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US Census Bureau  - Incorrect Use Of The Exponential Method
Methods and Materials of Demography  - Incorrect Use Of The Exponential Method

External Links:
Total Midyear World Population for 1950 to 2050 AD - US Census Bureau

Historical National Population Estimates: July 1, 1900 to July 1, 1999 - Population Estimates Program, Population Division, U.S. Census Bureau

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 Annual growth rate are correct, then the Actual population change is calculated as follows:

Population * Annual growth rate = Annual population change

Here's a few examples, according to US Census Bureau calculations:

1950: 2,555,948,654 * 1.47% = 37,803,324

Yet if the population and growth rate are correct, then the actual population change should be 37,572,445.

1951: 2,593,751,978 * 1.61% = 42,078,979

Yet if the population and growth rate are correct, then the actual population change should be 41,759,407.

1952: 2,635,830,957 * 1.71% = 45,357,074

Yet if the population and growth rate are correct, then the actual population change should be 45,072,709.

Calculating actual growth rates

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

Annual population change / Population = Annual growth rate

Here's a few examples, according to US Census Bureau calculations:

1950: 37,803,324  / 2,555,948,654  = 1.47%

Yet if the population and population change are correct, then the actual growth rate should be 1.48%

1951: 42,078,979 / 2,593,751,978  = 1.61%

Yet if the population and population change are correct, then the actual growth rate should be 1.62%.

1952: 45,357,074 / 2,635,830,957  = 1.71%

Yet if the population and population change are correct, then the actual growth rate should be 1.72%.

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):

Year Population Annual growth rate (%) Annual population change

 Two Possible Corrections

 
Actual population change  Actual growth rate (%) 
1950 2,555,948,654 1.47 37,803,324 37,572,445 1.48
1951 2,593,751,978 1.61 42,078,979 41,759,407 1.62
1952 2,635,830,957 1.71 45,357,074 45,072,709 1.72
1953 2,681,188,031 1.78 48,108,819 47,725,147 1.79
1954 2,729,296,850 1.87 51,610,647 51,037,851 1.89
1955 2,780,907,497 1.89 53,135,393 52,559,152 1.91
1956 2,834,042,890 1.96 56,014,775 55,547,241 1.98
1957 2,890,057,665 1.94 56,704,198 56,067,119 1.96
1958 2,946,761,863 1.77 52,541,912 52,157,685 1.78
1959 2,999,303,775 1.4 42,289,638 41,990,253 1.41
     
1960 3,041,593,413 1.34 40,994,462 40,757,352 1.35
1961 3,082,587,875 1.81 56,217,751 55,794,841 1.82
1962 3,138,805,626 2.20 69,752,139 69,053,724 2.22
1963 3,208,557,765 2.20 71,363,768 70,588,271 2.22
1964 3,279,921,533 2.09 69,235,562 68,550,360 2.11
1965 3,349,157,095 2.08 70,440,849 69,662,468 2.10
1966 3,419,597,944 2.02 69,941,891 69,075,878 2.05
1967 3,489,539,835 2.04 72,044,250 71,186,613 2.06
1968 3,561,584,085 2.08 74,830,927 74,080,949 2.10
1969 3,636,415,012 2.05 75,385,774 74,546,508 2.07
     
1970 3,711,800,786 2.07 77,658,331 76,834,276 2.09
1971 3,789,459,117 2.00 76,404,806 75,789,182 2.02
1972 3,865,863,923 1.94 75,861,208 74,997,760 1.96
1973 3,941,725,131 1.88 74,659,609 74,104,432 1.89
1974 4,016,384,740 1.79 72,642,033 71,893,287 1.81
1975 4,089,026,773 1.73 71,156,071 70,740,163 1.74
1976 4,160,182,844 1.71 71,699,274 71,139,127 1.72
1977 4,231,882,118 1.68 71,532,674 71,095,620 1.69
1978 4,303,414,792 1.71 74,259,028 73,588,393 1.73
1979 4,377,673,820 1.70 75,092,516 74,420,455 1.72
     
1980 4,452,766,336 1.70 76,202,670 75,697,028 1.71
1981 4,528,969,006 1.75 80,111,220 79,256,958 1.77
1982 4,609,080,226 1.76 81,842,485 81,119,812 1.78
1983 4,690,922,711 1.70 80,203,354 79,745,686 1.71
1984 4,771,126,065 1.69 81,448,402 80,632,030 1.71
1985 4,852,574,467 1.70 83,411,880 82,493,766 1.72
1986 4,935,986,347 1.73 85,973,518 85,392,564 1.74
1987 5,021,959,865 1.71 86,532,357 85,875,514 1.72
1988 5,108,492,222 1.68 86,380,841 85,822,669 1.69
1989 5,194,873,063 1.67 87,498,865 86,754,380 1.68
     
