US Census Bureau  - Incorrect Use Of The Exponential Method

"Freedom is the freedom to say that two plus two make four. If that is granted, all else follows." Winston Smith in 1984, by George Orwell (1949)

Note: This page is best printed in landscape and in colour.

Introduction

Take a look at the US Census Bureau's Total Midyear World Population for 1950 to 2050 AD. The compound growth rate varies pretty much every year, yet for each year you have an averaged compound growth rate that is effectively constant or fixed for the 12 months of each year. It looks very much like a classic example of variable rate compound interest.

For each row of data in the original US Census Bureau table, you have an apparently very simple calculation easily tested on any basic calculator. You will find that when the starting population is multiplied by the variable rate compound growth rate for each year, the resulting population change figures  do not tally with those published by the US Census Bureau. The US Census Bureau is in error. Why? What have they done wrong?

This article will examine the causes of the error and proposes that either the population change figures are listed incorrectly, or the growth rate figures are listed incorrectly. At this stage it is not necessary to resort to the exponential method, and basic decimal calculations will suffice.

The rest of the article will then reveal that it is the US Census Bureau's misuse of the exponential method in determining the growth rates which is the the cause of their miscalculations, and even exposes the source of the incorrect formula used by the US Census Bureau in their misuse of the exponential method.

Calculating actual population change

If the population and growth rate are correct, then the actual population change is calculated as follows:

Population * Average annual growth rate = Average annual population change

Calculating actual growth rates

If the population and population change are correct, then the actual growth rate is calculated as follows:

Average annual population change / Population = Average annual growth rate

Hence, applying the scientific method, each row of data in the US Census Bureau table is independently testable.

The US Census Bureau Figures (with two possible corrections)

Here are the US Census Bureau figures, with the two possible corrections added (in red):

Table 1: US Census Bureau figures compared with Exponentialist figures. For any given row, the starting population multiplied by the growth rate must result in the population change listed for that row. If the growth rates are correct, then the population change figures provided by the US Census Bureau are incorrect.

Source: US Census Bureau (figures in black, http://www.census.gov/ipc/www/idb/worldpop.html web site accessed 2nd December, 2008) and this Exponentialist web site (figures in red)

The Exponential Method

In 2005 I was advised by Peter D. Johnson of the International Programs Center (US Census Bureau) that:

"Demographers regularly use the exponential method to calculate the average annual growth rate because it works regardless of the time interval between the population figures:

r = 100 * 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 

The population change column is simply the difference in the population values:

PC(t to t+1) = P(t+1) - P(t) "

In short, the US Census Bureau records a mid-point estimate each year of world population, and uses the exponential method to "reverse engineer" the annual growth rate and annual population change for the previous year. So, to get the 1950 figures for Average annual growth rate (%) and Average annual population change, they need the figures for 1951. 

Mr Johnson argues that both sets of calculations are correct, and further states that:

"The difference between your growth rate and mine is like the difference between the APR and APY interest rates reported by banks."

Yet the Couttsian Growth figures are based on taking the starting population, then applying the same variable rates of compound interest (as given by the US Census Bureau) each year. The Couttsian Growth - Annual Increase is effectively the annual compound interest earned for the rate given for that year. Given that we are both using the same interest rates, this can have nothing to do with the difference between APR and APY interest. 

So Is This Variable Rate Compound Interest?

The problem, I suspected, was that the US Census Bureau did not seem to recognise that a variable rate compound interest model (what I call the Couttsian Growth Model) applies to populations. After all, nobody else does, so why should they?

I put it to the US Census Bureau:

"The only other explanation is that the US Census Bureau does not agree that population grows via variable rates of compound growth. Would you care to officially comment on that suggestion for my article?"

 Mr Johnson  replied:

"The results clearly show that the population is growing by variable rates of compound growth, as seen in the graph of the growth rates:

http://www.census.gov/ipc/www/img/worldgr.gif  "

So what's the issue, and who is right?

Different Perspectives

In fact, the issue is simple. It depends on whether you start knowing (or assuming) a growth rate and / or a population size. It depends on whether you have a census mentality, or a population modelling mentality. As I will prove, the issue is that the US Census Bureau have incorrectly calculated the growth rate for each year due to an error in their use of the exponential method.

