OF FREQUENCY OF THE BAROMETRIC HEIGHT AT DIVERS STATIONS. 44 L 
the mode and to say on which side of this tenth the mode lies. Thus in Table IV., 
for example, Glasgow 29"'9 30^^, means that the Glasgow modal height has 29 9 
for its nearest tenth, but lies on the 30^' side of this, 
A closer approximation to the position of the mode may be obtained by dealing 
with the three chief frequencies and finding the vertex (i.) of a parabola with vertical 
axis passing through their tops, or (ii.) of a normal curve of errors with vertical axis 
passing through the same three points, the latter, according to our experience, giving 
the better approximation to the true mode. 
Let c be the unit of grouping, let 3/3 be the maximum frequency, and and the 
frequencies on either side of it. Let z be the distance of the mode from y^ towards 
3 / 3 , then we have 
(i.) For the parabola : 
2 = c where A„ = y,- — ijs. 
^ \^21 ^ 32 / 
(ii.) For the normal curve ; 
2 
^21 + ^32 
2^21-^32)’ 
where = log y, — log y,. 
A third method, which is generally far more accurate, as it depends on ;di the 
observations, has been given in the memoir on skew variation in the Phil, lians. 
already cited (see pp. 375—6). This depends upon the principle that the distance of 
the mode from the mean is, with a close degree of approximation, thrice the distance 
of the median* from mean. It may be as well to illustrate these methods on an 
actual example. 
Modal Fleight of the Barometer at Southam'pton. 
(i.) By inspection of observation polygon. 30'''i30"‘0 
(ii.) By using a parabola through three ordinates . . . 30''*0G25 
(hi.) By using a normal curve through three ordinates . 30 ‘0615 
(iv.) By the principle of the modal third, as above . . 30 '0372 
(v.) By actual determination of the frequency curve. . 30"'0390 
By adding up the frequencies for Southampton in Table I., and then interpolating, 
it will be found that the median height of the barometer there is almost exactly 30 . 
But by Table III., the mean height is 29"'98l4, the third of the distance accordingly 
between mean and mode or the modal third = 0’0186, whence we obtain 30 0372 ror 
the mode. It is clear that this method gives a close approximation to the true result, 
and is one which can be used by any ordinary observer, dhe value obtained will be, 
* The median height oe the barometer ia the height given by that observation out of + 1 observa¬ 
tions, which has the heights of n observations less and the heights of n observations greater than its 
own height, i.e., it is the middle height of the series of ob.;ervations arranged in order of magnitude. 
3 L 
VOL. CXC.—A. 
