Simple Regression Slope Calculator
Fits the least-squares slope for a simple linear regression with one predictor. The form displays b1 = sum((xi−xbar)(yi−ybar)) / sum((xi−xbar)^2) beside regression slope, using a worked condition that can be recalculated with the labeled inputs.
Describe the observed data when the sample changes
Regression slope
The question behind regression slope
The simple regression slope page fits the least-squares slope for a simple linear regression with one predictor.
Regression slope is limited to the statistical quantity named by the result panel. The regression slope calculation does not silently add a population, time horizon, causal direction, or decision threshold that is absent from the fields.
How the inputs shape regression slope
- Predictor X: For regression slope, the displayed predictor x sequence is 12, 15, 18, 21, 24, 27. Preserve predictor x order when regression slope depends on pairing, lag, rank, or time position, and distinguish an observed zero from a missing predictor x entry.
- Response Y: For regression slope, the displayed response y sequence is 20, 24, 25, 31, 33, 38. Preserve response y order when regression slope depends on pairing, lag, rank, or time position, and distinguish an observed zero from a missing response y entry.
The entries used for regression slope must refer to one coherent analysis condition. Combining incompatible populations, periods, or measurement definitions can produce valid regression slope arithmetic for a nonexistent study.
Following the simple regression slope relationship
For regression slope, match every symbol in the relationship to a labeled field before substituting numbers. Regression slope is reported in Y units per X unit.
While checking regression slope, use predictor x observations from one defined analysis set rather than totals copied from incompatible groups.
A reproducible simple regression slope case
The default regression slope condition is Predictor X = 12, 15, 18, 21, 24, 27, Response Y = 20, 24, 25, 31, 33, 38.
The example slope is approximately 1.1714 response units per X unit.
The live calculator reports Slope 1.1714286 · Intercept 5.6571429. Repeating one intermediate step from b1 = sum((xi−xbar)(yi−ybar)) / sum((xi−xbar)^2) provides a fixed regression slope reference check for later code changes.
What the simple regression slope arithmetic assumes
The slope is a conditional model coefficient, not proof that changing X causes Y to change.
For simple regression slope, a fitted coefficient or association is conditional on the model and observed range; it does not by itself show that changing one variable will cause another to change.
Reading regression slope in context
When interpreting simple regression slope, inspect residual behavior, influential observations, nonlinearity, dependence, and extrapolation before carrying a regression result to a new setting.
As a second check for regression slope, outliers, ties, ordering, and missing entries can affect regression slope even when the number of observations stays unchanged.
Testing how stable regression slope is
Change predictor x while holding the remaining entries fixed, then state why the direction and size of the regression slope change are plausible from b1 = sum((xi−xbar)(yi−ybar)) / sum((xi−xbar)^2).
Repeat the regression slope exercise with response y. If a modest defensible change materially alters the interpretation, report both conditions rather than presenting that regression slope scenario as exact.
Reporting regression slope reproducibly
Report regression slope using b1 = sum((xi−xbar)(yi−ybar)) / sum((xi−xbar)^2), followed by the entered values, units, exclusions, and analysis date. Name the regression slope population or dataset boundary instead of leaving it implicit.
Keep the full calculator output with the record, including Slope 1.1714286 · Intercept 5.6571429. A later regression slope review can then distinguish a changed input from a different convention or software implementation.
The surrounding workflow may also require population covariance and simple regression intercept.
Questions about regression slope
Why could another program report a different regression slope?
A different convention for rounding, tails, ties, interpolation, parameterization, or missing values can change regression slope. Compare the printed regression slope formula and its input definitions before treating either output as wrong.
What does regression slope represent on this page?
It is the quantity produced by b1 = sum((xi−xbar)(yi−ybar)) / sum((xi−xbar)^2) from the displayed predictor x, response y. This page fits the least-squares slope for a simple linear regression with one predictor.
What should be saved with regression slope?
Save the entered values and units for predictor x, response y, along with the analysis date, exclusions, software or formula version, and the relationship b1 = sum((xi−xbar)(yi−ybar)) / sum((xi−xbar)^2). That record is sufficient to rebuild this specific regression slope calculation.
Does regression slope establish a causal or population conclusion?
No. The displayed simple regression slope value is conditional on the entered data and named method. The regression slope design, measurement process, and assumptions determine what can be concluded beyond those values.
How should regression slope be rounded?
Keep the unrounded regression slope for subsequent arithmetic, then report only the precision supported by the source measurements and the decision context. Extra digits in regression slope do not correct sampling or model error.