Every year the Bureau takes a new estimated figure (mid-year global population) as an input. For example, in 1950 the estimated mid-year population is  2,555,948,654 and the growth rate is given as 1.47%, so the resulting annual population increase should always be 37,572,445 (the Couttsian Growth calculation) and not 37,803,324 (the US Census Bureau's calculation). However, the 1951 mid-year population estimate then comes in - 2,593,751,978 - and the US Census Bureau subsequently derives - via the Exponential Method - both the Average annual growth rate (%) and the Average annual population change for 1950 (the year before). This is quite a reasonable approach for a census mentality, driven by annual population estimates.

Using Couttsian Growth, a stated growth rate for a given year is applied to a starting population for that same year to get an annual population increase, and this is added to the total to get the starting figure for the next year. All you need is the starting population, and the growth rates, with no need for subsequent population measurements. This is a population modelling approach, driven by variable compound interest growth rates.

Incorrect Use Of Exponential Method by US Census Bureau

However, the annual growth rates quoted by the US Census Bureau are incorrect (in that they do not yield the correct annual population increase for each year when applied to the stated starting population for each year). 

The solution to obtaining the correct annual growth rates is as follows:

[1] Obtain the growth ratio of this year's population to this last year's population. Example. for 1951 and 1950 respectively:

2,593,751,978 / 2,555,948,654 = 1.01479033

Hence, the growth rate for mid-year 1950 to mid-year 1951 is (1.01479033 - 1) = 0.01479033 or slightly more than 1.47 %.

So what does the answer provided by the Exponential Method represent? The answer is that the Exponential Method used by the US Census Bureau yields a growth factor expressed as a Natural Logarithm, but not as a percentage growth figure. 

For example, Mr Johnson's MS Excel formula is equivalent to:

[2] r = 100 * ln [ P(t+n) / P(t) ] / n 

But it should be:

[2a] ln(r) = 100 * ln [ P(t+n) / P(t) ] / n 

[2b] Thus r = e ^ln(r)

In other words, to convert a Natural Logarithm back into a real number it needs to be expressed as an exponent of e, the base of the Natural Logarithms.

[2c] But r is the Exponential Factor, or growth ratio, not the actual growth rate. To get the growth rate (let's call it R):

R = r -1

[2d] To convert this growth rate into a percentage, multiply it by 100

[3a] For the years 1951 and 1950, the figures translate into this formula as follows:

Ln(r) = 100 * Ln(2,593,751,978 / 2,555,948,654)

Ln(r) = 100* Ln(1.01479033)

Where Ln(1.01479033) = 0.0146820197

Hence, Ln(r) = 100 * 0.0146820197

[3b] Alternately, take the Natural Logarithm of each number (for the 1951 population and 1950 population respectively) and then, in accordance with standard Exponential Rules, subtract as follows:

 Ln(r) = 100 * (21.6763713 - 21.66168928) = 100 * 0.0146820197

 [3c] Multiplying by 100, either way  (3a or 3b) we get 1.46820197 which is, don't forget, a Natural Logarithm (and needs to converted back to a proper number).

r (with a value of 1.46820197 %) was rounded in the US Census Bureau's MS Excel spreadsheet to 2 decimal places (=1.47 %).

Effectively, this is as far as the US Census Bureau got. They seem to have forgotten that a calculation involving two Natural Logarithms results in another Natural Logarithm, and not a real number at all.

[4] The US Census Bureau forgot the standard exponential rule that:

[4a] From 3a, if Ln(1.01479033) = 0.0146820197

[4b] Then e ^{Ln(1.01479033)   =} e ^{0.0146820197  =} 1.0147903299782

Hence r = 1.0147903299782 and not 1.46820197

This is a growth factor, and not a growth rate.

[4c] However from [2b], R=r-1, and so

R = 1.0147903299782 - 1 = 0.0147903299782

which is 1.47903299782 % when expressed as a percentage

[4d] Thus, the growth rate R is 1.47903299782 %, and not 1.46820197 %

The Same Exponential Method Error Repeated in US National Figures

In fact, a quick check of the US Census Bureau's Historical National Population Estimates: July 1, 1900 to July 1, 1999 reveals that the same error exists there too. 

