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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...

A statistical measure of the dispersion of data points in a data series around the mean. It is calculated as follows: The coefficient of variation represents the ratio of the standard deviation to the mean, and it is a useful statistic for comparing the degree of variation from one data series to another, even if the means are drastically different from each other.

If data values are all equal to one another, then the standard deviation is zero. If a high proportion of data points lie near the  mean  value, then the standard deviation is small. An experiment that yields data with a low standard deviation is said have high  precision . If a high proportion of data points lie far from the mean value, then the standard deviation is large. An experiment that yields data with a high standard deviation is said to have low precision. The following quantities/equations are quantitative measures of precision. The equations provide precision measures for a limited number of repetitive measurements, i.e. between 2 and 20. The equation at the end is the  true standard deviation  for any number of repeat measurements. The mean or average ,  , is calculated from: