Another major issue deserves special mention. Correcting for biases that may be present in the data or data samples is a topic for the statistics of data analysis which extends beyond the scope of this paper. The problems associated with sample statistics are complex and involved, more so for some distance indicators than for others, although all indicators are affected to some degree. Unfortunately, there is no single prescription that is appropriate in all circumstances, and a complete discussion is better addressed in another review (Bertschinger 1992). For the purposes of this paper, we call the attention of the reader to a few common biases that afflict several techniques, and these should be kept in mind throughout this review.
A true Malmquist (1920) effect may be present such that any sample of objects becomes more and more restricted to brighter members as distance increases. Thus, a sample of distant galaxies will have a higher average luminosity than a nearby sample. This effect, in itself, does not mandate biased distances provided that any parameter used to predict the galaxy luminosity is not also biased.
A more subtle effect can arise if the galaxy sample is chosen from a set of biased parameters and those parameters are also used to estimate distances (e.g., magnitudes or diameters taken from a photographic catalog). If those same parameters, which have some measurement uncertainty, enter into the calculation of distances, then the sample will include galaxies near the selection limit that should have been rejected (i.e., they appear brighter or larger than the selection limit). The improper sample selection introduces biased distance estimates near the completeness limit which, in turn, biases the sample mean.
Biases can also result from differences in the environments of the samples. For example, most methods are calibrated locally but applied in large galaxy clusters. If the galaxies in clusters are somehow different from those in the Local Group, a very simple bias arises. Similarly, first-ranked galaxies may foster unusual objects (e.g., extremely bright planetary nebulae, globular clusters, red giants) which do not develop in the Local Group, again leading to a simple calibration bias.
Another subtle form of bias can be introduced in the analysis phase of distance determination by binning data. A histogram of globular clusters or planetary nebulae, for example, will have a bias which scatters objects from heavily populated bins into bins with fewer objects. Proper analysis techniques (e.g., maximum likelihood, numerical simulations) can account for this effect, first discussed by Eddington (1913, 1940), and more recently by Trumpler and Weaver (1962). A variant of the problem is discussed by Lynden-Bell et al. (1988) in the context of deriving the average distance to a cluster of galaxies. When the distance measuring technique has an uncertainty which is large enough that a significant number of background and foreground galaxies are inadvertently included in the sample, the mean distance for the cluster sample may be biased, depending on the spatial density of the sample and interloper galaxies.