Standard Deviation – Why 1.5 Standard Deviations Calculation Isn’t Straightforward

normal distributionprobabilitystandard deviation

The normal distribution problem

I don't understand why I can't simply add 1.5 standard deviations to get the answer.

If 1 standard deviation is 10kg and the mean is 400kg, then 415kg is 1.5 standard deviations.

So I calculated it like this: .3413 + ((.4772-.3413)/2) = 0.40925

This equation takes one half of the difference between two standard deviations and one standard deviation, then adds it to the first standard deviation.

Why does this not work? Why do I have to use the table provided?

Best Answer

The reason that we cannot (linearly) interpolate between 0.3413 and 0.4772 is because the pdf of the Normal distribution is not uniform (flat at a single value).

Consider this more simple example, where we can use geometry to find the areas.

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The total area of the plot is 1 (it's a square cut diagonally, with the two pieces rearranged to be a triangle). Using Base*Height/2 we can find that the area of region A is 0.5, and the total area of regions B and C is also 0.5.

But the areas of B and C are not equal. The area of region C is 0.5*0.5/2 = 0.125, and therefore the area of region B is 0.375. So even though regions B and C are equally wide along the x-axis, since the height is not constant, they have different areas.

The Normal distribution that you are dealing with in your exercise is similar, but with a more complicated function for the height instead of a simple triangle. Because of this, the area between two values can't be solved as simply - hence the use of Z-scores and a table to find probabilities.

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