the Church of Jesus Christ of Latter-day Saints

The cover of the November issue of *US News & World Report* announces an article about the Church of Jesus Christ of Latter-day Saints, popularly known as the Mormons. One of the article’s main points is the remarkable growth of the Utah-based religion, and the interesting observation that the emergence of Mormonism represents the first world-wide invention of a new religion since the prophet Mohamed gave the world Islam. Among the article’s more remarkable statements is a prediction about the growth in Church membership. It says: "If current trends hold, experts say Latter-day Saints could number 265 million worldwide by 2080, second only to Roman Catholics among Christian bodies. Mormonism, says Rodney Stark, professor of sociology and religion at the University of Washington, "stands on the threshold of becoming the first major faith to appear on Earth since the prophet Mohammed rode out of the desert."

This is a truly remarkable prediction. To put it in perspective, it says that worldwide Church membership, at about 11 million members today (2000), will grow to almost the population of the United States within a lifetime. These projections are sure to fuel the imaginations and testimonies of Latter-day Saints, but the real question is, where did this projection come from, and is it accurate?

A common method for making predictions about the growth of populations is to fit the measured data to a mathematical function and (if the fit is good enough) calculate the value of the function at some point in the future. This sounds simple enough, but like many things in life the work is in the details. For example, how might we decide which mathematical function best models the data? How do we know that a function that does a good job of modeling the data over the period when the data was collected will accurately predict future trends? Even if we find the right sort of function, how should we adjust the function’s coefficients to optimize the fit between the function and the measured data?

For any curve fitting method to work properly, each of these questions must be addressed and dealt with in a quantitative and logical manner. For some of the questions, the branch of mathematics we call statistics has done most of the work for us. We simply need to make prudent use of it. For some of the other questions, however, we are left with considerable leeway, and must make use of experience and a little intuition. As we shall see, this is the origin of much of the uncertainty in these sorts of projections.

Let’s start with the first question. How might we decide which mathematical function best models the data? For this question we draw on a long history of mathematical functions that model things as diverse as the growth in human populations to nuclear decay. The simplest of these simply states that the growth (or decay) in a quantity is directly proportional to the amount of the quantity. Take a population of rabbits, for example. On average, the number of new bunnies in a given period of time depends on the number of females (assuming there are enough bucks, of course). The same proportionality holds for the decay of nuclear elements. The number of radioactive particles emitted in a given time depends directly on the number of radioactive atoms that emit those particles. In the language of mathematics we express this growth formula as a differential equation:

In equation [1], a is the growth constant, N is the amount of the quantity under consideration, dN is the change in the quantity, and dt a small unit of time over which the change, dN, occurs. For example, at a given time we might have N = 20 rabbits, dN = 10 new bunnies, and dt = 3 months. You can work through equation [1] numerically, or (using the mathematical tools of calculus) you can solve equation [1] analytically. The solution to this differential equation is:

In equation [2], *e* is a special number that shows up all over the place in mathematical equations, physics, chemistry, and engineering. It’s a transcendental number, like p. The value of e is approximately 2.718. N_{0 }is the value of the quantity at time t_{0}.

One of the first problems we see with equation [2] is that the quantity N increases without limit. This is a problem because in real life things don’t increase without bound. Something inevitably steps in to stop the growth process. In the case of the rabbits it might be the farmer who soon realizes that he’s about to be either up to his ears in bunnies or spending a fortune on feed. He may choose to limit the population of rabbits to maximize his income within the physical constraints of the size of his farm.

Figure 1. Church membership by year. In this figure the red curve consists simply of line segments connecting the dots, which are the measured values for Church membership found on the official Internet site of the LDS Church.

You might think that Church growth would be governed by equation [2]. In fact, if you look at figure 1, which illustrates the Church’s growth through the year 2,000, you might think that it looks characteristically like an exponential curve. There are some good reasons to expect exponential growth. After all, the LDS Church is a missionary church, and we expect that the number of missionaries – and hence converts – should be roughly proportional to church population. To take an example, we expect that if the Church membership is 3 million and there are 30,000 missionaries who baptize 250,000 converts each year, then when the Church membership hits 6 million there should be 60,000 missionaries who baptize 500,000 converts each year. This is classical exponential growth.

