## On average, we die a decade earlier than expected

“Doctors say he’s got a 50/50 chance of living… though there’s only a 10% chance of that.”

I’ve lately had occasion to contemplate my own mortality. How long should I expect to live? The most recent life table published by the Centers for Disease Control (see the reference at the end of this post) indicates an expected lifespan of 76.5 years for a male. This is based on a model of age at death as a random variable $X$ with the probability density shown in the following figure.

Probability distributions of age at death based on the United States period life table for 2014.

The expected lifespan of 76.5 years is $E[X]$ (using the red curve for males). In other words, if we observed a large number of hypothetical (male) infants born in the reference period 2014– and they continued to experience 2014 mortality rates throughout their lifetimes– then their ages at death would follow the above distribution, with an average of 76.5 years.

However, I have more information now: I have already survived roughly four decades of life. So it makes sense to ask, what is my conditional expected age at death, given that I have already survived to, say, age 40? In other words, what is $E[X | X \geq 40]$?

This value is 78.8 years; I can expect to live to a greater age now than I thought I would when I was first born. The following figure shows this conditional expected age at death $E[X | X \geq x]$, as well as the corresponding expected additional lifespan $E[X-x | X \geq x]$, as a function of current age $x$.

Conditional expected age at death and expected additional lifespan, vs. current age.

For another example, suppose that I survive to age 70. Instead of expecting just another 6.5 years, my expected additional lifespan has jumped to 14.5 years.

Which brings us to the interesting observation motivating this post: suppose instead that I die at age 70. I will have missed out on an additional 14.5 years of life on average, compared to the rest of the septuagenarians around me. Put another way, at the moment of my death, I perceive that I am dying 14.5 years earlier than expected.

But this perceived “loss” always occurs, no matter when we die! (In terms of the above figure, the expected value $E[X-x | X \geq x]$ is always positive.) We can average this effect over the entire population, and find that on average males die 12.2 years earlier than expected, and females die 10.8 years earlier than expected.

Reference:

1. Arias, E., United States Life Tables 2014, National Vital Statistics Reports, 66(4) August 2017 [PDF]

Following are the probabilities $P(\left\lfloor{X}\right\rfloor = x)$ for the United States 2014 period life table used in this post, derived from the NVSR data in the above reference, extended to maximum age 120 using the methodology described in the technical notes.

