Sankhya: The Indian Journal of Statistics

1998, Volume 60, Series B, Pt. 3, 387--398

A TRANSFORMATION TO NORMALITY FOR THE SKEWNESS COEFFICIENT IN NORMAL SAMPLES

By

RAM\'ON ARDANUY, MAR\'IA M. SOLDEVILLA and
QUINT\'IN MART\'IN
* Universidad de Salamanca, Salamanca*

SUMMARY. D'Agostino's formula to transform g_{1} to normality,
in normal samples of size n, is
based on Johnson SU transformation. It can be used for n >= 8 but does not take into
account the fact that the range of g_{1} is bounded by

$A_n = \pm(n-2)/\sqrt{(n-1)}$,
so it loses
accuracy for extreme percentage points. In this paper we propose first to transform the
range of g_{1} to ±a by means of a logit function and then to make a Johnson SU
transformation to normality. The transformation proposed can be applied for n >= 4 and it
is slightly more accurate than D'Agostino's formula for n >= 8, avoiding the lateral effect
of that transformation. A polynomial improvement, tables and approximate formulae to
carry out the transformation are given in this paper, as well as some empirical
comparative results, between the mentioned transformations.

*AMS (1991) subject classification.* 62E15, 62E20, 62Q05, 62F03.

*Key words and phrases. *D'Agostino's formula, percentage points, skewness,
test for normality, transformations to normality.