Transactions and Proceedings of the New Zealand Institute/Volume 29/Article 2

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Art. II.—An Investigation into the Rates of Mortality in New Zealand during the Period 1881–91.

By C. E. Adams, B.Sc., A.I.A., F.S.S., Lecturer on Applied Mathematics, Lincoln College.

[Read before the Philosophical Institute of Canterbury, 6th May, 1896.]

Plates I.–IV.


The following tables, showing the rates of mortality in New Zealand during the period 1881–91, are deduced from the censuses for 1881, 1886, and 1891, and from the deaths for each year during the period.

Generally it may be said that the final tables show a comparison of the numbers living at each age with the deaths occurring at that age. It would have been possible to have computed the tables from one census and the deaths in that census year, but it was considered preferable to use average results, and for this purpose the average population as given by the three censuses has been employed, and the average number of deaths has also been used. In adopting this method there is a greater chance of the final results exhibiting correctly the general mortality of the colony than there would have been had the figures relating to one year only been employed.

The census has never been taken in the middle of a calendar year in New Zealand. In 1881 it was taken on the 3rd April, in 1886 on the 28th March, and in 1891 on the 5th April. This necessitates an assumption being made as to the population living on the 1st July in each year, for the numbers living in the middle of the year have to be compared with the deaths during the year. It was assumed that the population on the 1st July was the mean of the populations on the 1st January and the 31st December. The numbers living in each age-group, as given by the census, were increased in the same proportion as the whole population had increased from the date of the census to the 1st July. This adjustment was made for each of the three censuses, and the total for each agegroup found. One-third of these totals gives the average number of persons living in each of the age-groups, as shown in Table A.

No adjustments were necessary for the deaths. One-eleventh of the total number of deaths in each age-period for the eleven years 1881-91 was taken for the average number of deaths, and the results are given in Table A.

It will be observed that the average deaths and average population in Table A are given in groups of five years. The next step in the construction of the final tables is to ascertain the population and deaths at each year of age. The method now generally adopted is that known as Milne's Graphic Method. After a very careful consideration of this method it was decided not to adopt it, but to use instead a mathematical process of distribution based on the method employed by G. W. Berridge ("Journal of the Institute of Actuaries," xiii., 220, and xiv., 244; "Text-book of the Institute of Actuaries," Part ii., p. 465). The results of the distribution are given in Table B. As a test of the smoothness of the distribution, the results were drawn to scale on large diagrams, of which Plates I. and II. are reduced copies.

The population and deaths from 5 to 75 were treated in this way, the figures relating to the first five years of life requiring special treatment.

From Table B the ratio of deaths to population at each age () is at once obtained, and these ratios are given in Table C.

The probability of living a year at each age () is derived immediately from by means of the relation . The columns headed in Table E, from 5 to 75, were calculated by means of this formula.

The ages 0 to 5 now require consideration. Table D gives the annual births and deaths of children under five years of age for each of the years 1880–92. From these figures, by means of a modification of the method used by Dr. Farr ("Journal of the Institute of Actuaries," ix., p. 134), the probabilities of living a year at each age were determined. The results, after a slight adjustment to make them join smoothly on to the rest of the table, are given in column , ages 0–5, in Table E.

The probability of dying in the year at each age () is obtained from by subtracting from unity: thus, .

The next column in the order of formation is the column. Starting with an assumed 10,000 births (), the number surviving the year () is obtained from the relation . Similarly the number who reach the age of two alive, out of 10,000 born alive, is or generally for any year x, .

The difference between the number born, , and the number surviving the first year, , gives the number who die in the first year, , or . Similarly for the number who die in the second year, , , and generally for the number dying in the xth year . In this manner the column was formed.

It is not intended in the present investigation to carry these results past age 75, as the available data are insufficient to warrant satisfactory results. It must also be borne in mind that the colony dates from 1840, and the above tables terminate at 1891, consequently all results past age 51 cannot relate to native-born New-Zealanders.


General Explanation of Table E.

