Concept of Survivor Curve and Probable Life

Curve

From Chapter 9, Page 207 of Winfrey’s Text Book

“The survivor curve is a curve which shows the number of units of property that survive in service at given ages. The area under the curve is a direct measure of the average service life of the property units. The probable life of the surviving units at any age can also be calculated from the remaining area by diving the remaining area by the amount surviving at that age.”

As per Winfrey’s Text Book

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Probable Life = (Shaded area)/(% Survived at that age)

From Winfrey’s Reference 9-13*

“Service remaining at any age is equal to the area under the curve to the right of the ordinate erected at that age.”

*Statistical Analyses of Industrial Property Retirements,by Robley Winfrey, 1967.

As per Winfrey’s Reference 9-13

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Expectancy at 5 yrs = (Shaded area at 5 yrs)/(% Survived at 5 yrs)

From Winfrey’s Reference 9-13*

“The expectancy at any age is a function of the remaining service which is obtained by summing the areas for each age interval, starting at the age of the last survival (0% surviving) and working to the left to the age in question.”

*Statistical Analyses of Industrial Property Retirements,by Robley Winfrey, 1967.

From Winfrey’s Reference 9-13*

“The expectancy of life at any given age is then obtained by dividing the remaining service at that age by the percent surviving at the same age.”

“The probable average life of the survivors at any given age is equal to the sum of the expectancy and the age for which the expectancy is computed.”

*Statistical Analyses of Industrial Property Retirements,by Robley Winfrey, 1967.

Example Problem

100Total

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152520151052

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Number of Units RetiredAge in Years

Using the Individual Unit Method, draw the Frequency Curve, Survivor Curve and Probable Life Curve

Frequency Curve

• Choose X-axis for Age in Years (N) and Y-axis for Number of Units Retired (f)

• Draw the Frequency Distribution Curve

Frequency Distribution Curve

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A ge, N (Y ear s )Age, N (Years)

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f)

Average Service Life

Age in Years

(N)

Number of Units Retired

(f)

Cumulative Number of

Units Retired

Number of Units

Survived%

Survived N*f0 0 0 100 100 02 2 2 98 98 45 6 8 92 92 3010 15 23 77 77 15015 25 48 52 52 37520 20 68 32 32 40025 15 83 17 17 37530 10 93 7 7 30035 5 98 2 2 17540 2 100 0 0 80

Sum = 100 Sum = 1889

Average Service Life, M = = = 18.89(N*f)

f1889100

Survivor Curve

• Survivor Curve is drawn with X-axis for Age in Years and Y-axis for Percent Surviving

Survivor Curve100

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Total Remaining Service

• Total Remaining Service in %-Years is calculated at any age by considering a triangle under the Survivor Curve to the right of the ordinate at that age

• The area of a triangle is given by:

Area = ½*(Base)*(Altitude)• Total Remaining Service

= ½*(Number of Years Remaining)*(% Surviving)

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Survivor Curve and Average Life Line

Survivor Curve

Average Service Life=18.89

Calculation of Total Remaining Service (%-Years)

Total Remaining Service at any time, t

At=1/2*(Number of Years Remaining)*(% Surviving)

At t=0, A0=1/2*(40-0)*(100) = 2000

At t=2, A2=1/2*(40-2)*(98) = 1862

At t=5, A5=1/2*(40-5)*(92) = 1610

At t=10, A10=1/2*(40-10)*(77) = 1155

At t=15, A15=1/2*(40-15)*(52) = 650

Calculation of Total Remaining Service (%-Years)

Total Remaining Service at time t,

At=1/2*(Number of Years Remaining)*(% Surviving)

At t=20, A0=1/2*(40-20)*(32) = 320

At t=25, A2=1/2*(40-25)*(17) = 127.5 128

At t=30, A5=1/2*(40-30)*(7) = 35

At t=35, A10=1/2*(40-35)*(2) = 5

At t=40, A15=1/2*(40-40)*(0) = 0

Average Service Life

Average Service Life can be calculated by dividing the total area (%-Years) below the Survivor Curve (A0) by Total % (100%).

