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2.2.1 Estimating Service Life By Actuarial Methods

Since it is impractical to record the complete life

histories of a large number of units, a method analo-
gous to the population census can be used. A popula-

tion survey of a specified product is taken over a
comparatively short interval of time and the number of
units in the population at each age is recorded. The

purpose of the survey is to estimate the percentage of

units surviving to various ages, the mean service life
expectancy when new and the mean residual life of used

units of various ages. The removal rate of units from

the population can also be estimated. This information,

together with acquisition cost and repair cost data can
be used to determine the optimum age to replace units.

The validity of the actuarial method depends on a rather

strong assumption which is that the service life of a

unit is independent of the year in which the unit was

manufactured. When applied to human populations, the

actuarial method assumes that birth and death rates de

pend only on the age of individuals and not on the par

ticular calendar year in which they were born. This
assumption is generally reasonable for the items con-

sidered in this study.

23-615 0.73 - 13

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An illustration of this actuarial method is pre

sented in Table 9 and is based on a survey of curtain

and drapery service life conducted in 1957. Column

(1) lists the ages in years of units surveyed. Column (2) lists the number of units surviving in each age group. Column (3) lists the number of units removed

during each age interval. Column (4) is the sum of

units in columns (2) and (3) at each specified age and

is the number of units exposed to risk during each age interval. Column (5) is the ratio of units removed to

units exposed to risk in each age interval. This ratio

is called the removal rate or failure rate. Column (6)

lists the survival rate, which is units minus the removal rate. Column (7) is the number of units surviving to each age from a hypothetical population of 1000 units.

For example, the number surviving to age 1 year is 1000

in this Table. The number surviving 'to age 2 years is

1000 times the probability of surviving to age 2 (1.e.,

.8652). Hence approximately 865 units will survive to

age 2 years. To find the number surviving to age x + 1,

take the number surviving to age x and multiply by the probability of surviving through age interval x given in

column (6).

Numbers in column (7) divided by 1000 are

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TABLE 9

LITE TABLE FOR CURTAINS AND DRAPERIES IN THE LIVING

ROOM OF ONE OWNER
(197 households in the Wilmington urbanised area within Delaware, 1964)

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1,000
1,000

865
748
628
844
497
413
207
207
207
184
184
184
184
37
27
27
27
27
27
37
27
27
27
27

6,329 5,329 4,464 8,716 3,088 2,544 2,047 1,634 1,867 1,100 833 699 565 431 297 270 243 216 189

10 11 12 13 14 15. 16 17:

18 : 19

20 21 22

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135 108 81 54 27

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* All Items suaumed to bene been sexquired on January 1 dl tho your al acquisition.
• Number of curtain and drapery papela.
The mean vervloo-ilo apociancy is 6,820/1,000 + 5 or 6.8 years

When there is no inventory and no removal, the removal race is indocrminale. For this model prvival nie dl I was assumed, which yielda a maximum figura Il a survival rato al 0 had been numed, tbe mcan service life aspectancy would have boon 6.1 instead of 6.8 years, a decronus el 10 per cent. If the data bed boen smoothed. the climate would fall between there atromcl.

Source: Pennock, Jean L. and Jaeger, Carol M., "Estimating the

Service Life of Household Goods by Actuarial Methods,"
JASA, Vol 52 (1957).

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also the percentages of units surviving to various
ages. For example, approximately 54% of the units will
survive 5 years.

The residual life of a unit aged x
years is given by the following formula:

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For example, a unit aged 5 years has a mean life ex

pectancy of approximately 6.1 years.
Numbers in column (8) are obtained by cumulating numbers
in column (7) starting at the bottom. Hence the number
in column (8) corresponding to age 24 years is the sum
of the numbers in column (7) corresponding to ages 24

and greater.

2.2.2 Survival Distributions

A useful mathematical approximation to the life distribu-
tion for typical consumer products can be made using the
Weibull distribution
F(t)

-(at)
where F(t) is the percentage of items failing before
lifetime t. The parameter a will be ? 1 for items which
wear out with increasing age. If the value of a is close
to 3, the life distribution can be approximated by a normal

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probability distribution. The mean life y in terms

of the Weibull distribution parameters is

H = f(1/2) 1/2

where r is the gamma function and is tabulated in

statistical tables. The variance or variability of

observed lifetimes is given by

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The standard error of observations is the square root

2

and is denoted by the symbol o.

of o

The Weibull distribution parameters can be estimated
by plotting the percentage of units failing as a function

of age on Weibull probability paper.

If n is the sample size and is the estimated mean life,

ther the standard error of estimating the true population mean life is ori. In practice, the estimate for o is

substituted for o to estimate the standard error.

2.2.3 Application of Statistical Techniques

For purposes of illustration, the above statistical tech

niques were somewhat modified because of the limitations

of available data. These techniques have been applied to

previously presented automobile, home appliance, and con

struction equipment data.

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