This part details the data and methods used to derive the 2005–07 complete life tables presented in this report.
Data
The data used to construct the 2005–07 complete life tables were:
 deaths registered in New Zealand of people resident in New Zealand in the December years 2005–07, by singleyear of age, sex, and ethnicity
 live births registered in New Zealand to mothers resident in New Zealand in the December years 2004–07, by sex and ethnicity
 the estimated resident population of New Zealand at 30 June 2006, by singleyear of age, sex, and ethnicity.
Deaths numerator
The life tables were compiled from deaths registered, rather than deaths occurring. Most death statistics refer to registrations rather than occurrences for a given time period, because of the time lag between when the death occurred and when it is registered. Hence, for a given time period, the number of death registrations can be confirmed before the number of death occurrences. For periods of a year or more, the difference between registrations and occurrences is generally small, so death statistics referring to registrations are suitable for most purposes.
An allowance for ethnic nonresponse among deaths was also made. There was no response to the ethnicity question for 4.9 percent of deaths in 2005–07.
Because deaths in the first year of life are skewed towards the early part of this age, infant death rates were calculated from more detailed data. This involved the division of the first year of life into more detailed ages.
Population denominator (exposedtorisk population)
Usually the mean population for a period is used as the denominator to calculate death rates. However, mean population estimates are not available for all ethnic populations. To ensure consistency of method among all population subgroups, the estimated resident population at 30 June (the midpoint) was used. The impact of using ‘midpoint’ rather than ‘mean’ population estimates is generally insignificant.
The estimated resident population at 30 June 2006 is based on the census usually resident population count at 7 March 2006, with adjustments for:
 nonresponse to the census ethnicity question
 net census undercount
 residents temporarily overseas on census night
 births, deaths, and net migration between census night and 30 June of the census year
 reconciliation with demographic estimates at ages 0–4 years.
The 2006 Census asked people "Which ethnic group do you belong to? Mark the space or spaces which apply to you". The census usually resident population count of 4,027,947 included 565,329 who identified with the Māori ethnicity and 167,784 who gave no specific ethnic response. The 2005–07 life tables use as a population denominator the estimated resident population for each ethnic group of New Zealand, at 30 June 2006. New Zealand's estimated resident population of 4,185,000 included 624,000 who identified with the Māori ethnicity.
Because of changes to the census ethnicity question between 1996 and 2006, the 1996 and 2006 population estimates for ethnic groups are not necessarily comparable. Nevertheless, the derived mortality measures presented here are considered to give a statistically satisfactory description of Māori and nonMāori mortality experience during the 1995–97 to 2005–07 periods. Note that all population estimates used in the 1995–97 to 2005–07 life tables have been derived using the same methodology. In addition, the ethnicity question used in the 1996 Census is the same as that used in birth and death registration forms from September 1995. The use of population estimates based on the 1996 Census also allows the adjustment ratios presented in Ajwani (2003) to be incorporated.
For more information about the estimated resident population, refer to “Information about the population estimates” on the Statistics NZ website (www.stats.govt.nz).
Derived rates
The life tables are based on deaths averaged over three years. This is designed to reduce the impact of yeartoyear statistical variations, particularly at younger ages where there is a small number of deaths and at very old ages where the population at risk is small. In some cases the New Zealand data does not enable death rates to be reliably estimated at all ages. For most ages above 90 years, death rates of the total New Zealand population have been modelled on the mortality trends of other countries such as Australia, Canada, Japan, the United Kingdom and the United States. For the Māori and nonMāori populations, death rates have also been modelled at some younger ages.
There are some small observed numeratordenominator ethnic differences since 1995 in comparison with 1996, 2001, and 2006 census data. For the 2000–02 and 2005–07 life tables, these estimated differences are not significant enough to reliably adjust death numbers by age, sex, and ethnicity. For the 1995–97 life tables, the smooth adjustment factors presented in Ajwani (2003) have been applied to Māori deaths by age, to allow for underreporting of Māori deaths (relative to the Māori population). For the nonMāori life tables, corresponding adjustments have been applied to nonMāori deaths by age. These adjustment factors affect Māori life expectancy at birth by about 0.7 years, and nonMāori life expectancy at birth by about 0.1 years.
The construction of each complete life table involved two stages. First, central death rates (mx) were calculated for each age (x), except the first year of life, and were then smoothed to eliminate any apparent irregularities. Second, the smoothed rates were used to calculate a set of agespecific probabilities of death (qx), which were then used to derive other life table functions. The derivation of the mortality rate in the first year of life differed from all other ages and required special formulae, as detailed below in ‘Age 0 years’.
Life table notation
x 
Exact age (eg exact age 5 corresponds to 5 years and 0 days). 
l_{x} 
Number of people alive at exact age x from the original group of 100,000 (l0). 
L_{x} 
Average number of people alive in the age interval x to x + 1. 
d_{x} 
Number of deaths in the age interval x to x + 1. 
q_{x} 
Probability that a person at exact age x dies within a year. 
_{5}q_{x} 
Probability that a person at exact age x dies within 5 years. 
p_{x} 
Probability that a person at exact age x lives another year. 
_{5}p_{x} 
Probability that a person at exact age x lives another 5 years. 
_{5}m_{x} 
Central death rate for population in the age group x to x + 5. 
_{5}s_{x} 
Proportion of population in the age group x to x + 5 surviving another 5 years. 
e_{x} 
Expected number of years of life remaining at exact age x. 
Age 0 years
The probability of dying in the first year of life (q_{0}) required special treatment because infant deaths are skewed towards the early part of this age. The first year of life was divided into eight minor age intervals (n):
 less than 1 day
 from 1 day to less than 2 days
 from 2 days to less than 7 days
 from 1 week to less than 4 weeks
 from 4 weeks to less than 3 months
 from 3 months to less than 6 months
 from 6 months to less than 9 months
 from 9 months to less than 12 months.
For each of these age intervals the values of q_{0}(n), l_{0}(n), and d_{0}(n) were calculated. The following examples show the formula for calculating q0(n) for two of these age intervals for 2005–07, where the denominator reflects the exposedtorisk population:
For n = 3 

