(x^0\), obtained from all observations, the nature of the question is somewhat different.

Let us denote by \(\delta^0_1, \delta^0_2 \ldots\) the differences between the values \(x^0_1, x^0_2 \ldots\)—the arithmetic means, respectively, from \(p_1, p_2 \ldots\) observations—and the arithmetic mean from all observations. Then

\[ \delta^0_1=\frac{[l']}{p_1}-\frac{[l]}{p};\quad \delta^0_2=\frac{[l']}{p_2}-\frac{[l]}{p};\ldots \tag{k} \]

If, in the right-hand side, the true values are understood in place of \(l\), then \(\delta_i=0\), and therefore, for \(P=[p]\),

\[ m^2(0)=\delta^2_1= \left(\frac{1}{p_1}-\frac{1}{P}\right)^2 p_1\mu^2+ (P-p_1)\frac{1}{P^2}\mu^2; \]

\[ m^2(0)=\delta^2_1=\frac{P-p_1}{Pp_1}\mu^2, \]

or

\[ m^2(0)=\delta^2_i=\frac{P-p_i}{Pp_i}\mu^2, \]

whence

\[ [\delta^2]=\left[\frac{P-p_i}{Pp_i}\right]\mu^2 . \]

Then, for the root-mean-square error of a unit of weight, found from the deviations of the individual values \(x^0_1, x^0_2,\ldots\) from the common arithmetic mean \(x^0\), we obtain the new formula

\[ \mu=\sqrt{\frac{[\delta^2]}{\left[\frac{1}{p}\right]-\frac{n}{[p]}}}. \tag{2} \]

Formula (2), for \(n=1\), gives an indeterminacy, since in this case

\[ \left[\frac{1}{p}\right]=\frac{n}{[p]}=\frac{1}{p};\qquad \delta^2=0. \]

In the example (see A. S. Chebotarev, Method of Least Squares, Moscow, 1936, pp. 87–93), the error of a single measurement was obtained as \(\pm 2.1\); by the formula

\[ \mu=\sqrt{\frac{[p\Delta^2]}{n}}=\pm 1.22 \]

and by the formula

\[ \mu=\sqrt{\frac{[p\delta^2]}{n-1}}=\pm 1.69; \]

whereas by the formulas proposed by us,

\[ \mu=\sqrt{\frac{[\Delta^2]}{\left[\frac{1}{p}\right]}}=\pm 2.00 \]

and by formula (2) \(\mu=\pm 1.98\), which agrees well with the actual value of the root-mean-square error of a single measurement.

Conclusions

1. The use, in deriving the formula for the root-mean-square error of a single measurement, of the relation between the first powers of the corrections and the true errors makes it possible to trace that this formula is obtained for a very large number of measured quantities, i.e., when the sum of the products of their true errors is equal to zero.

2. The root-mean-square error of the mean of the error itself, therefore, depends on the number of measured quantities.

3. The derivation of the root-mean-square error of a single measurement obtained on the basis of the relation \([v^2] = -kw\) for an unlimited number of equations is the simplest and most general; consequently, it may be used in the preparation of textbooks.

4. The formula \(m^2(m) = \dfrac{1}{2r}\), which depends only on the number of equations, is subject to further investigation.

5. The formulas in use for the error of unit weight, based on the deviations of individual means from the general arithmetic mean and on the true errors, do not give the correct result.

1. Vol. 67, issue 2 of the Proceedings of TPI. ↩