Calculator Input
Enter x,y pairs line by line. Comma, space, tab, or semicolon separators are accepted.
Example Data Table
This sample resembles a decaying physics signal measured over time.
| Time (s) | Signal (V) |
|---|---|
| 0 | 12.00 |
| 1 | 9.84 |
| 2 | 8.05 |
| 3 | 6.60 |
| 4 | 5.39 |
| 5 | 4.43 |
| 6 | 3.63 |
Formula Used
The calculator fits the exponential model
y = A · e^(B·x).
ln(y) = ln(A) + Bxx and ln(y):
B = [nΣ(x ln y) - Σx Σln y] / [nΣx² - (Σx)²]
ln(A) = [Σln(y) - BΣx] / n
A = e^(ln(A))
If B < 0, the system decays. If B > 0, it grows.
For decay, half-life is ln(2)/|B|. For growth, doubling time is ln(2)/B.
How to Use This Calculator
- Enter the names and units for your x and y variables.
- Paste measured x,y data pairs into the textarea, one pair per line.
- Add an optional prediction x-value if you want interpolation or extrapolation.
- Choose decimal places and graph resolution.
- Click Calculate Exponential Fit.
- Review the equation, statistics, residual table, and Plotly graph.
- Download the analysis as CSV or PDF when needed.
Why this calculator is useful in physics
Exponential fitting is common in radioactive decay, capacitor discharge, damping, relaxation, attenuation, population growth, cooling approximations, and detector response analysis. This page converts raw measurements into a fitted equation, diagnostic statistics, and a clean graph for interpretation and reporting.
FAQs
1) What does an exponential fit calculator do?
It estimates the constants in a model where the response changes by a constant percentage rate. You get the equation, fitted values, residuals, and quality metrics.
2) When should I use exponential fitting in physics?
Use it when a quantity grows or decays proportionally to its current value. Typical examples include radioactive decay, charging, discharging, attenuation, and relaxation processes.
3) Why must all y-values be positive?
This calculator linearizes the model with a natural logarithm. Since ln(y) is only defined for positive y, zero or negative values cannot be processed here.
4) What does the exponent B mean physically?
B is the growth or decay rate. A positive value means growth, while a negative value means decay. Its magnitude sets how quickly the system changes.
5) What does R² tell me?
R² measures how well the fitted curve explains variation in the observed data on the original y-scale. Values closer to one usually indicate a stronger fit.
6) How is half-life or doubling time determined?
For decay, half-life equals ln(2) divided by the absolute decay rate. For growth, doubling time equals ln(2) divided by the positive growth rate.
7) Can I use noisy experimental data?
Yes. The regression averages the overall trend, and residual statistics help you judge scatter. Large residuals or a low R² suggest the model may need refinement.
8) What if my data are not truly exponential?
The calculator will still return a best fit, but the residual pattern and fit statistics may reveal a mismatch. In that case, another model may describe the system better.