Balance Repeatability Sample or Population for SD
Standard Deviation Formulas in Maths is fun is a nice article to review. It explains why we have use two different formulas to find standard deviation for the population and sample. However, I'd like to point out that your explanation of calculation is a bit off. The correct formula to calculate standard deviation is:

$$ \bbox[yellow,5px,border:2px solid red]
{
\sigma = \sqrt{\left(\frac{1}{N} \sum_{i=1}^N (x_i-\mu)^2\right)}\qquad (1)}$$

where $\sigma$ is the standard deviation of the population and $\mu$ is the Mean (the simple average) of the population of numbers:

$$\mu = \frac{1}{N} \sum_{i=1}^N x_i$$

According to the Math is Fun website:


  To calculate the standard deviation of those numbers:
  
  
  Work out the Mean (the simple average of the numbers)
  Then for each number: subtract the Mean and square the result
  Then work out the mean of those squared differences.
  Take the square root of that.
  


The correct formula to calculate standard deviation of a sample is:

$$ \bbox[white,5px,border:2px solid blue]
{
s = \sqrt{\left(\frac{1}{n-1} \sum_{i=1}^n (x_i-\chi)^2\right)}\qquad (2)}$$

where $s$ is the standard deviation of the sample (say you choose $n$ numbers out of total of $N$ numbers population) and $\chi$ is the mean (the simple average) of the chosen numbers, hence: $$\chi = \frac{1}{n} \sum_{i=1}^n x_i$$.

Note: In our old days when I was taking analytical chemistry, we were instructed to use equation (1), if the $N \gt 10$, but to use equation (2) whenever we have to deal with data where $N \lt 10$.