London's Pulse: Medical Officer of Health reports 1848-1972
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Hints from the Health Department. Leaflet from the archive of the Society of Medical Officers of Health. Credit: Wellcome Collection, London
London County Council 1906
[Report of the Medical Officer of Health for London County Council]
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11
The standard deviation is the error of the mean square. It is ascertained by adding up the
sum of the squares of the individual variations from the arithmetical average and after dividing by the
total number of observations, taking the square root of the result. It is usually called σ
| Dimensions. | Number of individuals in each group. | Deviation from centre. | Dimensions. | Number of individuals forming group. | Deviation from centre. | ||||
|---|---|---|---|---|---|---|---|---|---|
| — | — | — | |||||||
| — | — | — | |||||||
| — | — | — | |||||||
| — | — | — | |||||||
| — | _ | — | — | ||||||
| — | — | — | — | — | — | ||||
The dimensions must first of all be set out in the form of a seriatim table, the numbers of individuals
forming the group in each dimension must be recorded in Column Z. The average of this
series can be ascertained from inspection to be about 136 centimetres. Regarding this as centre
the other dimensions can be taken as units above, positive, or below, negative. This gives column
X. Then the products of ZX are calculated and added up. In this case there are 1,716 negative and
1,599 positive units, giving a total sum of 117 negative unit6. This number divided by the total
number of observations made, 635, gives a quotient .18, which has to be subtracted, because the
negative units were in excess, from the arbitrarily chosen centre, to give the arithmetical average
136 —.18=135.82 centimetres as average height. The products ZX are then once more multiplied by
X, the amount of deviation from the arbitrary centre, to obtain ZX2, all being now positive. The sum
of this column ZX2 divided bv the number of observations N (the sum of column Z) gives
As the deviation was from the arbitrarily chosen and not true average, to
correct this the square of the difference between the arbitrary centre and the true average must be
tracted. That is' the square of
in this case
So that
45*283 —•03 = 45*2511. This is the square of the standard deviation or σ2 and =45.2511 therefore
can be determined by the relation this bears to the value of the square root of the 6um of the
squares of the standard deviation of each divided by the total number in each group, this is in symbols
When the difference between the averages (A, A.) is just equal to this; that is A—A, =
the odds are that the difference is due to chance. When
the odds are 49 to 1 against chance or random selection, but with the difference A—A2 =
the odds are 1,000 to 1 against, and with four times they rise to 333,000 to 1 against a
chance selection being the explanation.
14968 B 2