Transactions and Proceedings of the New Zealand Institute 1896 [electronic resource]

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 age-group 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.

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The probability of living a year at each age (p_x) is derived immediately from m_x by means of the relation p_x = 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 (l₀), the number surviving the year (l₁) is obtained from the relation l₁=l₀ × p₀. Similarly the number who reach the age of two alive, out of 10,000 born alive, is l₂=l₁ × p₁, or generally for any year x, l_x=l_x-1 ×; p_x-1.

The difference between the number born, l₀, and the number surviving the first year, l₁, gives the number who die in the first year, d₀, or d₀=l₀-l₁. Similarly for the number who die in the second year, d₁, d₁=l₁-l₂, 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, l₂₅ (males)=8,112, and l₂₅ (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 column shows the deaths each year out of 10,000 born alive. Thus, d₂₅ (males)=43, and d₂₅ (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, p₂₅ (males)=9947, and p₂₅ (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, q₂₅ (males)=.0053, and q₂₅ (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	27,733	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	91·73
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

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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	7,726·6	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	5,320·9	22·32	5,428·6	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	4,328·2	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

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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	3,964·9	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	3,767·3	36·49	2,437·0	20·22
44	3,669·2	37·55	2,333·2	19·56

45	3,513·4	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	1,265·3	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

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Table C.—Ratio of Deaths to Population at each Age (m_x).
Ages.	Males.	Females.
5	.0048	.0042
6	.0039	.0033
7	.0032	.0026
8	.0027	.0022
9	.0023	.0020

10	.0021	.0019
11	.0021	.0020
12	.0021	.0021
13	.0023	.0023
14	.0025	.0026

15	.0029	.0030
16	.0033	.0033
17	.0038	.0037
18	.0042	.0040
19	.0046	.0043

20	.0050	.0044
21	.0052	.0047
22	.0054	.0049
23	.0055	.0050
24	.0056	.0052

25	.0053	.0054
26	.0053	.0056
27	.0053	.0058
28	.0054	.0059
29	.0055	.0061

30	.0059	.0063
31	.0061	.0065
32	.0063	.0067
33	.0065	.0070
34	.0067	.0073

35	.0068	.0077
36	.0071	.0080
37	.0073	.0082
38	.0076	.0083
39	.0080	.0084

40	.0083	.0082
41	.0087	.0082
42	.0092	.0082
43	.0097	.0083
44	.0102	.0084

45	.0110	.0085
46	.0116	.0088
47	.0123	.0091
48	.0129	.0095
49	.0135	.0101

50	0139	.0109
51	.0145	.0116
52	.0153	.0123
53	.0163	.0131
54	.0175	.0138

55	.0190	.0145
56	.0205	.0155
57	.0223	.0166
58	.0242	.0178
59	.0261	.0191

60	.0267	.0199
61	.0284	.0212
62	.0305	.0227
63	.0328	.0245
64	.0358	.0266

65	.0420	.0311
66	.0466	.0342
67	.0512	.0374
68	.0556	.0407
69	.0591	.0436

70	.0585	.0441
71	.0602	.0474
72	.0618	.0522
73	.0638	.0593
74	.0669	.0701

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Table D.—Births and Deaths of Children under Five Years of Age.
Years.	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	31
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

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Tables E.—New Zealand Life Table, 1881–91.
*Males*.
x	l_x	d_x	p_x	q_x
0	10,000	967	.9033	.0967
1	9,033	164	.9818	.0182
2	8,869	68	.9924	.0076
3	8,801	44	.9950	.0050
4	8,757	37	.9958	.0042

5	8,720	34	.9961	.0039
6	8,686	31	.9964	.0036
7	8,655	28	.9968	.0032
8	8,627	23	.9973	.0027
9	8,604	20	.9977	.0023

10	8,584	18	.9979	.0021
11	8,566	18	.9979	.0021
12	8,548	18	.9979	.0021
13	8,530	19	.9977	.0023
14	8,511	21	.9975	.0025

15	8,490	25	.9971	.0029
16	8,465	28	.9967	.0033
17	8,437	32	.9962	.0038
18	8,405	35	.9958	.0042
19	8,370	39	.9954	.0046

20	8,331	41	.9950	.0050
21	8,290	43	.9948	.0052
22	8,247	45	.9946	.0054
23	8,202	45	.9945	.0055
24	8,157	45	.9944	.0056

25	8,112	43	.9947	.0053
26	8,069	43	.9947	.0053
27	8,026	42	.9947	.0053
28	7,984	43	.9946	.0054
29	7,941	44	.9945	.0055

