We turn now to the comparison between D1 and D 5; the means and standard deviations of the distributions given by these doctors are shown in the accompanying table :

where x and x, are the widths of the "Normal" category as given by the two doctors. We take a supposed original population the age and sex groups of which have means and standard deviations as follows:

The two doctors having drawn samples of these we shall then have :

from which we obtain the best values of the ratios of the quantities M, M', M", M"", Σ, Σ', Σ", Σ""', X1, X5.

If we take the scale of nutrition to be such that x 3,* then we have these values for æ, and x, and for the means and standard deviations of the original popuX5 lation.

X1

* As is suggested in round numbers by the width of the "Normal" category for D 3 and D 4.

Let us compare these with the given means and standard deviations of D 1 and D 5, they are :—

We see here great differences between the means and standard deviations of the supposed original population and those of the D 5 and D 1 distributions showing that our assumption of the difference between the widths of the normal categories given by these two doctors is not sufficient to enable us to reduce their data to the same standard. But if we now assume in addition that the two doctors have different standards upon which nutrition is judged, i.e. that "mean nutrition " as judged by D 1 is different from "mean nutrition" as judged by D 5 we can allow for this by assuming that the means given by D 1 must be moved a distance μ bring them to the means of the original population. We have then these modified equations 54803 x − μ = M, ·63187 1 - μ = M', and the other ten equations being as before, we find :

to

The means and standard deviations of the given distributions are :—

When we compare these with the means and standard deviations of the original distributions we see that although we have allowed for a difference in the means of the two distributions and for a difference in the width of the normal categorythe standard deviations are not those of the original population. Consequently our assumptions cannot be the correct assumptions to be made in order to get rid of the differences between D 1 and D 5.

The probable errors of the means and standard deviations of the given distributions are:

Let us make the additional assumption that, in order to reduce D 1 to the same standard as D 5, we must spread out the distribution given by D 1, i.e. let us multiply the standard deviations given by D1 by a factor v; then we shall have these equations:

from which to obtain x1, x, μ, v, M, M', M", M", Σ, Σ', Σ", "", and we have these

=

Taking 5 30 again, x, is now 2:1446 and we have these means and standard deviations of the distributions given by D 5 and D 1 modified by μ and and we can compare these with the original populations, means and standard deviations.

v,

D 5.

8189 1.5152 9988 1-8711 5949 1.1798 9737 1.4760 Population 8190 1-5667 9988 1-8243 5949 1.1798 9737 1·4760 DI.

8195 1.6204 9988 1-7774

Now we see that the differences between these means and standard deviations are not great compared with the probable errors, so we conclude that compared with D 5 as standard D 1 is different in three ways:

(a) The "mean nutrition" of D1 is distant from that of D5 by 35 units (1188a) in the direction of better nutrition, i.e. D 1 is optimistic.

(b) The range of normal nutrition of D 1 is less than that of D 5,--the ratio of the two being '7.

(c) In spite of (b) D 1 has a much greater proportion of cases in the "Normal" category, and has few cases in the extreme categories, and consequently the standard deviation is one-third of the standard deviation of D 5.

When we compare the figures obtained from D3, D4 with those from D5 and D 1 we see that in the former case an assumption of 3 units as the width of the normal category gives us the following means and standard deviations for the original population.

and in the latter case the same assumption gives us :

8190 1.5667 ·9988 1.8243 5949 1.1798 9737 1.4760.

We cannot attempt to compare these figures except in a rough manner since they are for two different populations; but we see that there is some difference between D 5 and D 4, the two standards, although this difference is not as great as it was in the case of the teeth distributions.

We note that in both sets of data the means and standard deviations in the case of boys are less than those for girls and on the average, girls between 12 and 14, have better nutrition than boys of the same age.

V. TEETH AND OTHER FACTORS.

(i) Condition of Teeth and Cleanliness of Teeth.

In the annual report for 1911 on School Medical Inspection for our county we find on page 55 that the Senior School Medical Inspector writes: "During the year D3, D4 and D 1 recorded various conditions of cleanliness, discoloration, etc., of teeth, in addition to the fact of caries"; and later he defines what is meant by "uncleanness" and "discoloration" in the following words :

"Uncleanness. By this term is meant a condition in which particles cling to the teeth in such a manner that they could be rubbed off with a brush or the edge of a glass spatula.

Discoloration.-Perfectly sound, clean, and healthy looking teeth fall into three types (1) the ivory white, (2) the ivory yellow, (3) the nacreous white. These appearances may be marred by discolorations of various sorts, namely, stainings or

pigmentations of the teeth which either cannot be removed at all or not without strong scraping with a sharp edge. In one form of discoloration there is a general darkening or staining of the whole tooth. This is said to indicate death of the tooth pulp... it is not in any sense uncleanness."

"Another form of discoloration is where the surface of the teeth show coloured flecks or lines, brown or black. The flecks may coalesce into patches. . . . Sometimes there is a greenish slimy discoloration of the tooth which is said to be due to bacterial colonies. These flecks and lines, like the staining above mentioned should certainly not be taken as evidence of ordinary uncleanness, but they could no doubt be scaled off; and sedulous brushing might prevent their formation."

We therefore proceed to find the correlation between the carious condition of the teeth and the cleanliness of the teeth, for the distributions given by the doctors named above who are referred to in this paper as D 1, D 3, D 4. We note that the other two doctors also return some cases of uncleanness of teeth but these are so few that presumably D 2 and D 5 did not make the careful examination which the other three doctors before the inspection had undertaken to do. We have these results from the fourfold table method, the teeth being grouped as before for the carious condition, and into two groups clean and unclean, + sign meaning that less carious and cleaner teeth go together.

Mean r, found by weighting individual r's according to the P.E.'s, is

+1930·0223.

We note that four of these ten correlation coefficients, the whole of which are positive, are definitely significant, and that the mean correlation for the whole is also significant. This seems to indicate that there is a definite relation of a small amount between the fact of food particles remaining in the mouth and the carious state of the teeth. It is generally understood that before caries in a tooth can set in, a preliminary rotting of the tooth must have followed the chemical changes which take place in the remnants of food lodged in the interstices and crevices of the teeth. This seems to be borne out by the data examined above.

(ii) Condition of Teeth and Discoloration of Teeth.

The same three doctors' returns were examined to find the correlation between caries in the teeth and discoloration. The method adopted was like that used in