How to use (3 steps)
- Paste your data (t, y) or load a CSV. You can also use a preset example.
- Choose the model (auto compare/exponential/logistic) and display options (log axis, extrapolation).
- Click “Calculate” to view parameters, fitted curves, residuals, and model comparisons. Share the same settings with a share URL.
Recommended order
- First, fit the growth curve (this page)
- Then check doubling time (how fast?) → Go to doubling time
- If saturation exists, check logistic K (upper limit) → Go to K
Go deeper
- Linear regression & correlation
Understand “ln makes a line” for exponential growth
- Descriptive statistics
Check variability
Data input & options
Results (parameters, charts, residuals)
Results will appear here.
Fitted curves
Residual plot
Per-row output (yhat & residuals)
| line | t | y | yhat_exp | resid_exp | yhat_log | resid_log |
|---|
Model comparison (RMSE/AIC)
| model | k | RMSE | AIC | note |
|---|
Calculation steps (How it’s calculated)
Models (exponential & logistic) and assumptions
- Exponential growth uses ln(y), so y>0 is required.
- Logistic growth includes saturation (K) and can be unstable with few points.
Calculation overview
Doubling time
Time to double. For the exponential model it is ln(2)/r (r>0).
Logistic K (upper limit)
The upper limit (carrying capacity). Logistic is useful when saturation is visible in the data.
- Exponential: regress
ln(y)=ln(y0)+rtto estimaterandy0. Doubling time isln(2)/r. - Logistic: estimate
y=K/(1+exp(-r(t-t0)))via nonlinear least squares. - Compute predicted
yhatand residuals (y−yhat), and output charts and tables.
FAQ
How should I choose between exponential and logistic?
Use exponential when there is no saturation and logistic when a clear upper limit appears. Compare residuals and RMSE/AIC as well.
How is doubling time calculated?
It is ln(2)/r from the exponential growth rate r (r>0).
What if y is zero or negative?
The exponential model (ln) cannot use those values, so those rows are excluded. Try logistic or apply background correction.
Is extrapolation OK?
Predictions outside the data range are uncertain, so they are shown as pale dashed lines for reference.
What if the data does not fit or errors appear?
- Check that you have two columns (t and y) including headers and separators.
- If there are invalid rows, turn on “Remove missing/non-numeric” or fix the data.
- If the logistic fit is unstable, set the initial K to manual and enter a rough value.
- Exponential requires y>0 (rows with y=0 or negative are excluded).
How many points do I need?
The calculation runs with as few as 2 points, but logistic fits can be unstable with small n. If possible, use multiple points (e.g., 6–10) and check residuals.
When should I use the log axis?
In exponential regions, plotting y on a log axis can look close to linear (requires y>0).
Does the share URL include the data?
It saves settings (model/view), not the data itself.
Related tools
References (notes)
For understanding formulas and concepts. In research/education, check primary sources as needed.
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