As pointed out above, however, exponential growth cannot continue forever. In the case of Church membership, eventually everyone would join, and growth (at least from convert baptisms) would stop. In fact, it’s probably not realistic to assume that everyone will join. So there will be an upper limit to Church membership that is somewhere lower than the worldwide population. The real question is what is that limit?

A more accurate equation for describing the growth of real populations is a modification of equation [1], and is called the Verhulst or logistic equation. In its differential form, the logistic equation says that in the early stages of growth, when resources are essentially unlimited in respect to the small size of the growing population, the population increases in an almost exponential fashion. As the population increases it begins to slow down and approaches a number that represents the steady state or equilibrium carrying capacity. The differential form for the logistic equation is:

In equation [3], N is the number or quantity of the thing being considered (members of the Church, for example), a is the growth coefficient, and N_{c }is the carrying capacity. When the value of N is small, the growth increases almost exponentially, as in equation [2]. As the value of N gets nearer to the carrying capacity, however, the growth rate slows dramatically as the population approaches (but does not exceed) the carrying capacity. The solution to equation [3] is:

In equation [4] N_{0 }is the value of the quantity at time t = t_{0}.

There are a number of other growth equations that are similar to equation [4], such as the *arctangent-exponential* equation. However, these all provide similar results for predicting the future value of growing populations. In the analysis I shall present in this article, I’ll be using the logistic equation.

This sort of answers the question we asked earlier: "how might we decide which mathematical function best models the data?" The answer in this case is that there is a certain amount of evidence in the literature suggesting that the logistic equation is a good match for modeling the growth of some populations. I’ve presented some of the reasons that the logistic equation works, but it’s good to remember that this is still a rather simplistic approach and that there are many complicating factors that could throw off the extrapolated predictions. For example, a part of the world not previously open to Mormon missionaries might suddenly become available. This would offer a previously unavailable source of converts. We’d expect to see the carrying capacity increased, and that would put a kink in our curve fit. Technology might also play a role. The Internet, for example, might provide a mechanism for dispersing critical information about the Church to a worldwide audience, making it more difficult for Mormon missionaries to win over converts. By applying the logistic equation, we are assuming that the average forces for and against Church growth will remain stable over the period of extrapolation.

Recall the other questions: "How do we know that a function that models the data well over the period when the data was collected will model it well into the future?" The answer to this question is, quite frankly, "we don’t know." But there is good reason to expect that if the mathematical model accurately fits the data we have in hand, it will also make reasonable predictions for the future – at least if we don’t try to project too far. To test our fit, therefore, we shall compare the analytical prediction with the measured population statistics. If we find a good fit we can expect a relatively accurate prediction for future growth.

The third question was: "Even if we find the right sort of function, how should we adjust the function’s coefficients to optimize the fit between the function and the measured data?" This turns out to be the easiest question to answer. As luck would have it, there are precise mathematical tools that allow us to do just that, and those tools guarantee that the coefficients we calculate will give the absolute best possible match (for the function we choose) to the measured data.

The analysis used in this paper uses what’s called the "least-squares" method to fit equation [4] to the Church-growth data shown in figure 1. The results of this method are the numerical values of N_{c,} N_{0,} and a (these are the function’s coefficients) that give the best possible match between the measured population data and equation [4].

The least-squares method is a formal mathematical approach that finds the set of coefficients that minimize the error between the measured data and the analytical equation. The error is defined as the average of the squared values of the differences between the measured data and the values predicted by the equation. That sounds like a mouthful, but don’t let it scare you. Think of it this way. Imagine looking at a figure showing points that represent the measured data, and a line that represents the analytical function (take a peak ahead at figure 2). You would naturally judge how well the line fits the points by how close it comes to each of them. If the line missed most of the points, especially by a large margin, you’d be inclined to think that the line was a poor fit to the measured data. On the other hand, if the line hits each of the points you’d be inclined to think that the function describing the line also describes the measured data pretty well. That’s what we’re talking about here. First we take the difference between the line (the function) and the data points. Then we square those differences. That’s because we are interested in absolute differences – so we don’t want to be confused by positive and negative values. Finally, we take the average of the squared differences, and then we undo the squaring by taking the square root of the average. What we end up with is a quantity that statisticians call the standard deviation. Mathematically, the standard deviation is given by:

In equation [5], n is the number of measurements made, m_{i }is the i’th measurement (made at time t_{i}) and N(t_{i}) is the value predicted by equation [4] at the i’th time of measurement.

