  The reduced chi-squared statistic is a commonly used goodness-of-fit measure,
but it cannot easily detect features near the noise level, even when a large
amount of data is available. In this paper, we introduce a new goodness-of-fit
measure that we name the reduced psi-squared statistic. It probes the two-point
correlations in the residuals of a fit, whereas chi-squared accounts for only
the absolute values of each residual point, not considering the relationship
between these points. The new statistic maintains sensitivity to individual
outliers, but is superior to chi-squared in detecting wide, low level features
in the presence of a large number of noisy data points. After presenting this
new statistic, we show an instance of its use in the context of analyzing radio
spectroscopic data for 21-cm cosmology experiments. We perform fits to
simulated data with four components: foreground emission, the global 21-cm
signal, an idealized instrument systematic, and noise. This example is
particularly timely given the ongoing efforts to confirm the first
observational result for this signal, where this work found its original
motivation. In addition, we release a Python script dubbed $\texttt{psipy}$
which allows for quick, efficient calculation of the reduced psi-squared
statistic on arbitrary data arrays, to be applied in any field of study.
