- 1960 15.7 mil
- 1970 19.9
- 1980 23.7
- 1981 24.2
- 1982 24.8
- 1983 25.3
- 1984 25.8
- 1985 26.4
- 1986 27.1
- 1987 27.8
- 1988 28.5
- 1989 29.2
- 1990 29.9
- 1991 30.4
- 1992 30.9
- 1993 31.2
If we assume the growth is linear (constant number increase) and we use the regression tool, we get this as the fit:

If we assume the growth is exponential (constant percentage increase) and we use the exponential tool we get this as the fit:

Formally, the exponential fit, at 2.2% growth rate per year is a better fit than the linear fits although both representations look like they do a pretty good job.

Why is this?

Mathematically:

e

^{x}= 1 + x + (x^{2})/2 + (x^{3})/6 + (x^{4})/24 + ...so for small x, meaning for small growth rates, an exponential fit is like a linear fit because the terms in the above expansion which are of higher order than x

^{2}are very small so thate

^{x}~ 1 + xIf you don't want to deal with the mathematical explanation of this, you can think about it this way.

2.2% growth rate is a doubling time of 32 years. This is the same as the time period of the data. Thus, data that encompasses less than one doubling time will generally not reveal the exponential character of the growth pattern leading to ambiguity. This is a again where you need to use your brain.

Suppose you are the California Resource planner and you are charged with forming a comprehensive resource allocation system for California residents in the year 2025 and beyond.

If you use the linear fit to the population data you would get the following estimate for the population in the year 2025:

The linear fit is y = 0.48x - 933

y = population in millions; x = calendar year

Hence in the year 2025 your estimate would be:

y = 0.48(2025) - 933 = 39 million people

If instead, the exponential is the best fit then the estimate would be much higher:

At 2.2% growth rate the doubling time is 70/2.2 = 32 years; In 1993 the population was 31.4 million. In 2025, or 32 years later, the population will be 63 million --> considerably larger than 39 million!

Hence, you make a big mistake if you don't bother with fitting data with a possible exponential growth rate. As its difficult to establish early on if a resource use is happening exponentially there can be serious resource projection shortfalls. In other words, you have to use your brain to determine if a linear or exponential fit is the best way to describe the growth of the data.

Example of getting the doubling time from the data.

- Population in 1980 = 23.7 million
- Population in 1993 (14 years later) = 31.2 million
- ratio of final to initial population is 31.2/23.7 = 1.32
- Natural Log of 1.32 (ln 1.32) = 0.28
- 0.28/14 = 0.02 2% annual growth rate