Mean, variance, standard deviation — the shape of shake

The shape of shake

Two stocks. Both averaged 10% annual return for the last 20 years. Same average. But:

• Stock A: every year between +9% and +11%
• Stock B: most years roughly +10%, but in some years it lost 30% and in some years it gained 50%

Which one is "safer"? Obviously B is wilder. Same average, but the path is different. Stock A barely shakes. Stock B shakes a lot.

That difference — the shake — is what statistics calls variance (and its more readable cousin, standard deviation). The whole point of these statistics is to answer the question: how much does this thing wiggle around its average?

The strip picture

Picture the average as a horizontal line through the middle. The actual yearly returns are dots scattered around that line. Some above, some below.

For Stock A, the dots cluster tightly along the line — a narrow band, about 1% wide on each side.

For Stock B, the dots scatter wide — 30% below, 50% above. A much wider band.

The standard deviation (usually written σ, lowercase Greek sigma) is just the typical width of that band — how far away from the average a typical observation lives. Stock A might have σ = 1%. Stock B might have σ = 25%.

Same average. Different σ. Different shake.

Why we square things — the trick behind variance

"Variance" is technically the average of (distance from mean)². Why squared? Because positive deviations and negative deviations cancel out if you just take the average difference. (+5 above, −5 below → averages to zero, useless.)

Squaring kills the sign — both (+5)² = 25 and (−5)² = 25 contribute positively. Now we can average. That average is variance.

But variance has weird units (returns squared — what?). So we take the square root to get back to normal units. That's standard deviation. Standard deviation = √variance. Same shake, normal units.

Don't let any of that math intimidate you. The picture is the band. Wider band = bigger σ = more shake. That's the only thing you need to carry forward.

Where this becomes finance

Volatility — the word you hear constantly in finance — is just standard deviation of returns. Same thing, different name. When CNBC says "the VIX is at 25," they mean the market's expected annualized volatility (σ) is 25%. Wider band ahead.

This is also where the OOP-style polymorphism kicks in. The "shake" base class — variance, standard deviation, σ — gets specialized all over finance:

• In Track 3, σ is volatility of a single stock
• In Track 7, duration measures how much a bond shakes when rates move
• In Track 8, σ is the input that determines option prices (Black-Scholes)
• In Track 9, β (beta) measures shake of one stock relative to shake of the market

Every one of those is just "shake" in a different costume. If you've got the band picture, the rest is just naming conventions.

The takeaway

Average tells you where things tend to be. Variance (or σ, its square-rooted cousin) tells you how widely they spread around that center. Mean is the line. Variance is the width of the band. Most of the rest of finance is some specialization of these two concepts.