The Normal Distribution

The normal distribution is the bell-shaped curve you've seen a hundred times: symmetric, peaked in the middle, thinning out smoothly toward both tails. It matters so much because it shows up everywhere, whenever you add up lots of small independent effects, the result tends to look normal. That's the promise of the The Central Limit Theorem, and it's why the bell curve is the default model for noise, measurement error, and short-horizon returns.

What makes it easy to work with is that only two numbers describe it completely: the mean $\mu$, which sets where the peak sits, and the Standard Deviation $\sigma$, which sets how wide it is. Change $\mu$ and the whole curve slides; change $\sigma$ and it stretches or narrows. Nothing else about the shape can move.

The 68-95-99.7 rule

For any normal distribution, no matter its mean or width, the fractions of outcomes inside each band are fixed:

• within $\pm 1\sigma$ of the mean: about $68\%$,
• within $\pm 2\sigma$: about $95\%$,
• within $\pm 3\sigma$: about $99.7\%$.

Because the tails are so thin, anything past $3\sigma$ is rare, under $0.3\%$ of the time. This is why "a five-sigma event" is shorthand for "essentially never should happen," and why it raises eyebrows when such events keep occurring in real markets.

Drag the band below from $\pm 1\sigma$ out to $\pm 3\sigma$ and watch the shaded coverage climb toward the familiar 68 → 95 → 99.7. Move the mean and width too: the percentages don't budge, because they're a property of the shape, not of where it sits.

Worked example: standardize, then read the tail

A stock's daily return is roughly normal with mean $\mu = 0\%$ and standard deviation $\sigma = 1\%$. What's the chance it drops more than $2\%$ in a day?

First convert the threshold into standard-deviation units, its z-score:

$$z = \frac{x - \mu}{\sigma} = \frac{-2 - 0}{1} = -2 .$$

So a $-2\%$ day is exactly a two-sigma move. The rule says $95\%$ of outcomes fall within $\pm 2\sigma$, leaving $5\%$ outside, split evenly between the two tails. The lower tail alone is therefore about

$$\frac{5\%}{2} = 2.5\% .$$

A drop worse than $2\%$ should happen on roughly one day in forty. The z-score is the whole trick: it rescales any normal into the same standard curve so one set of percentages answers every question.

Common pitfalls

• Real returns have fatter tails. Markets produce far more extreme moves than a normal predicts. Using the bell curve for tail risk understates the danger, sometimes catastrophically.
• Symmetry is an assumption, not a fact. The normal treats gains and losses as mirror images. Many real quantities are skewed.
• A z-score needs the right $\sigma$. Feeding in a daily sigma while asking a weekly question gives nonsense; spread scales with the horizon (see The Random Walk).
• The rule is for the normal only. The 68-95-99.7 percentages don't transfer to other shapes. Check the distribution before quoting them.

The normal distribution is the workhorse of applied probability, the endpoint of the The Central Limit Theorem, one of the Common Distributions you must know cold, and the limiting shape of a The Random Walk as its steps pile up.