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Introduction

This article is a follow-up to my article US Census Bureau  - Incorrect Use Of The Exponential Method which explores errors in the use of the exponential method by the US Census Bureau on the following two web pages:

As noted in my article on the US Census Bureau , the source of their blunder is:

The Methods and Materials of Demography, edited by Jacob Siegel and David Swanson, Second edition, 2004. Elsevier Academic Press.

Chapter 11, by Stephen G Perz, is entitled Population Change (pp. 253-263 inclusive). The formula and notes for exponential method that features on page 259 is of the form:

r = ln [ P(t+n) / P(t) ] / n

where:

• r = average annual growth rate (in percent) between midyear t and midyear t+1
• P(t) = population at midyear t
• ln is the natural logarithm
• n is the number of years of growth

This page includes an example of its use as follows:

 r = ln(146,825,475 / 118,002,706)/11

 r = ln (1.244255) / 11

 r = 0.218537 / 11

 r = 0.019867 or an effective rate of 1.99% per year

This example is wrong, as I will now prove.

A Natural Logarithm of a Number

As explained in my other articles on the Exponential Method, it has been common knowledge for over 200 years that any calculation involving Natural Logarithms requires the resulting Natural Logarithm of a number to be converted back to the actual number. In fact, this is noted between steps [11.6] and [11.7] (which relates to the Geometric Method) on page 258 of The Methods and Materials of Demography. Hence, the actual formula should be:

ln (r) = ln [ P(t+n) / P(t) ] / n

To convert a Natural Logarithm of a number r back to a number it is necessary to express the irrational number e to the power of ln (r) using the following rule:

r = e ln(r)

Correcting The Example

Correcting the example, we have:

 ln (r) = ln(146,825,475 / 118,002,706)/11

 ln (r) = ln (1.244255) / 11

 ln (r) = 0.218537 / 11

 ln (r) = 0.019867

However, we need to convert the Natural Logarithm of r back to real number:

 r = e ln(r) = e 0.019867 = 1.020066

All this actually tells us is that the population grew by a ratio of 1.020066. Given that we have derived a growth ratio (or Exponential Factor) and not a growth rate at all, how do we derive the growth rate R? Easy - by subtracting 1 from the growth ratio:

R = r - 1

So

R = 1.020066 - 1 = 0.020066

Converting to a percentage (* 100) that's an average effective growth rate of 2.01 % per annum.

The Proof

The best way to prove that my calculation of 2.01% per annum for 11 years is correct, and the authors' calculation of 1.99 % per annum for 11 years is incorrect, is to test both calculations. Assuming a starting population of 118,002,706, the prediction in each case is a target population of 146,825,475 after 11 years of growth:

 Year Starting Population Growth Rate % New Population 1980 118002706 1.99 120350960 1981 120350960 1.99 122745944 1982 122745944 1.99 125188588 1983 125188588 1.99 127679841 1984 127679841 1.99 130220670 1985 130220670 1.99 132812061 1986 132812061 1.99 135455021 1987 135455021 1.99 138150576 1988 138150576 1.99 140899773 1989 140899773 1.99 143703678 1990 143703678 1.99 146563381

Table A - Example Growth at 1.99 % per annum - Methods and Materials of Demography example

The Methods and Materials of Demography prediction fails. Their calculation results in a population of 146,563,381 people, not 146,825,475 people - this is a difference of 262,094 people.

 Year Starting Population Growth Rate % New Population 1980 118002706 2.01 120370508 1981 120370508 2.01 122785821 1982 122785821 2.01 125249600 1983 125249600 2.01 127762815 1984 127762815 2.01 130326460 1985 130326460 2.01 132941547 1986 132941547 2.01 135609106 1987 135609106 2.01 138330192 1988 138330192 2.01 141105879 1989 141105879 2.01 143937261 1990 143937261 2.01 146825457

Table B - Example Growth at 2.01 % per annum - Exponentialist correction

The Exponentialist prediction succeeds. My calculation results in a population of 146,825,457 people, not 146,825,475 people - this is a difference of only 18 people, which is entirely due to rounding of the growth rate.

Conclusion

I am reminded of Richard Dawkins' open letter advice to his daughter then aged 10 (A Prayer For My Daughter from The Devil's Chaplain, Dawkins, 2003). Dawkins explains three bad reasons for believing, which are tradition, authority and revelation. I was referred to The Methods and Materials of Demography by the US Census Bureau because, I believe, it is considered by them to be an authoritative source on the exponential method (even more authoritative than the US Census Bureau itself!).

Well, as proven above, citations from authority do not necessarily prove anything. My understanding of the exponential method results in a much more accurate calculation than the authoritative source.