Some rejection limits of the absolute normal scores test (K) for symmetry are extended from sample size 10 to 17 and partially to 20, by essentially forming all possible values ofKfor fixed sample size. These limits are compared with normal and Edgeworth approximations, which seem to be quite satisfactory in that the percent error in test size for sample size greater than 20 should be less than 12% for the normal approximation and less than 2% for the Edgeworth approximation (see (3.3)), for test size ≥ .01. The powers of the absolute normal scores (K), Wilcoxon (W), binary (B), sign (S), andt(T) tests are estimated from Monte Carlo trials for shifts of symmetrical hypothesized populations which are normal, logistic, double exponential, and rectangular. The powers ofTandKare usually very close, withTmore powerful for normal shifts and large rectangular shift, andKmore powerful for the others, though the differences may not be substantial. In terms of decreasing power, the tests are ordered as follows, with a comma marking an apparent substantial difference: for normal shiftTKW, B, S;for logistic shiftKTW, S, B;for small double exponential shift and large test sizeSWKT, B;as the shift increases and test size decreasesW, SKT, BandW, KST, B;and for rectangular shiftB, KT, W, S, andB, TK, W, S.