In a groundbreaking article, T. L. Freeman discusses the relationship between actual age and effective age1. His conclusion is that the passing of the years goes faster as we grow older. This makes sense; for instance when you are 10 years of age, a year represents 10% of your life, and seems like a very long time. However, when you are 50 years old, one year has reduced to only 2% of your life, and hence seems only one-fifth as long.

Summarizing this work, Freeman comes to the conclusion that the actual age (AA) needs to be corrected for the apparent length of a year (AY). The apparent length of a year is inversely proportional to one person's actual age:

          AY= α/AA

The constant of proportionality α is rather loosely defined by Freeman as the age at which a year really seems to last a year, and it was arbitrarily set at 20 years (α=20).

Now Freeman determines the concept of Effective age, which is simply the integral over time of the Apparent Year from age 1 to the actual age (AA) of interest:

               AA            AA
          EA =  ∫ AY d(AA) =  ∫ 20/AA d(AA) = 20 ln(AA)
                1             1
Although this formula results in some interesting conclusions, there are several flaws with this concept. As mentioned above, the choice of the proportionality constant is rather arbitrary. There is no rational justification for the choice of this age, but it was solely chosen based on Freeman's own perception of (the passing of) time. Next, the evaluation of the integral seems incorrect, since its lower limit was set at 1, and not at 0. Obviously, the choice of zero as lower integration boundary yields can not be evaluated due to the logarithmic term in the expression. Because of the obvious problems with Freeman's concept of time perception, it is necessary to redefine the Effective Age on a sounder basis.

In the traditional concept of time perception, one person's Actual Age is proportional to the passing of time (t).

          AA = βt + γ

Note the occurrence of two parameters β and γ that are traditionally set to one and zero, respectively. However, each has a clear (though usually underappreciated) function in time perception. The β-parameter describes the rate at which one person ages; some persons remain annoying little crybabies during their life, while others become boring old farts at 20. The γ-parameter describes the origin of one person's time perception. Did you ever meet those proud parents boasting about their little one who is only x months old, and already walks, writes obfuscated C, or recently sold his first No, these youngsters aren't bright for their age; they simply have a high γ-factor.

It is clear that with this definition, one person's Actual Age may already be non-synchronous with time. However, analogous to Freeman's work, the apparent length of a year (AY) is not constant:

          AY= α/AA = α/(βt + γ)
We can remove one of the parameters by defining two parameters δ and ε.
          AY= α/(βt + γ) = (α/β)/(t + γ/β) = δ/(t + ε)
The actual values of δ and ε will become clear from the boundary conditions.

In order to obtain the Effective Age, the integral of AY is evaluated. Note that the integral is evaluated over time, and not over Actual Age, since AA is a function of time:

               t           t
          EA = ∫ AY d(t) = ∫ δ/(t + ε) d(t)
               0           0

          EA = δ ln(t + ε) - δ ln(ε)

The lower boundary condition (t=0) should yield an Effective Age of zero years (EA=0). Therefore ε = 1.

The upper boundary is less apparent. It should be chosen so that at t=tmax, EA = t. At death, the Effective Age and real time are again equal. However, no person knows for sure his or her personal life expectancy. This is clearly an issue for molecular biologists to address. However, if we assume for a person a life expectancy of 80 years (t=80, EA=80), we obtain:

          δ = 80/ln(81)

               80 ln(t + 1)
          EA = ----------
This formula can now be used to calculate the Effective Age (and the Effective percentage Completion of Life) as a function of time. This is shown in the following table:
time (yrs.)  EA (yrs.)  Life%
0            0.0      0
1           12.6      16
2           20.0      25
3           25.2      32
4           29.3      37
5           32.6      41
10          43.7      55
15          50.5      63
20          55.4      69
30          62.5      78
40          67.6      85
50          71.6      89
60          74.8      94
70          77.6      97
80          80.0      100

And thus, the bold statement in the title is justified. Life is half over at age ten, and three quarters over at age thirty. Note the rapid increase at very young ages: in the initial stages of life, life itself makes big strides forward. For instance, consider the concepts of speech, eating and walking; skills that are learned at a young age and are carried on throughout a person's life.

Another interesting observation that we can make is the age at which one year really seems to last one year. This can be calculated quite easily from the derivation above. For a life expectancy of 80 years, it is equal to 80/ ln(81) - 1 = 17.2 years. Quite close to Freeman's original assumption of 20 years.


The concept of Effective Age has far stretching implications. Some of these I have summarized below:
  • "Summer vacations lasted almost forever when I was in grammar school":
    True, they did. In fact, when you were six years old, an Apparent Year would be close to three years. That would make a three week summer vacation feel like almost nine weeks!
  • "Now that I am older, I can communicate better with my parents"
    Right. As you can see, you're catching up with them! Closing the "generation gap", so to speak.
  • "Life starts after 65"
    The credo of many people close to their pension age. Wrong: at 65, you only have about 5% of your Effective Age left. Choose your time wisely; start working late, and retire early.
  • "Old people are slow"
    That is such an insensitive comment. Old people aren't slow at all, they simply have a different time perception.
  • "Those annoying birthdays seem to roll around faster every year
    True, they do. Better start celebrating your Effective Age.

T. L. Freeman, Why it's later than you think, J. Irr. Res., 1983.