Before simply charging ahead and cranking the numbers on all these equations, it’s useful to stand back from the problem for a moment and digest what we are doing here. First, we have some limited data on Church membership. Second, we have a formula that we think might correctly describe how Church membership will continue to grow in the future – if we knew the coefficients for this formula. Third, we hope to determine the coefficients for the formula by fitting the curve to the measured data and adjusting the coefficients until we minimize the error between the theoretical curve and the measured data set. Finally, after we do all this, we expect to project our findings many years into the future to estimate the number of people in the Church 20, 40, or 80 years away.

You might feel discouraged by looking over all these assumptions. It might seem that the results we obtain in this fashion are totally unreliable. Indeed, this is a valid concern. Part of the reason for showing the mechanics behind the calculation is to give a feel for the degree of ambiguity involved in the process (you should be feeling skeptical about that 265-million figure by now). Still, the method is accepted and has a good track record of accurately predicting population growth in many situations. The further into the future that you try to project, however, the less accurate the projection becomes. With that in mind, let’s look at what the analysis predicts.

If we fit equation [4] to the data shown in figure 1 we obtain the following results for the coefficients, using the method of least squares

- a = 0.056842
- N
_{c }= 30.3 million - N
_{0 }= 2,505

You might convince yourself that these coefficients really do give the best fit to the measured data in figure 1 by trying some different coefficients and seeing if you can get a smaller value for the standard deviation (see equation [5]).

The coefficient N_{c }is called the carrying factor. It is the maximum value attained by the Church membership. In this case, you can see that the best-fit analysis predicts an ultimate stable Church membership of slightly over 30 million members. The coefficient N_{0 }is the membership in 1844 (for this analysis I used t_{0} = 1844). Figure 2 illustrates the measured Church-membership data (the red dots) and the logistic curve obtained by using the best-fit coefficients in equation [4], extrapolated out to the year 2100 (the blue curve). In the year 2080 this analysis predicts that Church membership will be about 29.8 million people – almost ten times smaller than the value prediction in the article in *US News & World Report*.

Figure 2. Projected Church membership. This figure shows the measured Church membership (red dots) and the projected Church membership determined by the least-squares method applied to equation [4]. This analysis predicts an ultimate stable Church membership of slightly over 30 million members, with about 29.8 million members in 2080.

There are always questions about statistical extrapolation, and the process is greatly susceptible to errors due to the fact that even small random fluctuations in the present can have dramatic effects on extrapolated numbers in the future. To get a flavor for how stable our prediction is, let’s look at the membership numbers we would have predicted for 2080, had we used only the data available in previous years. Clearly, if the analysis is reliable, we should expect to calculate the same carrying capacity if we use, for example, only the data between 1844 and 1996. After all, if the data between 1844 and 1996 gives a different carrying capacity than when we use all the data, right through to the year 2000, how do we know that there is not a different carrying capacity waiting for us when we do the analysis next year? Or the year after that? By comparing the carrying capacity that we would have calculated in previous years we can get an idea of how the carrying capacity might change in future calculations. And that should help us appreciate the reliability or our estimates. Table 1 shows the results.

The most obvious thing about table 1 is the way in which the predicted carrying capacity drops as we include more and more of the membership data. Had we done this analysis in the 1980s we would have predicted a carrying capacity in excess of the planet’s population – in the hundreds of billions. As more and more membership data is included in the analysis the estimated carrying capacity drops dramatically to relatively stable (but still declining) values in the tens of millions. This illustrates the difficulty in making predictions in the far-distant future based on a single analysis of membership data. The other notable thing about table 1 is the fact that, although gradually decreasing, the predicted carrying capacity has remained relatively stable based on data from the last few years. Table 1 also shows that actual membership in the year 2000 is below what would have been predicted by all previous least-squared approximations using equation [4] in prior years. In some cases the difference is quite significant. For example, using only membership data that includes 1991 and earlier years, we would have predicted that in the year 2000, Church membership would be nearly 12.5 million – or almost 1.5 million more than the current membership. We can anticipate, based on these observations, that the curve in figure 2 is similarly optimistic.