1990 5,282,371,928 1.57 83,336,869 82,933,239 1.58
1991 5,365,708,797 1.54 83,016,725 82,631,915 1.55
1992 5,448,725,522 1.48 81,261,903 80,641,138 1.49
1993 5,529,987,425 1.44 80,078,038 79,631,819 1.45
1994 5,610,065,463 1.43 80,916,563 80,223,936 1.44
1995 5,690,982,026 1.40 80,378,955 79,673,748 1.41
1996 5,771,360,981 1.37 79,461,215 79,067,645 1.38
1997 5,850,822,196 1.34 78,857,387 78,401,017 1.35
1998 5,929,679,583 1.30 77,850,002 77,085,835 1.31
1999 6,007,529,585 1.28 77,378,011 76,896,379 1.29
     
2000 6,084,907,596 1.26 77,372,980 76,669,836 1.27
2001 6,162,280,576 1.23 76,539,263 75,796,051 1.24
2002 6,238,819,839 1.22 76,421,588 76,113,602 1.22
2003 6,315,241,427 1.21 77,163,935 76,414,421 1.22
2004 6,392,405,362 1.21 77,935,074 77,348,105 1.22
2005 6,470,340,436 1.20 78,356,539 77,644,085 1.21
2006 6,548,696,975 1.20 78,852,010 78,584,364 1.20
2007 6,627,548,985 1.19 79,443,947 78,867,833 1.20
2008 6,706,992,932 1.18 79,751,007 79,142,517 1.19
2009 6,786,743,939 1.17 80,136,492 79,404,904 1.18
     
2010 6,866,880,431 1.17 80,635,963 80,342,501 1.17
2011 6,947,516,394 1.16 80,852,608 80,591,190 1.16
2012 7,028,369,002 1.14 80,780,430 80,123,407 1.15
2013 7,109,149,432 1.13 80,453,777 80,333,389 1.13
2014 7,189,603,209 1.11 79,923,047 79,804,596 1.11
2015 7,269,526,256 1.09 79,341,004 79,237,836 1.09
2016 7,348,867,260 1.07 78,752,715 78,632,880 1.07
2017 7,427,619,975 1.05 78,059,143 77,990,010 1.05
2018 7,505,679,118 1.02 77,255,840 76,557,927 1.03
2019 7,582,934,958 1.00 76,356,995 75,829,350 1.01
     
2020 7,659,291,953 0.98 75,481,209 75,061,061 0.99
2021 7,734,773,162 0.96 74,624,698 74,253,822 0.96
2022 7,809,397,860 0.94 73,683,056 73,408,340 0.94
2023 7,883,080,916 0.92 72,700,814 72,524,344 0.92
2024 7,955,781,730 0.90 71,708,461 71,602,036 0.90
2025 8,027,490,191 0.88 70,795,897 70,641,914 0.88
2026 8,098,286,088 0.86 69,968,084 69,645,260 0.86
2027 8,168,254,172 0.84 69,135,402 68,613,335 0.85
2028 8,237,389,574 0.83 68,296,484 68,370,333 0.83
2029 8,305,686,058 0.81 67,447,921 67,276,057 0.81
     
2030 8,373,133,979 0.79 66,641,869 66,147,758 0.80
2031 8,439,775,848 0.78 65,890,502 65,830,252 0.78
2032 8,505,666,350 0.76 65,125,109 64,643,064 0.77
2033 8,570,791,459 0.75 64,327,024 64,280,936 0.75
2034 8,635,118,483 0.73 63,484,706 63,036,365 0.74
2035 8,698,603,189 0.72 62,644,688 62,629,943 0.72
2036 8,761,247,877 0.70 61,821,348 61,328,735 0.71
2037 8,823,069,225 0.69 60,965,223 60,879,178 0.69
2038 8,884,034,448 0.67 60,069,171 59,523,031 0.68
2039 8,944,103,619 0.66 59,119,160 59,031,084 0.66
     
2040 9,003,222,779 0.64 58,180,174 57,620,626 0.65
2041 9,061,402,953 0.63 57,260,961 57,086,839 0.63
2042 9,118,663,914 0.62 56,290,226 56,535,716 0.62
2043 9,174,954,140 0.60 55,268,230 55,049,725 0.60
2044 9,230,222,370 0.59 54,200,541 54,458,312 0.59
2045 9,284,422,911 0.57 53,121,563 52,921,211 0.57
2046 9,337,544,474 0.56 52,037,683 52,290,249 0.56
2047 9,389,582,157 0.54 50,928,535 50,703,744 0.54
2048 9,440,510,692 0.53 49,804,646 50,034,707 0.53
2049 9,490,315,338 0.51 48,672,925 48,400,608 0.51
     
2050 9,538,988,263        

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/worldpoptotal.php 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:

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 Annual growth rate (%) and Annual population change, they need the figures for 1951. 

