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I am not a mathematician, I am a data analyst using logistic regression. I am trying to approximate the values of the Sigmoid over an interval --say [0, 0.5] -- with a single number and to check the accuracy of my approximation I want to compare the product of that value by the range --i.e. f(x)*0.5-- with the accurate measure of the area below the Sigmoid in that interval which is given by F(0.5) - F(0), where F is the antiderivative of the Sigmoid.

According to Wolfram MathWorld the antiderivative of Sigmoid is $ln(1 + e^x)$.

Therefore F(0) = 0.301 and F(0.5) = 0.423. Consequently area below the Sigmoid in the interval [0 , 0.5] is: F(0.5) - F(0) = 0.423 - 0.301 = 0.122.

This confuses me because the Sigmoid at x=0 is 0.5. Therefore the area of the rectangle formed by the value of the Sigmoid at 0 and the interval [0 , 0.5] is 0.5 * 0.5 = 0.25. How can this area be smaller than the area below the Sigmoid curve in the interval that I derive through the antiderivative function?

An image of the problem as I understand it

Your advice will be appreciated.

1 Answers 1

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You used base-10 logarithms to calculate $F$ when you should have used base $e$. It should be $$F(0) = \ln(2) \approx 0.69$$ and $$F(0.5) = \ln(1+e^{0.5}) \approx 0.97.$$