From the previous mountain entry, we learned the steps to calculate standard deviation in more complex cases.

The difference between Population vs Sample Standard Deviation hinges on one of these steps.

Steps For Calculating Standard Deviation (Population vs Sample)

Step 1: Find the mean of the data set.
- SAME.
Step 2: Subtract the mean from each number in the dataset to find the difference values.
- SAME.
Step 3: Add the square of the difference values together.
- SAME.
Step 4: Divide the sum of the difference values by the number of items in the dataset.
- DIFFERENT.
  - For $n$ items in a dataset...
    - Population SD: Divide by $n$
    - Sample SD: Divide by $(n-1)$
Step 5: Take the square root.
- SAME.

The Formulas

$$\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2}$$

Population Standard Deviation

$$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}$$

Sample Standard Deviation

NOTE: While the sample standard deviation is generally larger than the population standard deviation, this is not always the case (can you tell why?). Also, in very large datasets with many hundreds, thousands, or millions of entries, the standard and population deviation are essentially equal to each other.