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

by Charles Edward Adams

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 ($m_{x}$) is at once obtained, and these ratios are given in Table C.

The probability of living a year at each age ($p_{x}$) is derived immediately from $m_{x}$ by means of the relation $p_{x}={\frac {2-m_{x}}{2+m_{x}}}$. The columns headed $p_{x}$ 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 $p_{x}$, ages 0–5, in Table E.

The probability of dying in the year at each age ($q_{x}$) is obtained from $p_{x}$ by subtracting $p_{x}$ from unity: thus, ${q_{x}}={1-p_{x}}$.

The next column in the order of formation is the $l_{x}$ column. Starting with an assumed 10,000 births ($l_{0}$), the number surviving the year ($l_{1}$) is obtained from the relation $l_{1}=l_{0}\times p_{0}$. Similarly the number who reach the age of two alive, out of 10,000 born alive, is $l_{2}=l_{1}\times p_{1}$ or generally for any year x, $l_{x}=l_{x-1}\times p_{x-1}$.

The difference between the number born, $l_{0}$, and the number surviving the first year, $l_{1}$, gives the number who die in the first year, $d_{0}$, or $d_{0}=l_{0}-l_{1}$. Similarly for the number who die in the second year, $d_{1}$, $d_{1}=l_{1}-l_{2}$, and generally for the number dying in the xth year $d_{x-1}=l_{x-1}-l_{x}$. In this manner the column $d_{x}$ 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 $l_{x}$: This column shows how many out of 10,000 born alive reach each year of age up to 75. Thus, $l_{25}$ (males) = 8,112, and $l_{25}$ (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 $l_{x}$ 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 $d_{x}$: This colunm shows the deaths each year out of 10,000 born alive. Thus, $d_{25}$ (males) = 43, and $d_{25}$ (females) = 45, or 43 males die between the ages 25 and 26, and 45 females die between the ages 25 and 26.

Column $p_{x}$: This column gives the probability of living a year at each age. Thus, $p_{25}$ (males) = .9947, and $p_{25}$ (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 $q_{x}$: This column gives the probability of dying in a year at each age. Thus, $q_{25}$ (males) = .0053, and $q_{25}$ (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 $l_{x}$ 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.

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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 ($m_{x}$).

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.

$x$	$l_{x}$	$d_{x}$	$p_{x}$	$q_{x}$	$x$	$l_{x}$	$d_{x}$	$p_{x}$	$q_{x}$
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.

$x$	$l_{x}$	$d_{x}$	$p_{x}$	$q_{x}$	$x$	$l_{x}$	$d_{x}$	$p_{x}$	$q_{x}$
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