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The Pamphlet Collection of Sir Robert Stout: Volume 80a

Mortality from Five to Seventy-Five Years of Age

Mortality from Five to Seventy-Five Years of Age.

After considering the various methods avail-able for passing from the estimated mean population and the recorded deaths to the probability of living at each age, and being desirous of preserving all the well-marked features of the observations, the method employed by Mr. J, Milne in the construction of the well-known Carlisle table, and commonly known as the graphic method, was adopted. A description of the process, illustrated by a diagram of the population curve, is given in his treatise, vol. I., p. 101. In describing it we shall, as far as possible, use Mr. Milne's own words. Taking a sheet of sectional ruled paper we draw thereon a straight line of indefinite length, A 2 as a base for future operations.


And for the first group of ages assume A B at pleasure; then let B C, C D, D E, &c., be to A B in the same ratio of the 2nd, 3rd, 4th, &c., groups of ages respectively to the first interval A B. At each of the points A, B, C, &c, erect perpendiculars to A Z; then in the perpendicular that passes through B, assume Bbat pleasure, connect with the perpendicular on A and form parallelogram. In those that [unclear: pa] through C, D, E, &c, take the points c, d, e, [unclear: &c] so that the parallelograms Ab, Bc, Cd, &c, [unclear: be] completed, each of the others may be to [unclear: fi] first in the ratio of the numbers living in [unclear: f] corresponding group of ages to the number [unclear: b] the first interval. In this way the area of [unclear: th] parallelograms are made to represent the [unclear: num] her living in each group of ages.

Next let a line (as little curved as the [unclear: other] conditions will admit of) be described [unclear: through] these parallelograms, so that the point [unclear: describ] it, in its motion from the first ordinate Aa, [unclear: m] continually approach towards Z, and may [unclear: ne] change its direction abruptly, so as to form, [unclear: b] angle in its path. (In other words it must [unclear: be] a flowing line). Also, let the line thus [unclear: described] so cut each of the parallelograms, ahove-[unclear: me] tioned, that the area comprehended by the [unclear: bas] the two sides of the parallelogram [unclear: perpendicula] thereto and the portion of the line [unclear: which] intercepted between those sides, may be [unclear: equ] to the area of the same parallelogram. [unclear: Th] is, the curve in cutting the parallelograms [unclear: m] add an area to each equal to the area cut [unclear: of] so that the area of the parallelogram shall [unclear: be] exactly the same as before.

So shall the number of the living in [unclear: an] assigned year of age be to the given [unclear: number] the interval including that year, in the [unclear: rates] the area insisting upon the portion of the [unclear: bea] corresponding with the year assigned, to [unclear: be] area of the parallelogram in which it is [unclear: for] Similar parallelograms were set out for [unclear: f] deaths in the same groups of ages, and a [unclear: cen] drawn through them on the same condition.

We thus get two curves representing [unclear: res] tively the population and the deaths at [unclear: ea] nge, and dividing the deaths by the [unclear: popul] we obtain the function, called by Dr. [unclear: Farr] the "rate of mortality," and by Dr. [unclear: Sprag] the "central death rate," the symbol for [unclear: which] is mx. Then, in accordance with a [unclear: suggestion] by Mr. G. King (Journal Institute [unclear: Actual]vol. xxiv, p. 203), the values of mso [unclear: form] were plotted out on cross-ruled paper, [unclear: aa] carefully adjusted by drawing fresh [unclear: curre] to remove irregularities introduced by [unclear: famil] drawing of the first two curves, but [unclear: with] removing any of the characteristics of the [unclear: table]

Then in order to test the effect of the [unclear: grand] tion, the number living at each age [unclear: war] multiplied by mx, and the result gave the pected deaths. The sum of these should be [unclear: the] total number of deaths, according to the [unclear: org] nal facts. If there is any considerable [unclear: diff] ence the m's must be re-adjusted so [unclear: as] remove it. The results as finally adjusted [unclear: we] arranged in quinquennial groups, and will [unclear: be] found compared with the actual death Table D, hereunder :
Table D. Expected and Actual Death.
Males Females.
Ages, Expected. Actual. Expected. Actual Ages
5-9 135.02 135.85 111-29 112.38 [unclear: 5-10]
10-14 77.81 76.92 77.07 75.08 [unclear: 10-15]
16-19 105.82 105.62 98.50 102.08 [unclear: 15-20]
20-24 134.04 135.77 120.74 150.09 [unclear: 20-25]
26-29 130.60 129.31 117.27 112.62 [unclear: 25-30]
30-34 135.09 136.02 112.45 107.85 [unclear: 30-35]
35-30 150.44 149.92 111.09 115.31 [unclear: 35-40]
40-44 175.67 175.38 104.07 103.46 [unclear: 40-45]
45-49 196.13 l96.00 93.18 89.46 [unclear: 45-50]
50-54 204.67 204.54 87.04 90.15 [unclear: 50-55]
55-59 173.05 174.15 74 07 75.92 [unclear: 55-60]
60-64 177.44 171.08 79.18 77.62 [unclear: 60-65]
65-69 134.32 133.15 72.93 73.62 [unclear: 65-70]
70-74 1114.07 111.31 72.93 72.23 [unclear: 70-75]
Totals. 2040.97 2040.92 1332.16 1323.47 Total
page 7

From the m's we derive px, the probability of living a year by the formula.

Mathematical formula

and then starting from an arbitrary radix (in the present case 100,000 for each sex) by continuous multiplication we construct the column lxin each table, and this gives the number living at each age.