For n = 5 

where, for example:

probability of dying between 4 weeks and 3 months of life 

live births in 2007 

live births in the fourth (December) quarter of 2007 
k 
is an adjustment for deaths and migration made to the denominator to exclude people in the original birth cohort who died in an earlier age interval, and to allow for the effect of net migration, in order to give the correct 'exposedtorisk' population 
The values of q_{0}(n) were then used to calculate l_{0}(n) and d_{0}(n):
Given 

the radix of a life table 
then 

for n = 2, 3, ..., 8 





for n = 2, 3, ..., 8 



and 


The value of
was calculated as follows:

for n = 1, 2, ..., 7 

where 


where w(n) is the weight given by the fraction of the year covered by the age interval (n). For example, for n = 6:
Age 1 year and over
The central death rates (mx) were first calculated for each single year of age by dividing the average annual deaths of New Zealand residents for the period by the estimated resident population at the midpoint of the period. For 2005–07:
Some refinement of data was made in the very old ages, above 90 years, to offset the effects of age misreporting and small death numbers. The central death rates derived from actual data also showed minor fluctuations across other ages. To minimise these fluctuations the rates were smoothed using a cubic spline method (figure 3.01). For more details on this method see Benjamin and Pollard (1980) and Department of Statistics (1986).
Figure 3.01
The smoothed central death rates were then used to calculate the corresponding values of q_{x}, for each age, using the equation:
Each series of q_{x} was tested to ensure that the deviations between the actual and expected deaths were minimal. The values of q_{x} were then used to derive the remaining life table functions:

the radix of a life table 









where h is the highest age of a given population group. 
Supplementary functions for fiveyear age groups
In addition to the main life table functions, the following supplementary functions for fiveyear age groups have been calculated for 2005–07, and are contained in appendix 2:
Standardised death rates
Standardised death rates (SDRs) provide a summary measure of the mortality experience of an ethnic group, while allowing for the different agesex composition of each ethnic group. Using the direct method of standardisation, SDRs indicate the overall death rate (deaths per 1,000 population) if the observed agesex specific death rates were applied to a standard population. The SDRs presented in this report use the age and sex distribution of the estimated resident population of New Zealand at 30 June 1996 as the standard:
where
m_{a} is the agesex specific death rate of the ethnic group
P_{a} is the standard population at each age and sex
P is the total standard population
a are age groups 0, 1–4, 5–9, 10–15, ..., 80–84 and 85+ years