Predicted value (ŷ)

How to read it

OpenStax’s Introductory Statistics: “The ŷ is read “y hat” and is the estimated value of y. It is the value of y obtained using the regression line. It is not generally equal to y from data.” A fitted line is therefore written with ŷ on the left, as in OpenStax’s ŷ = −173.51 + 4.83x for final exam scores predicted from a third exam. Penn State’s STAT 501 course notes call ŷᵢ the predicted response, or fitted value, for experimental unit i, and write the line ŷᵢ = b₀ + b₁xᵢ. As on p̂, the hat says the value is an estimate.

Also searched as: predicted value symbol, ŷ, ŷ, y hat, y hat symbol.

The residual, y − ŷ

The gap between an observed value and the predicted one, y − ŷ, is the residual. OpenStax writes it y₀ − ŷ₀ = ε₀ and explains that it is called the “error” though it is not an error in the sense of a mistake. A point above the line has a positive residual. The least-squares line is the one that makes the sum of the squared residuals as small as possible.

Hats on parameters

NIST’s Engineering Statistics Handbook puts the hat on the estimated parameters of a model as well: its residual is the observed response minus the fitted function evaluated with β̂, the estimate of β.

Typing ŷ

Unlike x̄ and p̂, ŷ has a character of its own: U+0177, LATIN SMALL LETTER Y WITH CIRCUMFLEX, which Unicode marks as Welsh and defines as y followed by the combining circumflex accent (U+0302), the mark Unicode also names “hat”.