Scatter plot with linear regression

How to use it

Paste a table with at least two numeric columns and choose which is x and which is y. Put the variable you want to explain or predict on the y axis. Rows where either value is blank or not a number are skipped, and the page says how many. Enter an x value to read the line's prediction at that point.

The method: ordinary least squares

The line is the one that makes the sum of squared vertical distances from the points to the line as small as possible. With x̄ and ȳ the means of each column:

• slope b = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)²
• intercept a = ȳ − b·x̄
• r = Σ(x − x̄)(y − ȳ) / √(Σ(x − x̄)² · Σ(y − ȳ)²)
• r² = 1 − SSE / SST, the share of the variance in y explained by the line
• residual standard error = √(SSE / (n − 2)), the typical vertical miss, in y's units
• SE(b) = residual SE / √Σ(x − x̄)² and t = b / SE(b)

SSE is the sum of squared residuals, and SST is the total sum of squares of y around its mean. For a straight-line fit with an intercept, r² computed this way equals the square of r.

Worked example: Anscombe's quartet

The sample is the first of four small datasets the statistician Francis Anscombe published in 1973. Its 11 points give a mean x of 9 and a mean y of 7.50. The fitted line is y = 3.00 + 0.500x, with r = 0.816 and r² = 0.667, so the line accounts for two thirds of the variation in y. The slope's standard error is 0.118, giving t = 4.24 on 9 degrees of freedom. At x = 10, the line predicts 8.00.

Anscombe's point was that the other three datasets in the quartet give almost exactly the same line, the same r and the same r², while looking completely different when plotted. One is a smooth curve, one is a perfect line with one outlier, and one has every point but one at the same x. The statistics alone cannot tell them apart. That is the reason this tool always draws the points and never reports a fit without them.

Reading r

|r|	Label used here	r²
below 0.1	negligible	< 1%
0.1 to 0.3	weak	1–9%
0.3 to 0.5	moderate	9–25%
0.5 and above	strong	25% +

These cut-offs follow Jacob Cohen's widely used conventions for the behavioural sciences. They are a starting point for describing a result, not a verdict: what counts as strong depends on the field and on how noisy the measurements are.

Where it stops being reliable

• Curves. A straight line fitted to a curved pattern can still give a high r². Look at the plot: if the points bend away from the line at both ends, the line is the wrong model.
• Outliers. Least squares squares the distances, so a single far-off point can pull the line and inflate or destroy r. Try the fit with and without it.
• Extrapolation. The prediction is only supported inside the range of x you have. Beyond it, the line is an assumption.
• Causation. Correlation between two columns can come from a third variable that drives both. Nothing in the arithmetic can detect that.

For the distribution of either column on its own, use the summary statistics calculator or the histogram maker.