Calculator form
Use raw paired data for a full simple linear regression fit, or switch to summary mode to calculate the intercept from known regression statistics.
Example data table
This sample matches the default raw-data textarea so you can test the calculator immediately.
| X | Y |
|---|---|
| 1 | 2.1 |
| 2 | 2.9 |
| 3 | 4.2 |
| 4 | 5.1 |
| 5 | 6.0 |
Formula used
Intercept from means and slope
b₀ = ȳ − b₁x̄
Slope from paired data
b₁ = Sxy / Sxx, where Sxy = Σ(x − x̄)(y − ȳ) and Sxx = Σ(x − x̄)².
Intercept standard error
SE(b₀) = s × √(1/n + x̄²/Sxx), where s = √(SSE / (n − 2)).
Confidence interval
b₀ ± t* × SE(b₀)
In simple linear regression, the intercept is the predicted value of Y when X equals zero. The calculator reports both the direct intercept and related fit diagnostics when enough information is available.
How to use this calculator
- Choose Raw paired data when you have X,Y observations and want the full regression fit.
- Paste one pair per line in the textarea. Example:
3, 4.2. - Choose Summary inputs when slope, mean X, and mean Y are already known.
- Add optional summary statistics such as n, residual standard error, and Sxx to estimate uncertainty for the intercept.
- Enter a prediction X value if you want the fitted Y estimate at a specific X.
- Set the confidence level and decimal precision.
- Press Calculate intercept. The result appears above the form, directly below the header.
- Use the CSV and PDF buttons to export the result summary and processed dataset.
Frequently asked questions
1. What is the regression intercept?
It is the estimated value of Y when X equals zero. In the line equation Y = b₀ + b₁X, the intercept is b₀.
2. Why can the intercept be negative?
A negative intercept simply means the fitted line crosses the Y-axis below zero. That can happen even when all observed Y values are positive.
3. Does the intercept always have practical meaning?
No. If X = 0 lies far outside your observed range, the intercept may be mathematically correct but not meaningful for real interpretation.
4. What data format does the raw mode accept?
Enter one observation per line using two numeric values. Commas, spaces, semicolons, or tabs can separate X and Y.
5. Why is the confidence interval sometimes missing?
The interval needs enough information to estimate uncertainty. In summary mode, provide sample size, residual standard error, and Sxx.
6. Does this calculator support multiple regression?
No. This page is designed for simple linear regression with one predictor and one outcome variable.
7. What happens if all X values are identical?
The slope and intercept cannot be estimated from raw data because Sxx becomes zero. The calculator will return a validation message.
8. Which intercept formula is used here?
The main identity is b₀ = ȳ − b₁x̄. Raw mode first estimates b₁ from the data, then computes the intercept from sample means.