Average Service Life = A0/100 = (2000)/(100) = 20.0 YrsAverage Service Life as calculated from the Frequency Distribution Curve differs

a little (18.89 Yrs), since the area calculation gives approximate results.

Expectancy As Per Winfrey

• Expectancy at the start of any age is calculated by dividing Total Remaining Service by % Surviving

Expectancy at the Start of the YearExpectancy at time t,

Et= (Total Remaining Service, At)/(% Surviving)

At t=0, E0= (2000)/(100)=20.0

At t=2, E2= (1862)/(98)=19.0

At t=5, E5= (1610)/(92)=17.5

At t=10, E10= (1155)/(77)=15.0

At t=15, E15= (650)/(52)=12.5

At t=20, E20= (320)/(32)=10.0

At t=25, E25= (128)/(17)=7.5

Expectancy at the Start of the YearExpectancy at time t,

Et= (Total Remaining Service, At)/(% Surviving)

At t=30, E30= (35)/(7)=5.0

At t=35, E35= (5)/(2)=2.5

At t=40, E40= (0)/(0)=Undefined. However, considering 40 years as the end of service life after which no life is remaining, we consider E40= 0.

Probable Life

• Probable Life at the start of any age is the Sum of Expectancy and Number of Years of Life at that age

Probable Life at the Start of the Year

Probable Life at time t,

Pt= Expectancy (Et) + Number of Years of Life (N)

At t=0, P0= 20.0+0=20.0

At t=2, P2= 19.0+2=21.0

At t=5, P5= 17.5+5=22.5

At t=10, P10= 15.0+10=25.0

At t=15, P15= 12.5+15=27.5

At t=20, P20= 10.0+20=30.0

Probable Life at the Start of the Year

Probable Life at time t,

Pt= Expectancy (Et) + Number of Years of Life (N)

At t=25, P25= 7.5+25=32.5

At t=30, P30= 5.0+30=35.0

At t=35, P35= 2.5+35=37.5

At t=40, P40= 0+40=40.0

Probable Life Curve

• Probable Life Curve can be drawn by taking X-axis for Age in Years and Y-axis for Probable Life at any time

Probable Life Curve

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Summary of Calculations

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Survivor and Probable Life Curve

Survivor Curve Probable Life Curve

Average Service Life=18.89

Area Calculation by Integration• Enter the Data in MS Excel Spreadsheet

Area Calculation by Integration• Draw a Scatter Plot for % Surviving against Age using the following Excel option

Area Calculation by Integration• Scatter Plot in order to draw the Survivor Curve

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Area Calculation by Integration• We find the best fit curve by Regression using Excel• Right click on any dot and select Add Trend Line, which gives the following window• Select Polynomial of Order 2

• We get the Trend Line as it fits best with the data

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• Right click on Trend Line and select Format Trend Line• Click on Options and check the boxes as shown

• We find the 2nd Degree Polynomial Equation of the Survivor Curve as shown below

y = 0.0413x2 - 4.4574x + 107.74

R2 = 0.9829

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Area at any age can be calculated by integrating the equation of the Survivor Curve between two age limits of consideration

where y = 0.0413t2 – 4.4574t + 107.74

Area Calculation by Integration

Area at any age t1 = ydtt=t1

t=40

Area at age 0 yrs =

= [0.0413t2 – 4.4574t + 107.74]dtt=0

t=40

ydtt=0

t=40

= [0.0413t3/3 – 4.4574t2/2 + 107.74t]t=0

t=40

= [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)]

- [0.0413(0)3/3 – 4.4574(0)2/2 + 107.74(0)]

= 1624.75 %-Years

Area at age 2 yrs =

= [0.0413t2 – 4.4574t + 107.74]dtt=2

t=40

ydtt=2

t=40

= [0.0413t3/3 – 4.4574t2/2 + 107.74t]t=2

t=40

= [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)]

