  I present a critical review of techniques for estimating confidence intervals
on binomial population proportions inferred from success counts in
small-to-intermediate samples. Population proportions arise frequently as
quantities of interest in astronomical research; for instance, in studies
aiming to constrain the bar fraction, AGN fraction, SMBH fraction, merger
fraction, or red sequence fraction from counts of galaxies exhibiting distinct
morphological features or stellar populations. However, two of the most
widely-used techniques for estimating binomial confidence intervals--the
'normal approximation' and the Clopper & Pearson approach--are liable to
misrepresent the degree of statistical uncertainty present under sampling
conditions routinely encountered in astronomical surveys, leading to an
ineffective use of the experimental data (and, worse, an inefficient use of the
resources expended in obtaining that data). Hence, I provide here an overview
of the fundamentals of binomial statistics with two principal aims: (i) to
reveal the ease with which (Bayesian) binomial confidence intervals with more
satisfactory behaviour may be estimated from the quantiles of the beta
distribution using modern mathematical software packages (e.g. R, matlab,
mathematica, IDL, python); and (ii) to demonstrate convincingly the major flaws
of both the 'normal approximation' and the Clopper & Pearson approach for error
estimation.
