Standard Error of Slope Calculator

Calculator

Input method

Confidence level (%)

Hypothesized slope

Decimal places

X label

Y label

X unit

Y unit

Summary n

Summary slope b1

Summary Sxx

Summary SSE

Residual standard error

Separate X list

Separate Y list

Paired x,y rows

Example Data Table

This sample matches the default paired data.

x	y	Use
1	2.1	Observation 1
2	2.9	Observation 2
3	3.7	Observation 3
4	5.1	Observation 4
5	5.8	Observation 5
6	7.2	Observation 6

Formula Used

For a simple linear regression line, use y = b0 + b1x.

b1 = Sxy / Sxx

Sxx = Σ(x - x̄)²

Sxy = Σ(x - x̄)(y - ȳ)

SSE = Σ(y - ŷ)²

MSE = SSE / (n - 2)

Standard Error of Slope = sqrt(MSE / Sxx)

Confidence Interval = b1 ± t critical × SE(b1)

The t statistic is (b1 - hypothesized slope) / SE(b1).

How to Use This Calculator

Choose paired rows, separate lists, or summary values.
Enter at least three observations for raw data modes.
Use the same order for every x and y value.
Set the confidence level and hypothesized slope.
Add variable labels and units if needed.
Press Calculate to view results above the form.
Use CSV or PDF buttons to download the report.

Understanding the standard error

The standard error of slope measures uncertainty in a fitted line. It shows how much the slope may change in another sample. It is not the slope value itself. It is the expected sampling noise around it. Smaller values mean tighter slope estimates. Larger values require more caution.

Why the formula works

A regression line leaves residuals behind. Each residual is an observed value minus a predicted value. These residuals estimate random scatter around the line. The calculator turns that scatter into mean square error. Then it divides by the spread of x values. Wide x spacing lowers uncertainty. Narrow x spacing raises uncertainty.

What the result tells you

The slope standard error supports confidence intervals and t tests. A confidence interval gives a likely range for the true slope. A t test compares the slope with a chosen value. Many users compare it with zero. This checks whether a visible trend is statistically clear. A small p value suggests stronger evidence. A wide interval suggests weak precision.

Data choices matter

Good input data improves the answer. Use paired x and y values from the same observations. Avoid mixing units. Check copied values before calculating. Outliers can change the slope and its standard error. Curved patterns can make a straight line misleading. Unequal scatter can also distort conclusions.

Using raw and summary modes

Raw data mode is best for most work. It computes the slope, intercept, residuals, fitted values, and sums of squares. Separate list mode is useful for spreadsheet columns. Summary mode is helpful for textbook problems. It needs sample size, Sxx, and residual variation. You may enter SSE or residual standard error.

Reporting the analysis

Report enough detail for another person to verify your result. Include the slope, standard error, degrees of freedom, and confidence level. Also mention sample size and units. Add R squared when raw data is used. Do not report only the p value. The size and direction of the slope matter too. Use the output with subject knowledge.

Common mistakes

Do not use y spread alone. Do not ignore the x spread. Do not round too early. Keep extra digits until the report. Always match each x with its y.

FAQs

What is the standard error of slope?

It is the estimated standard deviation of the regression slope. It shows how uncertain the sample slope is due to random variation in the data.

How many points are required?

You need at least three paired observations. Two points create a perfect line, so residual error and slope uncertainty cannot be estimated.

What does Sxx mean?

Sxx is the sum of squared deviations of x values from their mean. A larger Sxx usually gives a more precise slope estimate.

Can I use summary values only?

Yes. Choose summary mode. Enter n, slope, Sxx, and either SSE or residual standard error. Raw residual tables will not be available.

What is the t statistic used for?

It compares the estimated slope with your hypothesized slope. It is often used to test whether the true slope could be zero.

Why is my standard error large?

It may be large because residual scatter is high, the sample is small, x values are too close, or outliers affect the fitted line.

Does a small standard error prove causation?

No. It only supports precision in a linear association. Causation needs study design, control of confounding, and subject knowledge.

Why is R squared missing in summary mode?

R squared needs total variation in y or equivalent summary data. This calculator keeps summary mode focused on slope standard error.