Descending into ML: Check Your Understanding

Mean Squared Error

Consider the following two plots:

A plot of 10 points. A line runs through 6 of the points. 2 points are 1 "unit" above the line; 2 other points are 1 "unit" below the line. A plot of 10 points. A line runs through 8 of the points. 1 point is 2 "units" above the line; 1 other point is 2 "units" below the line.

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Which of the two data sets shown in the preceding plots has the higher Mean Squared Error (MSE)?
The dataset on the left.
The six examples on the line incur a total loss of 0. The four examples not on the line are not very far off the line, so even squaring their offset still yields a low value: $$ MSE = \frac{0^2 + 1^2 + 0^2 + 1^2 + 0^2 + 1^2 + 0^2 + 1^2 + 0^2 + 0^2} {10} = 0.4$$
The dataset on the right.
The eight examples on the line incur a total loss of 0. However, although only two points lay off the line, both of those points are twice as far off the line as the outlier points in the left figure. Squared loss amplifies those differences, so an offset of two incurs a loss four times greater than an offset of one.
$$ MSE = \frac{0^2 + 0^2 + 0^2 + 2^2 + 0^2 + 0^2 + 0^2 + 2^2 + 0^2 + 0^2} {10} = 0.8$$

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