Column : This column shows how many out of 10,000 born alive reach each year of age up to 75. Thus, (males) = 8,112, and (females) = 8,316, or, out of 10,000 males born alive, 8,112 reach the age of 25, and out of 10,000 females born alive 8,316 reach the age of 25.

The two columns for males and females are not strictly comparable, for they do not represent the actual numbers born, but only numbers proportional to them. As is well known, the number of male births exceeds the number of female births. The columns show for each sex how, out of 10,000 born, the numbers gradually diminish.

Column : This colunm shows the deaths each year out of 10,000 born alive. Thus, (males) = 43, and (females) = 45, or 43 males die between the ages 25 and 26, and 45 females die between the ages 25 and 26.

Column : This column gives the probability of living a year at each age. Thus, (males) = .9947, and (females) = .9946, or 9,947 males out of 10,000 alive at age 25 survive the year; and 9,946 out of 10,000 females alive at age 25 reach age 26.

Column : This column gives the probability of dying in a year at each age. Thus, (males) = .0053, and (females) = .0054, or 53 males out of 10,000 alive at age 25 die in the year; and 54 out of the same number of females alive at 25 die in the year.

Plates III. and IV. show the results of the life tables graphically. From the column it will be observed that the males are reduced to half the number born between the ages 63 and 64, while it is not till between the ages 66 and 67 that the females are similarly reduced.

The whole of the calculation was done in duplicate, and every care has been exercised to insure accuracy. Some of the results have been checked graphically, results true to four significant figures being easily obtained by this process.

In conclusion, I have to express my thanks to my friend, Mr. Morris Fox, A.I.A., Actuary to the Government Life Insurance Department, for his ever-ready and valuable assistance in the preparation of this paper.


Table A.—Average Population and Average Deaths, 1881–91, in Five-year Periods.

Ages. Males. Females.
Population. Deaths. Population. Deaths.
 0–5  42,832 1234.20 41,766 1025.50
 5–10 40,348  137.18 39,551  113.09
10–15 34,771   77.09 34,323   74.45
15–20 27,783  103.36 28,167  102.82
20–25 25,172  134.64 24,674  119.36
25–30 24,171  129.64 19,778  113.73
30–35 21,736  136.91 16,227  109.45
35–40 20,258  149.09 14,302  116.00
40–45 19,165  176.64 12,601  104.27
45–50 16,433  200.55  9,897   90.64
50–55 13,365  205.73  7,491   91.78
55–60  7,938  175.00  4,533   74.91
60–65  5,520  168.09  3,409   77.55
65–70  2,968  147.73  2,026   74.45
70–75  1,761  108.82  1,350   72.09


Table B.—Population and Deaths for each Year from 5 to 75.

