Method of Least Squares (Approximation Theory)/Examples/Arbitrary Example 1

Examples of Method of Least Squares

Let $B$ be a false balance.

$2$ items are weighed on $B$: first individually and then together.

The recorded weights are:

$17 \, \mathrm g$ and $25 \, \mathrm g$ for the separate items

$40 \, \mathrm g$ for the combined weight.

The least squares estimates of the true weights are the values of $\hat {w_1}$ and $\hat {w_2}$ that minimize:

$L = \paren {w_1 - 25}^2 + \paren {w_1 - 17}^2 + \paren {w_1 + w_2 - 40}^2$

Differentiating with respect to $w_1$ and $w_2$ and equating the derivatives to zero, gives us:

\(\ds \hat {w_1}\)

\(=\)

\(\ds 16.33\)

\(\ds \hat {w_2}\)

\(=\)

\(\ds 24.33\)

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): least squares: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): least squares: 1.