Linear Regression Calculator
Least-squares slope, intercept, equation, r², standard error and a prediction for any x — from up to 10 data pairs, with the residual table shown.
Linear Regression Calculator: with the default inputs, slope (b) is 4.47619.
Only the first N (x, y) boxes are used. Pairs 6–10 live under More options.
The line is evaluated here. Staying inside the range of your data is far safer than going outside it.
How much y changes for a one-unit increase in x.
- Intercept (a)
- 46.857143The fitted y when x = 0.
- Regression equation
- y = 4.47619x + 46.857143
- r² (variation explained)
- 0.992363
- Correlation r
- 0.996174
- Standard error of the estimate
- 1.038925Typical size of a residual, in the units of y. Needs at least 3 points.
- Predicted y
- 87.142857
Assumptions
- Ordinary least squares with y regressed on x; all the error is assumed to be in y.
- The standard error of the estimate uses n − 2 degrees of freedom and needs at least three points.
- Residuals are assumed independent with constant variance — the usual regression conditions, which the calculator does not test for you.
| x | Observed y | Fitted y | Residual |
|---|---|---|---|
| 1 | 52 | 51.3333 | 0.6667 |
| 2 | 55 | 55.8095 | -0.8095 |
| 3 | 61 | 60.2857 | 0.7143 |
| 4 | 64 | 64.7619 | -0.7619 |
| 5 | 70 | 69.2381 | 0.7619 |
| 6 | 72 | 73.7143 | -1.7143 |
| 7 | 79 | 78.1905 | 0.8095 |
| 8 | 83 | 82.6667 | 0.3333 |
Least squares chooses the line that makes the sum of the squared residuals as small as possible. The residuals themselves always add up to zero.
How this is worked out
The formula
Slope: b = Σ(x − x̄)(y − ȳ) ÷ Σ(x − x̄)² Intercept: a = ȳ − b·x̄ Line: ŷ = a + b·x Residual sum of squares: SSE = Σ(y − ŷ)² = Syy − b·Sxy Standard error of the estimate: s = √( SSE ÷ (n − 2) ) r² = 1 − SSE ÷ Syy
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- How many pairs?
- Only the first N (x, y) boxes are used. Pairs 6–10 live under More options.from 2 to 10 · whole numbers only · defaults to 8
- x1
- A number.defaults to 1
- y1
- A number.defaults to 52
- x2
- A number.defaults to 2
- y2
- A number.defaults to 55
- x3
- A number.defaults to 3
- y3
- A number.defaults to 61
- x4
- A number.defaults to 4
- y4
- A number.defaults to 64
- x5
- A number.defaults to 5
- y5
- A number.defaults to 70
- x6(under More options)
- A number.defaults to 6
- y6(under More options)
- A number.defaults to 72
- x7(under More options)
- A number.defaults to 7
- y7(under More options)
- A number.defaults to 79
- x8(under More options)
- A number.defaults to 8
- y8(under More options)
- A number.defaults to 83
- x9(under More options)
- A number.defaults to 9
- y9(under More options)
- A number.defaults to 88
- x10(under More options)
- A number.defaults to 10
- y10(under More options)
- A number.defaults to 91
- Predict y at x =
- The line is evaluated here. Staying inside the range of your data is far safer than going outside it.defaults to 9
What you get back
- Slope (b)main answer
- How much y changes for a one-unit increase in x.
- Intercept (a)
- The fitted y when x = 0.
- Regression equation
- r² (variation explained)
- Correlation r
- Standard error of the estimate
- Typical size of a residual, in the units of y. Needs at least 3 points.
- Predicted y
What this assumes
- Ordinary least squares with y regressed on x; all the error is assumed to be in y.
- The standard error of the estimate uses n − 2 degrees of freedom and needs at least three points.
- Residuals are assumed independent with constant variance — the usual regression conditions, which the calculator does not test for you.
About this calculator
Simple linear regression finds the one straight line that comes closest to a cloud of points, and then lets you use it. It answers two different questions: how much does y change when x changes (the slope), and what y should I expect at this particular x (the prediction).
What "least squares" means
For any candidate line, each point has a residual — the vertical gap between the observed y and the line. Square those gaps, add them up, and choose the line that makes the total smallest. Squaring is what makes the answer a single tidy formula, and it is also why one far-away point can pull the whole line towards itself.
The residual table below the results shows every gap. Two things are always true and worth checking: the residuals sum to zero, and the line passes exactly through the point (x̄, ȳ).
How to read the results
- Slope is the headline. Its units are "y-units per x-unit" — dollars per square foot, points per study hour. A slope of 4.48 means each extra hour is associated with about 4.5 more points.
- Intercept is the fitted value at x = 0. It is often meaningless in itself (nobody has zero square feet), and it is fine for it to be.
- r² is the fraction of the variation in y that the line accounts for, from 0 to 1. It is not a verdict on whether the model is right; a curved relationship can have a high r² and still be the wrong shape.
- Standard error of the estimate is the most under-used number here: it is roughly the typical size of a prediction miss, in the units of y. If it is 1.04 points, your predictions are good to about ±2 points most of the time.
Where it misleads
- Extrapolation. The line is only evidence about the range of x you actually measured. The calculator warns you when the prediction x falls outside it, because that is where regression goes spectacularly wrong.
- Regression is directional. Regressing y on x and x on y give different lines (unless r = ±1). Put the thing you want to predict on the y side.
- Outliers. A single unusual point, especially one with an extreme x, has enormous leverage over the slope. Look at the chart.
- r² near 1 with two or three points is meaningless — two points always fit a line perfectly.
- Finally, the slope is an association, not a causal effect, unless the x values were assigned by an experiment.
Frequently asked questions
▸How do I find the line of best fit by hand?
Compute the two means, then the slope as Σ(x − x̄)(y − ȳ) ÷ Σ(x − x̄)². The intercept is ȳ − slope × x̄. The steps above show each of those sums for your data.
▸What does r² of 0.99 mean?
That the straight line accounts for 99% of the variation in y. It does not mean the relationship is causal, nor that a straight line is the right shape — only that the line fits these points closely.
▸What is the standard error of the estimate?
The typical vertical distance between the data points and the line, in the units of y. About two-thirds of points fall within one standard error of the line, and it is the right number for quoting prediction accuracy.
▸Can I predict y for an x outside my data?
You can, and the calculator will do it, but the line is only supported by the range you measured. Extrapolation assumes the relationship continues unchanged, and that assumption breaks more often than not.
▸Why do I get a different line if I swap x and y?
Least squares minimises vertical distances, so it treats y as the thing being predicted. Swapping the roles minimises horizontal distances instead and gives a different slope — the two agree only when the fit is perfect.
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