Application of the Limited Enthusiasm Model with Reversion, Written in 2022.

The Church of Scotland is a protestant Presbyterian church and is the established church in Scotland. It originated in the Reformation, following a very different path to its established neighbour – the Church of England. In common with most historic UK Christian denominations, it has steadily declined since 1960. This analysis investigates the likelihood of decline leading to denominational extinction.

Estimating Model Parameters

I will use the Church of Scotland’s membership data to estimate the parameters of the Limited Enthusiasm Model with Reversion. Using this cut-down version of the Demographics Model avoids estimating values for child retention among church members, hardening rates etc.

The reversion model has only five parameters:

  1. Reproduction Potential
  2. Duration Enthusiastic
  3. Loss Rate
  4. Fraction of Converts who Become Enthusiasts
  5. Initial Fraction of Enthusiasts in the Church

The key parameter is the reproduction potential. If this potential is less than one, then the Church of Scotland’s decline is to extinction.

Calibration Strategy

The best fit between data and model is found using least-squares. I was able to access data on the Church’s joining rates and loss rates (deaths and leaving) for most of the period. Thus, I set the loss rate (3) at the average value. I also used least-squares over the joining rates, determining realistic values of the initial fraction of enthusiasts (5). I optimised for the duration enthusiastic (2) only for a range of reproduction potentials (1) and the fraction of converts enthusiastic (4).

Membership 2000-2020

I took the membership data from figures published by the Church of Scotland.

Every optimisation indicates that the Church of Scotland will end up extinct.

One such optimisation is given in figure 1. The reproduction potential chosen was the more optimistic of the possible range, 0.74, still well under the extinction threshold of 1.

Figure 1: Data Fit for Church of Scotland Membership and the Limited Enthusiasm with Reversion Model

Range of Optimised Parameters

The optimisations indicate a range of possible parameter values:

Reproduction Potential0.54 – 0.73
Duration Enthusiastic6 – 9.5 years
Loss Rate
(Reversion and Deaths)
Fraction of Converts who Become EnthusiastsAbout 5%
Initial Fraction of Enthusiasts in the Church0.5% – 1%
Model Parameters for the Church of Scotland

Values of 1% or higher for the fraction of converts who become enthusiasts degraded the data fit. Likewise, for a fraction of the church who are enthusiasts.

My conclusion is that the Church of Scotland contains around 1000 enthusiasts, down from 3000 at the beginning of the century. They remain effective for a good interval of time, but few of their converts go on to make more converts. Consequently, the Church of Scotland cannot add sufficient people to make up for losses. Thus, extinction is inevitable.

Possible Extinction Date

Although the model predicts extinction, it does not indicate an extinction date. The model contains exponential decline because it does not capture the effects of ageing. By 2060 it forecasts a church of 70,000 members and 15,000 by the end of the century. These results are probably too optimistic.

Fitting a straight line to the data gives a more realistic estimate of the extinction date, as decline through ageing is a linear process. The Church of Scotland is estimated to be extinct by 2038, figure 2. A similar fit to the data from 1970 gives the same date.

Figure 2: Linear Fit to the Church of Scotland Membership Data

The church’s decline has slowed slightly in the last four years. Using the four most recent data points extends the extinction date to 2042. If the decline continues to slow, that is good news for the Church, but it will need several more years to be sure any slow down is sustained. The church’s future will likely lie somewhere between linear and exponential decline due to the presence of some city-based strong congregations.

The Church of Scotland’s decline pattern is similar to other historic denominations, such as the Church of England and the Scottish Episcopal Church.

Long–Term View

The membership of the Church of Scotland from 1840 to the present day shows the classic institutional lifecycle, figure 3[1]. In 1843 the church split into two groups, known as the “Disruption”. I have added their figures together until the various reunifications in 1900 and 1929 to keep continuity. The church grew steadily until the twentieth century. Viewing the church numbers as a percentage of the population indicates that the church’s growth was mainly in line with the population after 1880, figure 4. Church growth ended after the last national revival ended.

Figure 3: Membership of the Church of Scotland (and Constituents) 1840-2020.
Figure 4: Membership of the Church of Scotland (and Constituents) 1840-2020 as a Percentage of the Population.

After 1900, church numbers remained steady, with a slight boom in the 1950s. Some people believe this was a post-war baby boom effect; however, it is more pronounced in the Church of Scotland than in other denominations. Indeed, there was an increase in the church’s proportion in the late 50s, a possible sign of a renewal or revival movement.

Since the 1960s, the decline has been relentless.


  1. At the disruption, 1843, the church split into the Church of Scotland and the Free Church of Scotland. There was already a separate United Presbyterian Church resulting from earlier divisions. The United Presbyterians and most of the Free Church joined together in 1900 to form the United Free Church. Most of this denomination re-joined the Church of Scotland in 1929. Between 1843 and 1929, I used the combined numbers of these denominations

    In each reunion, a remnant did not join the national church. There is still an existing Free Church and United Free Church. I have not included the figures for these groups from 1929.