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I read this sentence about the sum of squared error SSE: "The square ensures the error is always positive and larger errors are penalized more than smaller errors". I want to understand what is meant by "larger errors are penalized more than smaller errors", and how to prove that?

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    $\begingroup$ If you square small and bigger values, then the latter will change more then the first one. Plot $x$ vs $x^2$ to convince yourself. $\endgroup$ – Tim Aug 16 '17 at 5:38
  • $\begingroup$ Consider accepting @Tim answer. If the error is $1$ its square is still $1$, but if the error is $100$ its square is $10000$ which is significantly larger. $\endgroup$ – Gnattuha Aug 20 '17 at 19:08
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    $\begingroup$ Let an error be $\epsilon$, so that its size is $|\epsilon|$. Then its square can be factored into two terms, $$\epsilon^2 = \omega \times |\epsilon|,$$ with $\omega = |\epsilon|$. This shows how the square of any number can be considered to be its size multiplied by a weight $\omega \ge 0$. Notice that these weights are directly proportional to the sizes of the errors. That's all there is to the quotation--it is a mathematical triviality, merely involving a reinterpretation of the square of a number. $\endgroup$ – whuber Aug 20 '17 at 21:05
  • $\begingroup$ "The sum of squared error" is not a question. You have to actually ask a question. $\endgroup$ – smci Aug 21 '17 at 0:17
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Plot $x$ against $x^2$. Smaller values squared change less, then larger values squared. The steepness of the slope of $x^2$ increases as $|x|$ grows. So if you square small errors, the penalty is smaller, then if you squared large errors. There is nothing to prove in here, it is just how the square function works.

x vs x^2

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