Normal

It Had To Be You

Some others I've seen might never be mean
Might never be cross or try to be boss,
But they wouldn't do.
For nobody else gave me a thrill.
With all your faults, I love you still,
It had to be you, wonderful you,
It had to be you.

So what does the Mean have to be for things to be “normal”.

Nerd alert. 

If there is an absolute truth, then the number of samples can be as small as one and you should still  find the same value, i.e. truth. The Standard Error is defined as the square root of the Variance divided by the square root of the sample size. Thus if the value is an absolute, and there is no error, then the Variance must also be zero.

Scientists try to reduce error and establish absolute values. In doing so they are reducing the Variance of those observations of the absolute. If the Variance is one billionth of a percent then, this is not zero, but it is very, very close to zero. The Standard Deviation is the square root of the Variance. If the Variance is one-billionth, then the Standard Deviation is 32 one-millionth of a percent. 

While reducing the Standard Error also means reducing the Variance and the Standard Deviation, knowing the upper bound of the Standard Deviation can also give the upper bound of the Skew of the distribution. If you know the Standard Deviation, then you know what percentage of the values that fall within multiples of that Standard Deviation. For example, in a normal distribution, 68.27 % of the values fall within one Standard Deviation from the Mean; 95.45% of the values fall within 2 Standard Deviations from the Mean; 99.73% fall within 3 Standard Deviations from the Mean. The Three Sigma (three Standard Deviations), rule is generally sufficient in the physical sciences. In particle physics, the standard is Five Sigma  (99.99994%). 

This rule requires that a distribution be somewhat normal. While the usual intent is to lower the variance, given a Mean and the requirement that the minimum value of a distribution is zero, it is possible to estimate the required Standard Deviation. If 99.7% of the values fall within 3 Standard Deviations of the Mean, and the minimum value must be zero, then the Standard Deviation must be approximately the Mean divided by three. The are several distributions that are somewhat normal but allow for a non-zero skew.    

A common one is the exponentially modified Gaussian (normal) distribution. In probability theory, an exponentially modified Gaussian distribution describes the sum of independent normal and exponential random variables. It will not allow any non-zero values. An exponentially modified Gaussian distribution will have a skew between 0.0 ( closest to a pure normal distribution) and 0.31 (closest to a pure exponential distribution). In an exponentially modified Gaussian distribution, the mean will always be greater than the median, but the closer the mean is to the median, the smaller the skew. Given a maximum Skew of 0.31, Pearson’s Second Coefficient of the Skew, and a Standard Deviation of Mean/3, then the Median must be greater than 0.897 times the Mean, which is equivalent to saying that Mean must be less than 1.15 times the Median in order for this to be an exponentially modified Gaussian (normal) distribution. 

As noted above, normal distributions follow the 68/95/99.7 rule.  That is 68% of the values are in the range Mean ± 1 Standard Deviation;  95% of the values are in the range Mean ± 2 Standard Deviations; and 99.7 % of the values are in the range Mean ± 3 Standard Deviations.  As noted above, in particle physics the standard is  99.99994% of the values are in the range Mean ± 5 Standard Deviations.  By making the Standard Deviation as small as possible the range of possible values becomes the Mean.  In normal distributions, the Mean is equal to the Median.  In slightly Skewed “normal” distributions, the 68/95/99.7 rule this means that 100% is approximately equal to 99.7%. This means that 100% is in the range Mean ± 3 Standard Deviations and 50% is in the range 0 to the Median. This means that a distribution is "normal" if (Mean- 3 * Standard Deviations)/2 = Median.  The Standard Deviation must be 1.5 divided by the Median multiplied by the Mean . If the ratio of the Mean to the Median exceeds 1.5, then the underlying distribution can not be “normal”.  A distribution can be made up of several “normal” distributions, but it might itself not be "normal".