30	7,897	46	.9941	.0059
31	7,851	48	.9939	.0061
32	7,803	49	.9937	.0063
33	7,754	50	.9935	.0065
34	7,704	52	.9933	.0067

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

40	7,376	61	.9917	.0083
41	7,315	63	.9913	.0087
42	7,252	67	.9909	0091
43	7,185	69	.9904	.0096
44	7,116	72	.9899	.0101

45	7,044	77	.9891	.0109
46	6,967	81	.9885	.0115
47	6,886	84	.9878	.0122
48	6,802	87	.9872	0128
49	6,715	90	.9866	.0134

50	6,625	92	.9862	0138
51	6,533	94	.9856	.0144
52	6,439	97	.9848	.0152
53	6,342	103	.9839	.0161
54	6,239	108	.9826	.0174

55	6,131	116	.9812	.0188
56	6,015	122	.9797	0203
57	5,893	130	.9780	0220
58	5,763	138	.9761	.0239
59	5,625	144	.9743	.0257

60	5,481	145	.9736	.0264
61	5,336	149	.9720	.0280
62	5,187	156	.9700	0300
63	5,031	162	.9677	.0323
64	4,869	172	.9649	.0351

65	4,697	193	.9589	.0411
66	4,504	205	.9545	.0455
67	4,299	214	.9501	.0499
68	4,085	221	.9459	.0541
69	3,864	222	.9426	.0574

70	3,642	212	.9418	.0582
71	3,430	201	.9414	.0586
72	3,229	194	.9401	.0599
73	3,035	187	.9382	.0618
74	2,848	185	.9353	.0647
75	2,663

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		Females.
0	10,000	817	.9183	.0817
1	9,183	155	.9831	.0169
2	9,028	64	.9929	.0071
3	8,964	42	.9953	.0047
4	8,922	33	.9964	.0036

5	8,889	29	.9967	.0033
6	8,860	27	.9970	.0030
7	8,833	22	.9974	.0026
8	8,811	20	.9978	.0022
9	8,791	18	.9980	0020

10	8,773	16	.9981	.0019
11	8,757	18	.9980	.0020
12	8,739	18	.9979	.0021
13	8,721	20	.9977	.0023
14	8,701	23	.9974	.0026

15	8,678	26	.9970	.0030
16	8,652	28	.9967	.0033
17	8,624	32	.9963	.0037
18	8,592	35	.9960	.0040
19	8,557	36	.9957	.0043

20	8,521	38	.9956	.0044
21	8,483	40	.9953	.0047
22	8,443	41	.9951	.0049
23	8,402	42	.9950	.0050
24	8,360	44	.9948	.0052

25	8,316	45	.9946	.0054
26	8,271	46	.9944	.0056
27	8,225	48	.9942	.0058
28	8,177	48	.9941	.0059
29	8,129	50	.9939	.0061

30	8,079	50	.9937	.0063
31	8,029	52	.9935	.0065
32	7,977	54	.9933	.0067
33	7,923	55	.9930	.0070
34	7,868	58	.9927	.0073

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

40	7,498	62	.9917	.0083
41	7,436	61	.9918	.0082
42	7,375	61	.9918	.0082
43	7,314	60	.9917	.0083
44	7,254	61	.9916	.0084

45	7,193	61	.9915	.0085
46	7,132	63	.9912	.0088
47	7,069	65	.9909	.0091
48	7,004	66	.9905	.0095
49	6,938	69	.9900	.0100

50	6,869	75	.9892	.0108
51	6,794	78	.9885	.0115
52	6,716	82	.9878	.0122
53	6,634	86	.9870	.0130
54	6,548	90	.9863	.0137

55	6,458	93	.9856	.0144
56	6,365	98	.9846	.0154
57	6,267	103	.9835	.0165
58	6,164	109	.9824	.0176
59	6,055	114	.9811	.0189

60	5,941	117	.9803	.0197
61	5,824	122	.9790	.0210
62	5,702	128	.9776	.0224
63	5,574	135	.9758	.0242
64	5,439	143	.9738	.0262

65	5,296	162	.9694	.0306
66	5,134	172	.9664	.0336
67	4,962	182	.9633	.0367
68	4,780	188	.9607	.0393
69	4,592	191	.9585	.0415

70	4,401	191	.9566	.0434
71	4,210	195	.9537	.0463
72	4,015	204	.9492	.0508
73	3,811	219	.9424	.0576
74	3,592	243	.9323	.0677
75	3,349