Calculating the Standard Deviation in Complex Cases is relatively easy, if you follow the process.

But first of all, what do we mean by "in complex cases"? We just mean a dataset that consists of three or more numbers that are not all equal to each other. For example, the dataset below would be a more complex case:

$$1, 2, 3, 4, 5$$

Calculating Standard Deviation - The Process

What is the standard deviation of the dataset below?

$$1, 4, 9, 16, 25$$

Step 1: Find the average of the dataset.

$$\frac{1+4+9+16+25}{5} = 11$$

Step 2: Find the difference between each number in the dataset and the average. For example, our average is $11$ right? Notice how the first number in the dataset, $1$, is $10$ less than our average, so we would give it a difference value of $-10$. Here they are for all five numbers in the dataset:

$$(1-11), (4-11), (9-11), (16-11), (25-11) = -10, -7, -2, 5, 14$$

Step 3: Calculate the sum of the square of each number found in step 2.

$$(-10)^2 + (-7)^2 + (-2)^2 + 5^2 + 14^2 = 374$$

Step 4: Divide the number from step 3 by the number of items in the dataset (in our case $5$). Take the square root of that result.

$$\sqrt{\frac{374}{5}} = \sqrt{74.8}$$

$$\sigma \approx 8.65$$

If You Want the Formula

It's ugly, but here ya go:

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