  (Abridged) We develop median statistics that provide powerful alternatives to
chi-squared likelihood methods and require fewer assumptions about the data.
Applying median statistics to Huchra's compilation of nearly all estimates of
the Hubble constant, we find a median value H_0=67 km/s/Mpc. Median statistics
assume only that the measurements are independent and free of systematic
errors. This estimate is arguably the best summary of current knowledge because
it uses all available data and, unlike other estimates, makes no assumption
about the distribution of measurement errors. The 95% range of purely
statistical errors is +/- 2 km/s/Mpc. The statistical precision of this result
leads us to analyze the range of possible systematic errors in the median,
which we estimate to be roughly +/- 5 km/s/Mpc (95% limits), dominating over
the statistical errors. A Bayesian median statistics treatment of high-redshift
Type Ia supernovae (SNe Ia) apparent magnitude versus redshift data from Riess
et al. yields a posterior probability that the cosmological constant Lambda > 0
of 70 or 89%, depending on the prior information used. The posterior
probability of an open universe is about 47%. Analysis of the Perlmutter et al.
high-redshift SNe Ia data show the best-fit flat-Lambda model favored over the
best-fit Lambda = 0 open model by odds of 366:1; corresponding Riess et al.
odds are 3:1 (assuming prior odds of 1:1).Median statistics analyses of the SNe
Ia data do not rule out a time-variable Lambda model, and may even favor it
over a time-independent Lambda and a Lambda = 0 open model.
