linear regression and least squares
Simple linear regression finds the line that best describes a scatter of points, by minimizing the sum of squared residuals, the ordinary least squares (OLS) method. This page shows how slope and intercept come out of the same data, what the standard error measures, and why it needs at least 3 points to make sense.
The line predicts y = 0.6×6 + 2.2 = 5.8 for an x = 6 outside the sample, an extrapolation: the line assumes the same linear trend continues past the observed data, which is not always true.
With exactly 2 points the line passes exactly through both of them (zero residual at each), so SSR = 0 and the standard error comes out zero; that is not a perfect fit earned on merit, it is that no degree of freedom is left to measure any deviation.
When y does not vary with x, the numerator of the slope formula (the covariance) vanishes, and the best-fit line becomes a horizontal line at y's mean, predicting the same value for any x.
Simple linear regression is a statistical model that describes the relationship between an independent variable X and a dependent variable Y as a line, Ŷ = β₀ + β₁X. The slope β₁ tells you how much Y changes for each one-unit increase in X, and the intercept β₀ is the predicted value of Y when X is zero. It is the foundation of much of data analysis, used both to understand a relationship and to predict values.
Correlation (r) measures the strength and direction of the linear relationship as a single number between -1 and 1; regression goes further and delivers the line's equation (slope and intercept), which lets you predict y from a new x, something correlation alone does not provide.
Because fitting a line to n points spends 2 degrees of freedom, one for the slope and one for the intercept; only n-2 degrees of freedom remain free to estimate the spread of the residuals, and dividing by that number instead of by n avoids underestimating the real error.
Not necessarily: the least-squares line is only calibrated within the observed x range; predicting y for an x far beyond the collected data assumes the same linear trend continues indefinitely, an assumption that rarely holds up in real-world phenomena.
It is the predicted value of y when x = 0, even when x = 0 does not make practical sense in the data's context (like weight or age); a negative intercept does not invalidate the line within the observed range, it just means the line, extrapolated to x = 0, crosses the y-axis below zero.
One x, y pair per line. Maximum 200 pairs.
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