# 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.

A B C D E F G Z

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 B*b*at 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 A*b*, B*c*, C*d*, &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 A*a*, [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 *m*so [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]

*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 :

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 |

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

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