Calculator Input
Enter one x,y pair per line. Commas, spaces, tabs, or semicolons are accepted.
Example Data Table
This sample is nonlinear because its first differences change. Its second differences remain constant.
| x | y | First difference | Pattern note |
|---|---|---|---|
| 0 | 1 | - | Start value |
| 1 | 4 | 3 | Difference increased |
| 2 | 9 | 5 | Curve continues |
| 3 | 16 | 7 | Quadratic behavior |
| 4 | 25 | 9 | Second difference is stable |
Formula Used
First difference: Δy = y₂ - y₁
Interval slope: m = Δy / Δx
Linear model: y = mx + b
Quadratic model: y = ax² + bx + c
Exponential model: y = Ae^(Bx), fitted by using ln(y).
Power model: y = Ax^B, fitted by using ln(x) and ln(y).
Logarithmic model: y = a ln(x) + b.
Goodness of fit: R² = 1 - SSE / SST. Lower RMSE means smaller average prediction error.
How to Use This Calculator
- Enter x and y values in the table box.
- Use one pair per line. You may separate values with commas or spaces.
- Choose auto mode for model comparison.
- Set a tolerance for linearity checks.
- Add extra x values if you want predictions.
- Press the calculate button.
- Review the detected pattern, graph, residuals, and formulas.
- Download the CSV or PDF report for later use.
Linear and Nonlinear Tables Explained
What the table shows
A value table can hide a simple rule. Each row gives an input and an output. The input is usually called x. The output is usually called y. When x changes by equal steps, the y pattern becomes easier to inspect. A linear table has a steady rate of change. That means the slope stays the same from one interval to the next.
Why differences matter
First differences compare neighboring y values. If these differences stay constant, the table is usually linear. If they change, the table is nonlinear. Second differences help with curved tables. Constant second differences often point to a quadratic rule. This calculator checks both ideas. It also checks slopes when x spacing is uneven.
Regression gives deeper insight
Real data is often messy. Values may include rounding, measurement error, or missing steps. For that reason, the calculator also fits several models. It compares linear, quadratic, exponential, power, and logarithmic forms when possible. It reports R squared, adjusted R squared, RMSE, MAE, AIC, and BIC. These measures show how closely each model follows the data.
Reading the result
A high R squared means the model explains most output variation. A low RMSE means predictions are close to the original values. Residuals show row level error. Small residuals suggest a useful model. Large residuals show rows that may be outliers. The graph gives a quick visual check. Use the model comparison table before choosing a formula.
Best use cases
This tool is useful for algebra, precalculus, statistics, science labs, and data review. It can test homework tables. It can inspect experiment results. It can also create quick predicted values. Use clean data when possible. Sort x values for clearer differences. Always confirm the model with context, not only numbers.
FAQs
What is a linear table?
A linear table has a constant rate of change. When x increases by equal steps, y changes by the same amount each time. Its graph forms a straight line.
What is a nonlinear table?
A nonlinear table does not keep a constant slope. The y values may curve, grow faster, slow down, or follow another changing pattern.
Why are first differences useful?
First differences show how much y changes between neighboring rows. Constant first differences usually suggest a linear rule when x spacing is equal.
Why are second differences useful?
Second differences compare the first differences. If they are constant, the table often follows a quadratic pattern, especially when x spacing is even.
Can this calculator handle uneven x values?
Yes. It calculates interval slopes using Δy divided by Δx. This gives a better linearity check when x values are not equally spaced.
Which model should I choose?
Use auto mode first. Then compare R², adjusted R², RMSE, residuals, and the graph. Choose the model that fits well and makes sense.
Why are some models unavailable?
Exponential and power models need positive y values. Power and logarithmic models also need positive x values. The calculator skips unsafe transformations.
Can I export the results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a formatted report with summaries and model comparison tables.