Math · Introductory statistics · Worked example
Fit an exponential model to growth data
A culture has 100, 205, 390, 810 and 1600 cells per microliter at t = 0, 1, 2, 3 and 4 hours. Fit an exponential model and find the doubling time.
See the curve
The counts roughly double each hour: the ratios are 2.05, 1.90, 2.08 and 1.98. A nearly constant ratio per step means exponential growth, so a straight line would leave curved residuals.
Fit the model
Least squares on the original scale gives y ≈ 100.494e^(0.692204t), with R² ≈ 0.99985. A straight-line fit to ln y gives a growth rate of 0.692 per hour as well, to three decimal places.
Find the doubling time
The count doubles when e^(0.692204t) = 2, so t = ln 2 ÷ 0.692204.
Result
y ≈ 100.5e^(0.692t) cells per microliter, doubling about every 1.00 hour.
Your turn
A model y = 50e^(0.35t) describes a population. How long does it take to double?
Show the answer and explanation
About 1.98 time units.
Solve e^(0.35t) = 2: t = ln 2 ÷ 0.35 ≈ 1.98.
Keep exploring
In Statistics, switch the fit model to Linear: R² drops to about 0.871, and the residuals curve, positive at both ends and negative in the middle.
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