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 Annual growth rate (%) and the 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.

Annual Percentage Yield and Continuous Compounding

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 in comparing my Actual Population Growth figures with the US Census Bureau's Annual Population Change figures (see Table 1 above) note that we are both taking the same starting population, then applying the same variable rate of compound interest (as provided by the US Census Bureau) for each year. Hence, if we both assume that these interest rates are correct, and we are both using these same interest rates, then the discrepancies between my Annual Population Change figures and the US Census Bureau's Annual Population Change figures can have nothing to do with the difference between APR and APY interest. 

Alternately, if we assume that the Annual Population Change figures are accurate, then it is not possible to obtain these figures unless we correct the Annual Growth Rate for each year.

As explained in my article Calculating APY and Continuous Compounding, for any given nominal interest rate the effective interest rate (also known as Annual Percentage Yield) increases as the number of compounding periods per year increases. When the number of compounding periods per year is one then the effective interest rate is identical to the nominal interest rate. When the number of compounding periods is infinite then you have continuous compounding, and the effective interest rate is maximised for the same nominal interest rate.

The formula used by the US Census Bureau effectively assumes an infinite number of compounding periods per year and therefore obtains the maximum effective interest rate for a given nominal interest rate. This is all very clever, but entirely irrelevant, as the target Annual Population Change displayed for each year can only be achieved by assuming just one compounding period per year and applying an effective interest rate equal to the nominal interest rate. Sure, the US Census Bureau calculations of the Annual Growth Rate are correct but that does not mean that their Annual Growth Rate figures are of any use in calculating the Annual Population Change for each row. In other words, when it comes to the US Census Bureau's Annual Growth Rate, they correctly calculate the wrong thing.

A key issue with the data as presented by the US Census Bureau therefore is that it is not easy for the average person to understand. There is no mention of terms such as nominal interest rate, effective interest rate (or annual percentage yield), compounding periods or even continuous compounding. Most people who agree to various financial contracts (mortgages, pension schemes, shares and investments and so on) would expect to see such terms actually used if they are relevant. Here, the US Census Bureau prefers to obscure our understanding of the data through omission rather than clarify our understanding of the data through the use of appropriate terminology.

I believe most people will read the table of data as presented by the US Census Bureau as follows:

Population * Annual growth rate = Annual population change

Most people will therefore be confused by the data, as this simple calculation fails for every row. The language used by the US Census is deceptively plain - annual growth rate - but who would have thought that the data displayed under this column is an effective interest rate (for an unspecified nominal interest rate) that (irrelevantly) assumes an infinite number of compounding periods and hence continuous compounding?

Is this mathematical arrogance on the part of the US Census Bureau, just plain laziness or sloppiness? You decide.

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 of a number back into the actual 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 the actual 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 the target 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. 

Date Population  Annual Increase Growth Rate
July 1, 1998 270,248,003   2,464,396  
July 1, 1999 272,690,813  2,442,810  0.90 

[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 of a growth rate and not the growth rate itself.

[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 of a number back into the actual 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 of a percentage growth rate as a percentage (rounded to two decimal places) for each row. In other words, they failed to convert the Natural Logarithm of a number to a the actual number, in this case 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).

Addendum (9th May 2011)

I recently noticed that the US Census Bureau have (again) changed the URL of their page Total Midyear World Population for 1950 to 2050 AD. Looking more carefully, I noticed that they have now finally conceded defeat and amended their annual growth rate figures to the correct figures. It has taken since mid 2005 to persuade them to do so! Sadly, they still claim to use the formula  r = 100 * ln [ P(t+n) / P(t) ] to derive their growth rates. However, if you try to use this formula with their data you will not derive the growth rates listed by the US Census Bureau. Hence, this US Census Bureau page is still in error. I have just submitted feedback online, asking them to correct the formula as per my article (above).

Addendum (24th May 2011)

It's over, at last! Without ever once acknowledging that they were wrong the US Census Bureau have capitulated, and have now amended their article to show the formula they use as r(t) = [ ( P(t+1) - P(t) ) / P(t) ] * 100. In other words, they've abandoned use of the exponential method (and logarithms) altogether. I can't believe how obstinate they have been, refusing for so many years to accept that they made a simple mistake. In due course I will amend this article to better reflect the fact that they are no longer wrong, but were wrong historically.