Growth model critique
Objective. Compare linear and exponential models against evidence.
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Learn: the key idea
Growth model critique compares whether data fits a linear model (constant increase) or an exponential model (constant percentage increase) by examining successive differences or ratios. If the differences between consecutive values stay roughly constant, a linear model y = mx + c fits well; if the ratios between consecutive values stay roughly constant, an exponential model y = ab^x fits better. For data 10, 20, 40, 80 the ratio is constantly 2, showing exponential growth rather than linear.
Worked example
A population is recorded as 100, 150, 225, 337.5 over four years. Critique whether a linear or exponential growth model fits better.
- 1Find successive differences: 50, 75, 112.5 — these are not constant, so linear growth is a poor fit
- 2Find successive ratios: 150/100 = 1.5, 225/150 = 1.5, 337.5/225 = 1.5
- 3Since the ratio is constant at 1.5, the data multiplies by 1.5 each year
- 4This constant ratio confirms an exponential model fits the growth
Answer: Exponential model fits best, with common ratio 1.5
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Start practisingCommon mistake. A common mistake is fitting a linear trend line to data that is actually growing exponentially, which badly underestimates future values. Always check whether differences or ratios are constant before choosing which model to fit.
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