Establishing the upper and lower bounds of normal using standard deviation I understand the concept of standard deviations and z-values, but I'm trying to figure out if standard deviations alone are good for establishing the upper and lower bounds for normal. For example, if I have the following dataset:



$x = 1,4,1,10,112,6,22,7,18,113,1,4,1,10,112,6,22,7,18,113,1,4,1,10,112,6,22,7$

$\mu = 26.82$

$std(x) = 41.16$



I have been establishing the range for normal values like this:



$lowerNormal = 26.82 - 41.16 = -14.78$

$upperNormal = 26.82 + 41.16 = 67.98$



So, when a new value comes along:

$a = 72$

I consider that value to be abnormal, since it is above the $upperNormal$ value.

My question is whether or not this is a statistically sound way for determining whether a value is normal. The real issue I'm seeing is that none of the values in $x$ will ever be negative, so it seems that at least the lower bound is somewhat arbitrary.

Anyways, I apologize if I am missing something simple. I'm learning statistics on my own and it's a little puzzling sometimes. Thanks for your help
Your problem is that you are subtracting the full standard deviation from the mean. When you do that, it means that you are multiplying the standard deviation by 2, since you are subtracting $1*std(x)$ and adding $1*std(x)$. 

Instead, add/subtract $\frac{1}{2}*std(x)$. 

Thus, your new normals are:

$$lowerNormal=26.82−\frac{41.16}{2}=6.24$$
$$upperNormal=26.82+\frac{41.16}{2}=47.4$$
If a negative number is not a possible/feasible data point than your lower limit is zero in this particular example. It will stay zero until you get more data that shifts your mean higher.