Ages. Males. Females.
Population. Deaths. Population. Deaths.
 5 8,356.2 40.16 8,210.6 33.32
 6 8,240.9 32.06 8,082.3 26.17
 7 8,098.3 25.81 7,935.6 20.96
 8 7,926.6 21.19 7,763.4 17.42
 9 7,726.0 17.98 7,568.1 15.22
10 7,497.9 15.96 7,352.2 14.15
11 7,244.4 14.95 7,119.4 13.94
12 6,969.8 14.77 6,873.4 14.42
13 6,679.2 15.23 6,618.5 15.35
14 6,379.7 16.19 6,359.5 16.59
15 5,995.8 17.50 6,043.7 17.97
16 5,729.0 19.03 5,810.5 19.38
17 5,502.9 20.68 5,605.2 20.72
18 5,820.9 22.32 5,423.6 21.90
19 5,184.4 23.87 5,279.0 22.85
20 5,147.3 25.56 5,243.7 23.33
21 5,076.0 26.60 5,105.6 23.81
22 5,021.9 27.29 4,952.1 24.08
23 4,980.3 27.59 4,780.7 24.14
24 4,946.5 27.56 4,591.9 24.00
25 4,951.2 26.25 4,828.2 23.40
26 4,908.4 25.96 4,130.2 23.06
27 4,850.5 25.76 3,942.7 22.73
28 4,775.8 25.74 3,768.1 22.40
29 4,685.1 25.89 3,608.8 22.14
30 4,527.1 26.62 3,470.8 21.78
31 4,427.7 26.97 3,343.2 21.71
32 4,336.8 27.34 3,230.9 21.78
33 4,256.6 27.76 3,133.2 21.95
34 4,187.8 28.21 3,048.9 22.23
35 4,140.1 28.32 2,988.1 23.14
36 4,090.5 28.92 2,921.6 23.35
37 4,047.2 29.68 2,858.7 23.38
38 4,008.4 30.57 2,797.6 23.24
39 3,971.8 31.61 2,736.0 22.89
40 3,904.9 33.01 2,686.6 22.10
41 3,914.6 34.19 2,613.7 21.52
42 3,849.0 35.36 2,530.5 20.87
43 3,707.3 36.49 2,437.0 20.22
44 3,669.2 37.55 2,333.2 19.56
45 3,613.4 38.64 2,191.2 18.67
46 3,399.5 39.53 2,081.4 18.23
47 3,286.3 40.28 1,975.1 17.97
48 3,173.5 40.87 1,873.4 17.86
49 3,060.3 41.28 1,775.9 17.91
50 3,011.6 41.76 1,716.7 18.79
51 2,868.8 41.70 1,615.2 18.77
52 2,698.6 41.39 1,505.5 18.56
53 2,502.2 40.81 1,388.3 18.12
54 2,283.8 40.04 1,205.3 17.49
55 1,948.8 36.94 1,090.4 15.85
56 1,737.9 35.70  980.1 15.20
57 1,555 9 34.73  887.8 14.77
58 1,406.3 34.02  814.7 14.54
59 1,289.1 33.61  760.0 14.55
60 1,286.0 34.32  769.1 15.31
61 1,199.4 34.10  729.6 15.47
62 1,109.3 33.76  686.5 15.57
63 1,013.6 33.29  638.5 15.62
64  911.7 32.63  585.3 15.58
65  760.2 31.93  496.5 15.42
66  664.1 30.93  443.8 15.19
67  580.6 29.73  397.9 14.90
68  510.1 28.33  359.2 14.61
69  453.0 26.78  328.6 14.33
70  430.4 25.14  320.0 14.10
71  389.3 23.42  295.7 14.01
72  350.5 21.68  270.8 14.14
73  313.5 20.03  245.2 14.53
74  277.3 18.53  218.3 15.31


Table C.—Ratio of Deaths to Population at each Age ().

Ages. Males. Females. Ages. Males. Females.
 5 .0048 .0042 40 .0083 .0082
 6 .0039 .0033 41 .0087 .0082
 7 .0032 .0026 42 .0092 .0082
 8 .0027 .0022 43 .0097 .0083
 9 .0023 .0020 44 .0102 .0084
10 .0021 .0019 45 .0110 .0085
11 .0021 .0020 46 .0116 .0088
12 .0021 .0021 47 .0123 .0091
13 .0023 .0023 48 .0129 .0095
14 .0025 .0026 49 .0135 .0101
15 .0029 .0030 50 .0139 .0109
16 .0033 .0033 51 .0145 .0116
17 .0038 .0037 52 .0153 .0123
18 .0042 .0040 53 .0163 .0131
19 .0046 .0043 54 .0175 .0138
20 .0050 .0044 55 .0190 .0145
21 .0052 .0047 56 .0205 .0155
22 .0054 .0049 57 .0223 .0166
23 .0055 .0050 58 .0242 .0178
24 .0056 .0052 59 .0261 .0191
25 .0053 .0054 60 .0267 .0199
26 .0053 .0056 61 .0284 .0212
27 .0053 .0058 62 .0305 .0227
28 .0054 .0059 63 .0328 .0245
29 .0055 .0061 64 .0358 .0266
30 .0059 .0063 65 .0420 .0311
31 .0061 .0065 66 .0466 .0342
32 .0063 .0067 67 .0512 .0374
33 .0065 .0070 68 .0556 .0407
34 .0067 .0073 69 .0591 .0430
35 .0068 .0077 70 .0585 .0441
36 .0071 .0080 71 .0602 .0474
37 .0073 .0082 72 .0618 .0522
38 .0076 .0083 73 .0638 .0593
39 .0080 .0084 74 .0669 .0701


Table D.—Births and Deaths of Children under Five Years of Age.

