Confidence Interval Slope Calculator

Calculator

Calculation Mode Regression Summary Raw Paired Data

Confidence Level (%)

Load Summary Example Load Raw Example

Slope Estimate (b1)

Standard Error of Slope

Sample Size (n)

Degrees of Freedom

Leave blank to use n - 2.

Raw x,y Pairs 1,2.1 2,3.5 3,4.2 4,6.0 5,7.1 6,8.4

Accepted separators include commas, spaces, semicolons, or tabs.

Calculate Confidence Interval

Example Data Table

This sample data can be used in raw mode.

Summary example: slope = 1.4200, standard error = 0.2100, sample size = 18, confidence level = 95.

Formula Used

The slope confidence interval is built with this expression:

b1 ± t × SE(b1)

Where b1 is the estimated slope, SE(b1) is the standard error of the slope, and t is the critical value from the t distribution using the selected confidence level and degrees of freedom.

For raw paired data, the calculator first fits a simple linear regression. It then computes:

• Slope: b1 = Sxy / Sxx
• Intercept: b0 = ȳ − b1x̄
• Residual mean square: MSE = SSE / (n − 2)
• Standard error of slope: SE(b1) = √(MSE / Sxx)

This gives a practical interval for the true population slope. If the full interval stays above zero or below zero, the slope direction is more convincing at that confidence level.

How to Use This Calculator

1. Select Regression Summary if you already know the slope estimate and its standard error.
2. Select Raw Paired Data if you want the calculator to fit the line from x,y observations.
3. Enter the confidence level, such as 90, 95, or 99.
4. For summary mode, enter slope, standard error, and sample size. Leave degrees of freedom blank to use n − 2.
5. For raw mode, enter one x,y pair per line.
6. Click the calculate button to see the interval, margin of error, and graph.
7. Use the export buttons to save the result as CSV or PDF.

Frequently Asked Questions

1. What does the slope confidence interval show?

It gives a plausible range for the true population slope. Narrow intervals suggest stable estimates. Wide intervals suggest more uncertainty in the relationship.

2. Why is the t distribution used here?

Regression slope intervals usually use the t distribution because slope uncertainty is estimated from sample data. It adjusts for sample size through degrees of freedom.

3. Can I calculate the interval from raw x,y pairs?

Yes. Enter one x,y pair per line. The calculator fits a simple linear regression, computes the slope, standard error, and interval automatically.

4. What if zero lies inside the interval?

Then the data do not show a clearly nonzero slope at the chosen confidence level. The linear effect may be weak or uncertain.

5. Why might the interval be wide?

Wide intervals often come from small samples, high residual scatter, weak x spread, or a large slope standard error.

6. Does this work for multiple regression?

No. This page is for a single predictor and one slope estimate. Multiple regression needs coefficient-specific inputs from a broader model.

7. What happens if all x values are nearly the same?

The slope cannot be estimated well because Sxx becomes tiny or zero. You need meaningful variation in x values.

8. Which confidence level should I choose?

Ninety-five percent is common. Higher levels give wider intervals, while lower levels give narrower intervals but less coverage.