Lesson 1.2: Standard Deviation (The "Intuitive" Spread)

Welcome to Lesson 1.2. In our last lesson, we learned about Variance (σ²), the 'average squared distance from the mean.' It's a great mathematical tool, but its units are weird (e.g., 'dollars-squared'). In this lesson, we will learn about its famous sibling, the Standard Deviation, which 'fixes' this problem.

The Standard Deviation is the single most important number for measuring risk in all of finance. It's often called "Volatility".

The Fix: Take the Square Root

The "fix" for Variance's weird units is simple: we just take its square root. This returns our measure of "spread" to the same units as our Mean.

Standard Deviation (σ)

\sigma = \sqrt{\text{Variance}} = \sqrt{\sigma^2}

The "So What?" (The Physical Meaning)

The "Typical" Distance

The Standard Deviation is the "typical" or "standard" distance that a data point is from the average.

If a stock's daily return has a Mean ( $\mu$ ) of 0.05% and a Standard Deviation ( $\sigma$ ) of 2%, it means:

The stock tends to drift up by 0.05% per day.
But on any given day, a "typical" random move will be about ±2% around that drift.

So, a "normal" day for this stock would be anywhere between -1.95% and +2.05%. A day where the stock moves +7% is a "3-sigma event" (3 standard deviations away from the mean) and is very unusual.

What's Next?

We now have our two most important numbers: the Mean ( $\mu$ ) for the "center" and the Standard Deviation ( $\sigma$ ) for the "spread." These are the only two "ingredients" we need to build our next tool.

In Lesson 1.3, we will use $\mu$ and $\sigma$ to create the most important "recipe" in all of statistics: the Normal Distribution, or "Bell Curve."

Key Concepts - Mean and Variance

The Normal Distribution N(μ, σ²)