Year through which the analysis is performed |
Predicted carrying capacity (millions) |
Predicted membership in 2080 (millions) |
Predicted membership in 2000 (millions) |
Average error (in percent) |

2000 |
30.3 |
29.8 |
11.20 |
6.6 |

1999 |
33.2 |
32.5 |
11.26 |
6.9 |

1998 |
37.3 |
36.2 |
11.34 |
7.1 |

1997 |
47.1 |
45.0 |
11.48 |
7.2 |

1996 |
59.2 |
55.5 |
11.60 |
7.5 |

1995 |
94.3 |
83.5 |
11.80 |
7.6 |

1994 |
376 |
230 |
12.10 |
7.9 |

1991 |
207,000 |
622 |
12.47 |
9.8 |

1989 |
333,000 |
578 |
12.27 |
10.3 |

1986 |
38,300 |
488 |
11.84 |
10.1 |

1982 |
45,400 |
491 |
11.86 |
11.8 |

Table 1. Projections based on earlier subsets of membership data. Suppose we had estimated future Church membership in earlier years. What might our predictions have been? As you can see, analysis performed in earlier years result in much greater estimates for Church membership in 2080, and in the ultimate carrying capacity. Earlier estimates also exaggerate Church membership in the year 2000.

Before leaving table 1, consider one further important bit of information. The last column shows the average error (in percent) between the measured membership data and the best-fit logistic equation using the limited data for different years. Notice that, as we include more and more of the membership data the average error decreases. This is encouraging, and along with the fact that the carrying capacity also seems to become relatively stable with more recent data, suggests that the future population estimates made on data through the year 2,000 are considerably more reliable than analysis performed on data up through earlier years. We can also anticipate that when the actual data becomes available in future years we will be able to make even more accurate predictions. This also leads us to suspect that analysis in earlier years is likely to overestimate projected Church membership.

Of course, no amount of mathematical analysis can (or ever should) be a substitute for real data. So, to see how accurate this attempt has been, keep it around and compare it with future growth as actual numbers roll in. Table 2 has the projected Church-membership data for each year through 2005, and in five-year increments from 2005 through 2080. If the Church is on target for meeting it’s eventual equilibrium membership of just over 30 million members, membership should be pretty close to these numbers in each of the listed years. If Church membership falls significantly below these numbers, look for Church membership to stabilize below 30 million. If they greatly exceed these numbers, look for membership to stabilize well above 30 million.

Year |
Projected Church Membership |

2000 |
11.203 |

2001 |
11.607 |

2002 |
12.016 |

2003 |
12.431 |

2004 |
12.849 |

2005 |
13.271 |

2010 |
15.414 |

2015 |
17.546 |

2020 |
19.584 |

2025 |
21.460 |

2030 |
23.128 |

2035 |
24.565 |

2040 |
25.770 |

2045 |
26.757 |

2050 |
27.552 |

2055 |
28.182 |

2060 |
28.676 |

2065 |
29.059 |

2070 |
29.354 |

2075 |
29.580 |

2080 |
29.752 |

So what have we learned about the projected 265-million Mormons that are predicted for the year 2080 in the article in *US News & World Report*? Well, scan through table 1 one last time. Notice that one of the predictions – the one based on membership data prior to 1995 – projects 230 million members in 2080. This is suspiciously close to the 265-million-member figure, which leads to the conjecture that the projected membership date in *US News* was based on a least-squares analysis of earlier data. One thing should be clear by now, however, the most current data does not support a population in 2080 that is anywhere near 265 million. But who knows what tomorrow’s calculations will tell?

Appendix

Mathcad document used to analyze Church Membership

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