- [0.0413(2)3/3 – 4.4574(2)2/2 + 107.74(2)]

= 1624.75 – 197.7605= 1426.989 %-Years

Area at age 5 yrs =

= [0.0413t2 – 4.4574t + 107.74]dtt=5

t=40

ydtt=5

t=40

= [0.0413t3/3 – 4.4574t2/2 + 107.74t]t=5

t=40

= [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)]

- [0.0413(5)3/3 – 4.4574(5)2/2 + 107.74(5)]

= 1624.75- 484.70 = 1140.05 %-Years

Area at age 10 yrs =

= [0.0413t2 – 4.4574t + 107.74]dtt=10

t=40

ydtt=10

t=40

= [0.0413t3/3 – 4.4574t2/2 + 107.74t]t=10

t=40

= [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)]

- [0.0413(10)3/3 – 4.4574(10)2/2 + 107.74(10)]

= 1624.75- 868.297 = 756.453 %-Years

Area at age 15 yrs =

= [0.0413t2 – 4.4574t + 107.74]dtt=15

t=40

ydtt=15

t=40

= [0.0413t3/3 – 4.4574t2/2 + 107.74t]t=15

t=40

= [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)]

- [0.0413(15)3/3 – 4.4574(15)2/2 + 107.74(15)]

= 1624.75- 1161.05 = 463.645%-Years

Area at age 20 yrs =

= [0.0413t2 – 4.4574t + 107.74]dtt=20

t=40

ydtt=20

t=40

= [0.0413t3/3 – 4.4574t2/2 + 107.74t]t=20

t=40

= [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)]

- [0.0413(20)3/3 – 4.4574(20)2/2 + 107.74(20)]

= 1624.75- 1373.453 = 251.297%-Years

Area at age 25 yrs =

= [0.0413t2 – 4.4574t + 107.74]dtt=25

t=40

ydtt=25

t=40

= [0.0413t3/3 – 4.4574t2/2 + 107.74t]t=25

t=40

= [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)]

- [0.0413(25)3/3 – 4.4574(25)2/2 + 107.74(25)]

= 1624.75- 1515.667 = 109.083%-Years

Area at age 30 yrs =

= [0.0413t2 – 4.4574t + 107.74]dtt=30

t=40

ydtt=30

t=40

= [0.0413t3/3 – 4.4574t2/2 + 107.74t]t=30

t=40

= [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)]

- [0.0413(30)3/3 – 4.4574(30)2/2 + 107.74(30)]

= 1624.75- 1598.07 = 26.68 %-Years

Area at age 35 yrs =

= [0.0413t2 – 4.4574t + 107.74]dtt=35

t=40

ydtt=35

t=40

= [0.0413t3/3 – 4.4574t2/2 + 107.74t]t=35

t=40

= [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)]

- [0.0413(35)3/3 – 4.4574(35)2/2 + 107.74(35)]

= 1624.75- 1630.988= - 6.238%-Years

Area at age 40 yrs =

= [0.0413t2 – 4.4574t + 107.74]dtt=40

t=40

ydtt=40

t=40

= [0.0413t3/3 – 4.4574t2/2 + 107.74t]t=40

t=40

= [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)]

- [0.0413(40)3/3 – 4.4574(40)2/2 + 107.74(40)]

= 1624.75- 1624.75 = 0 %-Years

Summary of Calculations

Age in Years

% Survived

Remaining Service by Integration (%-Years)

Expectancy at any age

Probable Life at any age

0 100 1624.75 16.25 16.252 98 1426.989 14.56 16.565 92 1140.05 12.39 17.39

10 77 756.453 9.82 19.8215 52 463.645 8.92 23.9220 32 251.297 7.85 27.8525 17 109.083 6.42 31.4230 7 26.68 3.81 33.8135 2 -6.238 -3.12 31.8840 0 0 0 40.00

Survivor Curve and Probable Life Curvey = 0.0413x2 - 4.4574x + 107.74

R2 = 0.9829

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