Several authors have employed analytical formulae to estimate errors in the measurement of line-strength indices (Rich 1988; Brodie & Huchra 1990; Carollo et al. 1993; González 1993). However, the derived error expressions have always been obtained from approximated versions of Eqs. (3 (click here)) and (4 (click here)), in which integrals are replaced by averaged number of counts in the bandpasses. Although this approximation works properly under certain circumstances, some additional assumed simplifications seriously constrain the suitability of such formulae. The expressions derived by Rich (1988) do not take into account readout noise (which can be important at low count levels) nor the effect of the sky subtraction. In the formulae employed by Brodie & Huchra (1990) and Carollo et al. (1993), although these factors are considered, important reduction steps such as flatfielding (in particular slit illumination) are not taken into account. All these problems are readily settled with a parallel reduction of data and error frames (González 1993, Sect. 3 (click here) this paper). Although this approach requires a more elaborate reduction, it yields the most confident results since it allows the application of analytical formulae which consider the final error in each pixel of the reduced spectra.
The most accurate set of analytic formulae published so far for the
computation of errors in line-strength indices are those presented by
González (1993). As a reference, we reproduce here his equations:
where
and
is the variance of 
at the wavelength
.
In order to check the accuracy of Eqs. (11 (click here)) and (12 (click here)) we compared the predictions of such expressions with the results obtained from numerical simulations. We took spectra and associated error spectra of a large homogeneous sample of 350 standard stars from the Lick library (observed with RBS --Richardson Brealy Spectrograph-- at the JKT --Jacobus Kapteyn Telescope-- of the Roque de los Muchachos Observatory, La Palma, February 1995). The error spectra were obtained by following a parallel reduction of data and error frames, as it has been previously described. As it is apparent from Fig. 2 (click here), there is an excellent agreement between González's formulae and the simulations.
Figure 2: Comparison of the relative errors employing González's
formulae with those obtained from numerical simulations. The symbols
corresponds to the measurement of six different line-strength
indices in a sample of 350 stars from the Lick library
Unfortunately, the showed concordance between the analytical formulae and the
simulations can not be extrapolated to any general situation. In particular,
one of the approximations used to derive Eqs. (11 (click here))
and (12 (click here)) is the assumption that the mean signal-to-noise ratio
in each bandpass can be computed as the quadratic sum of the individual
signal-to-noise ratio in each pixel
(Eqs. 16 (click here)-18 (click here)). In other words
where 
refers to the number of pixels involved in the
measurement of a particular bandpass. The results presented in
Fig. 2 (click here) were obtained employing data and error spectra in
which Eq. (19 (click here)) worked properly.
However, it is straightforward to see that, in this
expression, a simultaneous combination of large and small 
values would lead to a poor agreement between both terms.
For example, discrepancies between González's formulae and numerical
simulations are apparent in the measurement of the Mg2 index in the
spectra of the central dominant galaxy of the cluster Abell 2255 (observed
with the TWIN spectrograph at the 3.5 m Telescope of Calar Alto, August
1994), as it is shown in Fig. 3 (click here). These differences are
due to the fact that a bright sky-line falls within the central bandpass of
the Mg2 index. Sky subtraction during the reduction process introduces a
larger error in the pixels where sky-lines are present. As a result, the
reduced error spectra exhibit, simultaneously, pixels with very different
values, and Eq. (19 (click here)) is a poor
approximation.
Figure 3: Comparison of the Mg2 relative errors employing González's
formulae and numerical simulations. Filled triangles refer to a sample of 40
bright stars from the Lick Library, whereas open triangles correspond to
Mg2 measurements along the radius of the cD galaxy in Abell 2255. When
using González's expressions, there is a clear underestimation of the
relative errors in the galaxy due to the subtraction of a bright sky-line in
the Mg2 central bandpass during the reduction process