[1]   272,690,813 / 270,248,003 = 1.009039141

Hence, the growth rate for mid-year 1998 to mid-year 1999 is 1.009039141 - 1 = 0.009039141.

[2] Ln(r) = 100 * ln [ P(t+n) / P(t) ] / n 

[3a]  Ln(r) = Ln (272,690,813 / 270,248,003 ) = Ln 1.009039141 = 0.00899853

[3b] Ln(r) = Ln (272,690,813) / Ln (270,248,003 ) =  19.42384915 - 19.41485062 = 0.00899853

Either way (3a or 3b), after multiplying by 100, we get 0.899853.

r (with a value of 0.899853) was probably rounded by the US Census Bureau again to 2 decimal places (=0.90 %).

Again, this figure is a Natural Logarithm not a real number.

[4] The US Census Bureau forgot the standard exponential rule that:

[4a] If Ln 1.009039141 = 0.00899853 

[4b] Then e ^{Ln (1.009039141)}  = e ^0.00899853 = 1.009039141

[4c] Thus, the growth rate is 1.009039141 - 1 = 0.9039141

That's 0.9039141 %, and not 0.899853  %

Again, to convert a Natural Logarithm back into a real number it needs to be expressed as an exponent of e, the base of the Natural Logarithms.

The Source of The US Census Bureau's Blunder

On 7th May 2008 I was advised by Peter D. Johnson that the US Census Bureau uses the following citation source for the formula published on their website:

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

 r = 100 * ln [ P(t+n) / P(t) ] / n

Chapter 11, by Stephen J Perz, is entitled Population Change (pp. 253-263 inclusive). The formula itself features on page 259, together with an example of its use. I explore this example further on The Methods and Materials of Demography - Incorrect Use Of Exponential Method and prove that the formula stated therein is incorrect.

The US Census Bureau Total Midyear World Population for 1950 to 2050 AD page assumes that n = 1 and thus they use a simplified version of this formula:

 r = 100 * ln [ P(t+n) / P(t) ]

As I explained to Mr Johnson, providing a citation does not prove anything except perhaps that the source of the citation  - no matter how weighty a tome - is also wrong.

Conclusion

The US Census Bureau almost did the Exponential Method calculation correctly, but then incorrectly presented a Natural Logarithm as a percentage (rounded to two decimal places) for each row. They forgot to convert the Natural Logarithm back to a real number, the Exponential Factor. Then subtracting 1 from the Exponential Factor results in the true growth rate for the year.

The Exponential Method used (as follows) by the US Census Bureau is wrong:

 r = 100 * ln [ P(t+n) / P(t) ]

It should read:

ln(r) = 100 * ln [ P(t+n) / P(t) ]

Thus r = e ^ln(r)

But r is the Exponential Factor, or growth ratio. Then it is necessary to derive the growth rate (R):

R = r -1

The easy compound interest calculations presented by the US Census Bureau in Table 1 are all wrong. They are wrong because the US Census Bureau did not test them, and yet present them as fact. They did not test these easy compound interest calculations because they do not appear to see variable compound interest resulting in a growth model in its own right (the Couttsian Growth Model), even though they recognise that the compound growth rate varies from year to year.

Each row of data in Table 1 is a mathematical statement of fact, and is thus subject to scrutiny via the scientific method. If the test fails, then the statement of fact is incorrect. Hence, as soon as the US Census tested these statements they would have seen the discrepancies revealed (between the black and red figures in Table 1), and they would have discovered their error in the usage of the Exponential Method.

On several occasions I have tried to explain the Couttsian Growth Model to scientists and mathematicians. Most don't seem to get it, arguing that this sort of growth can result in any kind of growth at all (which they then incorrectly call "growth"). Exactly! That is the whole point, to provide a self-consistent growth model that applies universally to all populations of all species for all time. 

Any combination of variable positive and negative rates can be converted to Natural Logarithms and added together to get the Exponential Factor  that can then be applied to any sized population of any species for whatever time period applies. All populations of all species grow and shrink in the same way, via variable rate compound interest (see the Couttsian Growth Model, and  The Scales of e).