Age   P(all)              P(male)             P(female)
===========================================================
0 0.005831            0.006325            0.005313
1 0.000367843         0.000391508         0.000343167
2 0.000246463         0.000276133         0.000216767
3 0.000182814         0.000206546         0.000157072
4 0.000156953         0.000183668         0.000129216
5 0.000141037         0.000160804         0.000120255
6 0.000125127         0.000142914         0.000106328
7 0.000112203         0.000128008         0.0000963806
8 0.000100276         0.000112117         0.0000884231
9 0.0000913317        0.0000992073        0.0000834481
10 0.0000883454        0.0000932456        0.0000824477
11 0.0000952854        0.000103156         0.0000874072
12 0.000119095         0.000137857         0.000101304
13 0.000164729         0.000203286         0.000124134
14 0.000227209         0.000294457         0.000155893
15 0.000293617         0.000391501         0.000190617
16 0.000362946         0.000491412         0.000227306
17 0.000442117         0.000609009         0.000265957
18 0.000529115         0.000743227         0.000302594
19 0.000616971         0.000881116         0.000338207
20 0.00070566          0.00101964          0.000373784
21 0.000786255         0.00114394          0.000408332
22 0.000847887         0.0012343           0.000438875
23 0.000886654         0.00128494          0.000465417
24 0.000909534         0.00130883          0.000490933
25 0.000929392         0.0013228           0.0005174
26 0.000952147         0.00134161          0.000545804
27 0.000977786         0.00136426          0.00057515
28 0.0010063           0.00139366          0.000605432
29 0.00103864          0.00142878          0.00063566
30 0.00107285          0.0014657           0.000668788
31 0.00110792          0.00150147          0.000705793
32 0.00114483          0.0015361           0.000745674
33 0.00118454          0.00156959          0.000792356
34 0.00122898          0.00160582          0.000845811
35 0.00128495          0.00165443          0.000908956
36 0.00135141          0.00171632          0.000981746
37 0.00142538          0.00178654          0.00106021
38 0.00150387          0.0018631           0.00114136
39 0.00158781          0.00194786          0.00122516
40 0.00168489          0.00204935          0.00131843
41 0.00179886          0.00217314          0.00142304
42 0.00192761          0.00231898          0.00153401
43 0.00207581          0.00249327          0.00165615
44 0.00224899          0.00270038          0.00179419
45 0.00243725          0.00292846          0.00194116
46 0.00265083          0.0031884           0.00210855
47 0.0029103           0.00350311          0.00231349
48 0.00321588          0.00387243          0.0025555
49 0.0035468           0.00427362          0.00281771
50 0.00387592          0.00467212          0.00307957
51 0.00420287          0.00507013          0.00333606
52 0.00454693          0.00549737          0.00359937
53 0.00492128          0.00597165          0.0038758
54 0.00532664          0.00649003          0.00417147
55 0.00575619          0.00703581          0.00448667
56 0.00619215          0.00758275          0.00481225
57 0.00662626          0.00813029          0.00513737
58 0.00705499          0.00866988          0.00545972
59 0.00748745          0.00921066          0.00578812
60 0.00794918          0.00978879          0.0061402
61 0.0084469           0.010399            0.00653116
62 0.0089597           0.010994            0.00696556
63 0.00947691          0.0115436           0.00744933
64 0.0100035           0.0120603           0.00797895
65 0.0105466           0.0125691           0.00854801
66 0.0111425           0.0131347           0.00916566
67 0.0118165           0.0137895           0.00985298
68 0.0126025           0.0145881           0.010625
69 0.0135386           0.0155721           0.0115183
70 0.014622            0.016711            0.0125556
71 0.0157853           0.0179169           0.0136863
72 0.0169733           0.0191484           0.0148415
73 0.0181664           0.0203416           0.0160437
74 0.0193544           0.0214907           0.0172763
75 0.0205581           0.0226235           0.0185559
76 0.0219039           0.023887            0.0199909
77 0.0233782           0.0252875           0.0215559
78 0.0249405           0.0266573           0.0233399
79 0.0266659           0.0281283           0.0253501
80 0.0283006           0.0295587           0.0272207
81 0.0298041           0.0307938           0.0290306
82 0.0311707           0.0318902           0.0307088
83 0.0326118           0.0329808           0.0325375
84 0.0338734           0.0336728           0.0344093
85 0.0348103           0.0342521           0.0357896
86 0.0356915           0.0345144           0.0373244
87 0.036144            0.0342741           0.0384714
88 0.0361034           0.0334914           0.0391388
89 0.0355212           0.0321521           0.0392438
90 0.0343716           0.0302738           0.0387212
91 0.0326583           0.0279093           0.0375332
92 0.0304192           0.0251463           0.0356786
93 0.0277276           0.0221028           0.0331989
94 0.02469             0.0189177           0.030181
95 0.0214386           0.0157381           0.0267542
96 0.0181203           0.0127037           0.0230805
97 0.0148823           0.00993297          0.0193396
98 0.011857            0.00751144          0.0157101
99 0.00914934          0.00548606          0.0123497
100 0.00682791          0.0038652           0.00937893
101 0.0049216           0.00262443          0.0068711
102 0.00342273          0.00171608          0.00484973
103 0.00229461          0.00108015          0.00329442
104 0.00148199          0.000654343         0.00215221
105 0.000921789         0.000381551         0.00135156
106 0.000552114         0.000214241         0.000815772
107 0.00031851          0.000115918         0.00047333
108 0.000177059         0.0000604924        0.000264139
109 0.0000949139        0.0000304833        0.000141878
110 0.0000491106        0.0000148533        0.0000734294
111 0.0000245565        0.00000700891       0.0000366663
112 0.0000118822        0.00000320819       0.0000176914
113 0.00000557214       0.00000142697       0.00000826206
114 0.00000253659       0.000000617876      0.00000374136
115 0.00000112287       0.000000260924      0.00000164594
116 0.0000004842        0.00000010766       0.00000070483
117 0.000000203758      0.0000000434809     0.000000294373
118 0.0000000838243     0.0000000172192     0.000000120143
119 0.0000000337717     0.0000000066978     0.0000000480077
120 0.0000000216956     0.00000000409582    0.0000000304297

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### One Response to On average, we die a decade earlier than expected

1. David A says:

You are certainly quite negative!
(-;
So a minor positive adjustment if you live longer then any age on the chart perhaps the average premature death age drops.

The trouble with averages is the average person has one ball and one breast.

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