The conventional Q statistic, using estimated inverse-variance (IV) weights, underlies a variety of problems in random-effects meta-analysis. In previous work on standardized mean difference and log-odds-ratio, we found superior performance with an estimator of the overall effect whose weights use only group-level sample sizes. The Q statistic with those weights has the form proposed by DerSimonian and Kacker. The distribution of this Q and the Q with IV weights must generally be approximated. We investigate approximations for those distributions, as a basis for testing and estimating the between-study variance (T[superscript 2]). A simulation study, with mean difference as the effect measure, provides a framework for assessing accuracy of the approximations, level and power of the tests, and bias in estimating T[superscript 2]. Two examples illustrate estimation of T[superscript 2] and the overall mean difference. Use of Q with sample-size-based weights and its exact distribution (available for mean difference and evaluated by Farebrother's algorithm) provides precise levels even for very small and unbalanced sample sizes. The corresponding estimator of T[superscript 2] is almost unbiased for 10 or more small studies. This performance compares favorably with the extremely liberal behavior of the standard tests of heterogeneity and the largely biased estimators based on inverse-variance weights.