When standard deviation is zero, you will not have a curve with a shape, but just the middle line that represents the mean/median/mode
For any distribution, the smallest possible value for the standard deviation is zero. From the definition of the normal distribution centered at 0,
, we can't just set 
, because we can not divide by zero.
Instead, we should examine what happens as
. We know that 
is fundamentally related to what people interpret as the 'width' of the distribution. So, as know that the 'width' of the distribution will get narrower. The area under the distribution always has to integrate to zero, so the 'height' of the distribution will get taller to compensate for decreasing 'width.'
Taking this to its logic conclusion, we have a distribution function that is zero everywhere except that the center (in the above example, at x=0) and at x=0 is..... well.... that is kind of the rub. Infinity isn't really a number that it makes sense for a function to have as a value. We want to say the distribution has a value of infinity at the point it was originally centered around. This is not strictly correct, but it is correct enough to be the useful way to think about it in applications like physics and signals processing.
For any distribution, the smallest possible value for the standard deviation is zero. From the definition of the normal distribution centered at 0,
, we can't just set , because we can not divide by zero.Instead, we should examine what happens as
. We know that is fundamentally related to what people interpret as the 'width' of the distribution. So, as know that the 'width' of the distribution will get narrower. The area under the distribution always has to integrate to zero, so the 'height' of the distribution will get taller to compensate for decreasing 'width.'Taking this to its logic conclusion, we have a distribution function that is zero everywhere except that the center (in the above example, at x=0) and at x=0 is..... well.... that is kind of the rub. Infinity isn't really a number that it makes sense for a function to have as a value. We want to say the distribution has a value of infinity at the point it was originally centered around. This is not strictly correct, but it is correct enough to be the useful way to think about it in applications like physics and signals processing.
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