Year Births. Deaths.
0–1. 1–2. 2–3. 3–4. 4–5.
Males.
1880  9,893  986 183 60 54 31
1881  9,590  987 204 60 49 49
1882  9,712  934 178 82 63 56
1883  9,843 1,079 206 72 57 35
1884 10,131  870 145 77 55 36
1885 10,020  970 176 74 45 31
1886  9,872 1,027 162 56 50 81
1887  9,725  987 154 86 53 27
1888  9,641  752 140 57 36 33
1889  9,514  798 134 57 34 47
1890  9,293  775 114 54 45 42
1891  9,377  942 160 59 31 43
1892  9,101  910 132 77 41 42
Females.
1880  9,448  819 174 72 46 33
1881  9,142  744 187 65 57 38
1882  9,297  744 155 71 54 50
1883  9,359  916 190 61 43 36
1884  9,715  703 156 81 41 30
1885  9,673  786 124 57 47 35
1886  9,427  872 152 74 38 30
1887  9,410  808 157 63 43 29
1888  9,261  584 117 58 42 37
1889  8,943  658 116 45 41 23
1890  8,985  663 100 43 29 29
1891  8,896  725 122 47 36 28
1892  8,775  684 112 60 44 31


Table E.—New Zealand Life Table, 1881—91.

Males.

 0 10,000 967  .9033  .0967  40 7,376  61  .9917  .0083
 1  9,033 164  .9818  .0182  41  7,315  63  .9913  .0087
 2  8,869  68  .9924  .0076  42  7,252  67  .9909  .0091
 3  8,801  44  .9950  .0050  43  7,185  69  .9904  .0096
 4  8,757  37  .9958  .0042  44  7,116  72  .9899  .0101
 5  8,720  34  .9961  .0039  45  7,044  77  .9891  .0109
 6  8,686  31  .9964  .0036  46  6,967  81  .9885  .0115
 7  8,655  28  .9968  .0032  47  6,886  84  .9878  .0122
 8  8,627  23  .9973  .0027  48  6,802  87  .9872  .0128
 9  8,604  20  .9977  .0023  49  6,715  90  .9866  .0134
10  8,584  18  .9979  .0021  50  6,625  92  .9862  .0138
11  8,566  18  .9979  .0021  51  6,533  94  .9856  .0144
12  8,548  18  .9979  .0021  52  6,439  97  .9848  .0152
13  8,530  19  .9977  .0023  53  6,342 103  .9839  .0161
14  8,511  21  .9975  .0025  54  6,239 108  .9826  .0174
15  8,490  25  .9971  .0029  55  6,131 116  .9812  .0188
16  8,465  28  .9967  .0033  56  6,015 122  .9797  .0203
17  8,437  32  .9962  .0038  57  5,893 130  .9780  .0220
18  8,405  35  .9958  .0042  58  5,763 138  .9761  .0239
19  8,370  39  .9954  .0046  59  5,625 144  .9743  .0257
20  8,331  41  .9950  .0050  60  5,481 145  .9736  .0264
21  8,290  43  .9948  .0052  61  5,336 149  .9720  .0280
22  8,247  45  .9946  .0054  62  5,187 156  .9700  .0300
23  8,202  45  .9945  .0055  63  5,031 162  .9077  .0323
24  8,157  45  .9944  .0056  64  4,869 172  .9649  .0351
25  8,112  43  .9947  .0053  65  4,697 193  .9589  .0411
26  8,069  43  .9947  .0053  66  4,504 205  .9545  .0455
27  8,026  42  .9947  .0053  67  4,299 214  .9501  .0499
28  7,984  43  .9946  .0054  68  4,085 221  .9459  .0541
29  7,941  44  .9945  .0055  69  3,864 222  .9426  .0574
30  7,897  46  .9941  .0059  70  3,642 212  .9418  .0582
31  7,851  48  .9939  .0061  71  3,430 201  .9414  .0586
32  7,803  49  .9987  .0063  72  3,229 194  .9401  .0599
33  7,754  50  .9935  .0065  73  3,035 187  .9382  .0018
34  7,704  52  .9933  .0067  74  2,848 185  .9353  .0047
 75  2,663 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
35  7,652  51  .9932  .0068
36  7,601  54  .9929  .0071
37  7,547  55  .9927  .0073
38  7,492  57  .9925  .0075
39  7,435  59  .9920  .0080


