The following is the population standard deviation formula:

Where:
σ = population standard deviation
x1, ..., xN = the population data set
μ = mean of the population data set
N = size of the population data set

If we're in front of a normal distribution, a standard deviation of 1 (σ = 1) means that 68.27% of the distribution values lie within one standard deviation from the mean. In mathematical notation: Pr( μ-1σ ≤ X ≤ μ+1σ) ≈ 68.27%.

The formula to find the standard deviation of the sample mean is:

σX̄ = σ/√n

where:

• σX̄ – Standard deviation of the sample mean;
• σ – Population standard deviation; and
• n - Sample size.

We previously said that if we know the mean of the sampling distribution (μX̄), we also know the population mean (μ), as they're equal (μX̄ = μ). In practice, we never know μX̄, but we can estimate it using the sample mean (X̄).

σX̄ indicates how X̄ approximates μ. The smaller σX̄, the nearest μ can be from our estimate. As σ is constant, increasing the sample size is the only way to diminish σX̄. Therefore, increasing n is a way to reduce our estimations' sampling error.