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=== Expanded Formulas ===
The above equations are efficient to use if the mean of the x and y variables (<math>\bar{x} \text{ and } \bar{y}</math>) are known. If the means are not known at the time of calculation, it may be more efficient to use the expanded version of the <math>\widehat\alpha\text{ and }\widehat\beta</math> equations. These equations
<math>\begin{bmatrix}
n & \sum_{i=1 }^nx_i \\
\sum_{i=1}^nx_i & \sum_{i=1}^nx_i^{2}
\end{bmatrix}
\begin{bmatrix}
\widehat\beta \\
\widehat\alpha
\end{bmatrix}
=
\begin{bmatrix}
\sum_{ i=0 }^ny_i \\
\sum_{ i=0 }^ny_ix_i
\end{bmatrix}
</math>
Stand-alone equations for <math>\widehat\alpha\text{ and }\widehat\beta</math> may be derived by expanding the matrix equations above. The resultant equations are algebraically equivalent to the ones shown in the prior paragraph, and are shown below without proof.<ref>{{Cite web |title=Numeracy, Maths and Statistics - Academic Skills Kit, Newcastle University |url=https://www.ncl.ac.uk/webtemplate/ask-assets/external/maths-resources/statistics/regression-and-correlation/simple-linear-regression.html |url-status=live |access-date=30 Jan 2024 |website=Simple Linear Regression}}</ref><ref name=":1">{{Cite web |last=Muthukrishnan |first=Gowri |date=17 Jun 2018 |title=Maths behind Polynomial regression, Muthukrishnan |url=https://muthu.co/maths-behind-polynomial-regression/ |url-status=live |access-date=30 Jan 2024 |website=Maths behind Polynomial regression}}</ref>
<math>\begin{align}
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