Standardized t

Abstract

STUDENTS' t may be standardized by expressing it in units of its own standard deviation; thus, ts = t√{(ν - 2)/ν}, where ν is the number of degrees of freedom. The procedure reduces the variability of the tails with ν for a given probability, P, and in particular, for P = 0.05 and ∞ ≥ ν ≥ 4, ts lies between 1.96 and 2.0 whereas t lies between 1.96 and 2.776. This finding extends the applicability of certain 'large' sample methods down to 4 degrees of freedom. The standardizing is extremely simple to apply - merely divide the error sum of squares by ν - 2 instead of by ν and use this value to calculate the appropriate standard error (SE). For example, if d is the difference between two means and d ≥ 2 SE, P ≤ 0.05, that is, the difference is significant at the 5 per cent level. If d < 1.96 SE, P > 0.05 and the difference is not significant. Between these two limits of d, P ≃ 0.05; actually 0.0535 > P > 0.0455. Again, ± 2SE will give a close approximation to 95 per cent confidence limits (actual value between 95 and 95.45 per cent). When the formula is applied to the correlation coefficient, r, we find that r is significant at the 5 per cent point if r ≥ 2/√(ν + 2). ö © 1960 Nature Publishing Group.