The test for the significance of the difference of two means, when the standard errors of one observation are unequal, has been the subject of much recent discussion (Fisher 1935; Bartlett 1936; Welch 1937; Daniels 1938), but the appropriate treatment remains in doubt. A significance test for the difference of two means, on my principles, has already been given (Jeffreys 1937
a
), but is not altogether satisfactory, for two reasons. The result was, for large numbers of observations,
K
=
P(q|θh
) /
P(~Q|θH
) = (2/
π
σ
2
+
T
2
/ σ
2
/
m
+
T
2
/
n
)
1/2
exp (-1/2
x
-
-
y
-
)
2
/σ
2
/
m
+
T
2
/
n
) (1) where
x
-
and
y
-
are the means in the two series,
m
and
n
the numbers of observations, σ and
T
the (estimated) standard errors of one observation. The most serious practical defect of this formula is that the numbers of observations are supposed large enough for the uncertainty of the standard errors to be neglected. This was due to a premature approximation and could be corrected easily; the resulting change would be similar to the difference between the normal layer and "Student's" formula, as has already been shown in other cases. There is, however, an other anomaly, less serious in practice, but of theoretical importance. We noticed at if
T
= 0, when the observations in the second series are exact, the first factor reduces to (2
m
/
π
)
1/2
, which is the usual form for the test of one new parameter, and is satisfactory. But if σ =
T
, and
n
is very large, so that the uncertainty of the true value in the second series is again negligible, should again expect the outside factor to reduce to (2
m
/
π
)
1/2
, since we are again comparing the mean of the first series with an accurate value. Actually it reduces to (4
m
/
π
)
1/2
. This is not of much practical importance, since if formula (1) gives
k
= 1, the correct formula would give
k
= 1/√2, and the result would still be indecisive, though slightly in favour of ~
q
.
The GRIMMER test: A method for testing the validity of reported measures of variability