Table E.—New Zealand Life Table, 1881—91—continued.

Females.

 0 10,000 817  .9183  .0817  40 7,498  62  .9917  .0083
 1  9,183 155  .9831  .0169  41  7,436  61  .9918  .0082
 2  9,028  64  .9929  .0071  42  7,375  61  .9918  .0082
 3  8,964  42  .9953  .0047  43  7,314  60  .9917  .0083
 4  8,922  33  .9964  .0036  44  7,254  61  .9816  .0084
 5  8,889  29  .9967  .0033  45  7,193  61  .9915  .0085
 6  8,860  27  .9970  .0030  46  7,132  63  .9912  .0088
 7  8,833  22  .9974  .0026  47  7,069  65  .9909  .0091
 8  8,811  20  .9978  .0022  48  7,004  66  .9905  .0095
 9  8,791  18  .9980  .0020  49  6,938  69  .9900  .0100
10  8,773  16  .9981  .0019  50  6,869  75  .9892  .0108
11  8,757  18  .9980  .0020  51  6,794  78  .9885  .0115
12  8,739  18  .9979  .0021  52  6,716  82  .9878  .0122
13  8,721  20  .9977  .0023  53  6,634  86  .9870  .0130
14  8,701  23  .9974  .0026  54  6,548  90  .9863  .0137
15  8,678  26  .9970  .0030  55  6,458  93  .9856  .0144
16  8,652  28  .9967  .0033  56  6,365  98  .9846  .0154
17  8,624  32  .9963  .0037  57  6,267 103  .9835  .0165
18  8,592  35  .9960  .0040  58  6,164 109  .9824  .0176
19  8,557  36  .9957  .0043  59  6,055 114  .9811  .0189
20  8,521  38  .9956  .0044  60  5,941 117  .9803  .0197
21  8,483  40  .9953  .0047  61  5,824 122  .9790  .0210
22  8,443  41  .9951  .0049  62  5,702 128  .9776  .0224
23  8,402  42  .9950  .0050  63  5,574 135  .9758  .0242
24  8,360  44  .9948  .0052  64  5,439 143  .9738  .0262
25  8,316  45  .9946  .0054  65  5,296 162  .9694  .0306
26  8,271  46  .9944  .0056  66  5,134 172  .9664  .0336
27  8,225  48  .9942  .0058  67  4,962 182  .9633  .0367
28  8,177  48  .9941  .0059  68  4,780 188  .9607  .0393
29  8,129  50  .9939  .0061  69  4,592 191  .9585  .0415
30  8,079  50  .9937  .0063  70  4,401 191  .9566  .0434
31  8,029  52  .9935  .0065  71  4,210 195  .9537  .0463
32  7,977  54  .9933  .0067  72  4,015 204  .9492  .0508
33  7,923  55  .9930  .0070  73  3,811 219  .9424  .0576
34  7,868  58  .9927  .0073  74  3,592 243  .9323  .0677
 75  3,349 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
35  7,810  60  .9923  .0077
36  7,750  62  .9920  .0080
37  7,688  63  .9918  .0082
38  7,625  63  .9917  .0083
39  7,562  64  .9916  .0084