2a THEORY OF STATISTICS. 
square root (Chap. XIII. § 12), and this implies a standard error 
of about 5 units at the centre of the table, 3 units for a frequency 
of 9, or 2 units for a frequency of 4: such fluctuations might 
cause wide divergences in the corresponding contour lines. 
Using the suffix 1 to denote the constants relating to the 
distribution of stature for fathers, and 2 the same constants for 
the sons, 
4=1078 M,=6770 M,= 6866 i 
c= ooh ot Tol i. = OEE 
Hence we have from equation (7) 
¥15=267 
and the complete expression for the fitted normal surface is 
y= 00 Jo SEER EEE 
The equation to any contour ellipse will be given by equating 
the index of e to a constant, but it is very much easier to draw 
the ellipses if we refer them to their principal axes. To do this 
we must first determine 6, 2, and 2, From (9), 
tan 20 = — 46-49, 
whence 26=91° 14’, §=45" 37’, the principal axes standing very 
nearly at an angle of 45° with the axes of measurement, 
owing to the two standard-deviations being very nearly equal. 
They should be set off on the diagram, not with a protractor, but 
by taking tan 6 from the tables (1:022) and calculating points on 
each axis on either side of the mean. 
To obtain 2, and 2, we have from (10) and (11) 
224+ 22=14961 
22, 2,=12-868 
Adding and subtracting these equations from each other and 
taking the square root, 
2, 4-2,=5275 
2, — 2, =1-447 
whence 2, =3-36, 2,=1'91; owing to the principal axes stand- 
ing nearly at 45° the first value is sensibly the same as that found 
for oz in § 10. The equations to the contour ellipses, referred to 
the principal axes, may therefore be written in the form 
Grae 
sept [Tore 
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