Exponential Regression

Model exponential growth or decay from your data.

Enter your paired data points (x, y) to calculate the exponential regression model y=ab^x. The calculator provides the equation, coefficients 'a' and 'b', and the coefficient of determination (R²).

Examples

Click on an example to load the data and see how the calculator works.

Bacterial Population Growth

Population Growth

Modeling the exponential growth of a bacterial colony over several hours.

1 2
2 4.1
3 7.9
4 16.2
5 33.0

Compound Interest Investment

Finance

An investment's value over time with compounding interest, showing exponential growth.

0 1000
1 1050
2 1102.5
3 1157.6
4 1215.5

Radioactive Decay

Physics

Modeling the decay of a radioactive substance over time.

0 100
10 82
20 67
30 55
40 45

Moore's Law

Technology

Approximating the number of transistors on a microchip over time.

1971 2300
1982 134000
1993 3100000
2000 42000000
2011 2600000000
Other Titles
Understanding Exponential Regression: A Comprehensive Guide
Dive deep into the concepts, applications, and mathematics behind exponential regression to model growth and decay phenomena.

What is Exponential Regression?

  • Defining the Model
  • The 'a' and 'b' Coefficients
  • Linear vs. Exponential Models
Exponential regression is a statistical method used to find the 'best fit' exponential model (y = ab^x) for a given set of data points. Unlike linear regression, which models a straight-line relationship, exponential regression is ideal for situations where the rate of change is proportional to the current value, leading to a curved trend line.
The Core Equation: y = ab^x
In this equation, 'y' is the dependent variable, 'x' is the independent variable, 'a' is the initial value (the value of y when x=0), and 'b' is the growth or decay factor. If b > 1, the model represents exponential growth. If 0 < b < 1, it represents exponential decay.
When to Use It
This model is appropriate when your data's scatter plot suggests a curve that either rises or falls at an increasingly rapid rate. It's crucial that all y-values are positive, as the model cannot handle zero or negative values.

Step-by-Step Guide to Using the Calculator

  • Data Entry Format
  • Interpreting the Results
  • Making Predictions
1. Enter Your Data
Input your (x, y) data pairs into the 'Data Points' text area. Each pair should be on a new line, with the x and y values separated by a space or a comma. For example: '2 150'. You need at least three points for a meaningful regression.
2. Calculate and Analyze
Click the 'Calculate' button. The tool will output the regression equation (y = ab^x), the specific values for 'a' and 'b', and the Coefficient of Determination (R²).
3. Understanding R-squared (R²)
R² measures how well the exponential model fits your data. Its value ranges from 0 to 1. A value closer to 1 indicates a better fit, meaning the model accurately explains the variance in the data. For example, an R² of 0.95 means that 95% of the variation in 'y' is explained by the model.
4. Make a Prediction
To find the expected 'y' for a specific 'x' that's not in your original data, enter the 'x' value into the prediction field. The calculator will use the derived equation to compute the predicted 'y' value.

Real-World Applications of Exponential Regression

  • Biology and Epidemiology
  • Finance and Economics
  • Physics and Engineering
Exponential regression is not just an academic exercise; it models many real-world phenomena.
Population Growth
Biologists use it to model the growth of bacterial colonies, cell cultures, or even animal populations in ideal conditions with unlimited resources.
Compound Interest
In finance, an investment that earns compound interest grows exponentially. This model can predict the future value of such an investment.
Radioactive Decay
Physicists model the decay of radioactive isotopes using exponential regression. The 'half-life' concept is derived directly from an exponential decay model.
Disease Spread
In the early stages of an outbreak, the number of infected individuals often grows exponentially, a pattern that epidemiologists model to predict the spread and plan interventions.

Mathematical Derivation and Formulas

  • Linearization of the Model
  • Calculating the Coefficients
  • The Correlation Coefficient
The core trick to solving for 'a' and 'b' is to transform the exponential equation into a linear one. This process is called linearization.
1. Transformation
Start with y = ab^x. By taking the natural logarithm (ln) of both sides, we get: ln(y) = ln(a) + ln(b^x), which simplifies to ln(y) = ln(a) + x * ln(b). This is a linear equation in the form Y = A + Bx, where Y = ln(y), A = ln(a), and B = ln(b).
2. Linear Regression
Now we perform a standard linear regression on the transformed data points (x, ln(y)). The formulas for the slope (B) and intercept (A) are:
Slope (B) = (nΣ(xY) - ΣxΣY) / (nΣ(x²) - (Σx)²)
Intercept (A) = mean(Y) - B * mean(x)
3. Re-transformation
Once A and B are calculated, we reverse the transformation to find the original coefficients 'a' and 'b': a = e^A and b = e^B.
4. Coefficient of Determination (R²)
R² is calculated on the linearized data (x, Y). It is the square of the Pearson correlation coefficient 'r' for the (x, Y) pairs, indicating the proportion of variance in Y that is predictable from x.