The Normal Distribution
10 min read
The normal distribution is the most important object in probability. Sums of many small independent effects converge to it, a fact proved at the end of this module, which is why it appears wherever aggregation happens: measurement errors, averages of samples, totals of long games. This lesson builds fluency with its shape, its parameters, and the handful of numbers every practitioner knows by heart.
The normal density
A symmetric bell centred at $\mu$ with spread set by $\sigma$. Written $X \sim N(\mu, \sigma^2)$: the second parameter is the variance.
The formula is used far less often than its consequences. The density is symmetric about , so mean, median, and mode coincide; it never touches zero, so no value is impossible, but the tails die extremely fast, faster than any exponential; and the two parameters are exactly the mean and standard deviation, so locating and scaling the bell is the whole game.
Standardisation
The normal with mean 0 and variance 1, written . Its CDF is denoted and has no closed form; values come from tables, software, or memorised landmarks.
The z-score
Subtracting the mean and dividing by the standard deviation converts any normal to the standard one, so a single table serves every normal problem.
The z-score is a change of units: it measures distance from the mean in standard deviations. Every normal probability question reduces to "how many sigmas away is this value?", followed by a lookup. The lookups worth carrying in memory:
The 68-95-99.7 rule
One, two, and three standard deviations around the mean. Equivalently, the one-sided tails: $P(Z > 1) \approx 0.16$, $P(Z > 2) \approx 0.023$, $P(Z > 3) \approx 0.0013$.
Scores are . What fraction exceed 130?
About 2.3 percent. Standardise, then read the landmark. With the rule's numbers memorised, most normal questions need no table at all.
Closure under addition
The normal family's second superpower is algebraic: any linear combination of independent normals is itself normal, with mean and variance given by the rules from Probability I. Nothing like this holds for most distributions, the sum of two uniforms is not uniform, and it is what makes normal models so tractable.
Sums of independent normals
Means add, variances add, and the shape stays normal. Differences work the same way: the variance still ADDS, $\mathrm{Var}(X - Y) = \sigma_X^2 + \sigma_Y^2$.
Two independent measurements are each . Their difference is
The probability the measurements disagree by more than 8.5 is about . Subtracting noisy quantities makes noise add, a fact that surprises until the variance rule is second nature.
Common trap
Standard deviations do not add; variances do. Two independent variables sum to standard deviation , not 20. And the normal's dominance has limits: it arises from sums of many SMALL independent effects, and data driven by a few large shocks can be far heavier-tailed than any normal fit suggests, a theme the statistics course takes up properly.
Quick check
X is normal with mean 0 and standard deviation 1. Approximately what is ? Answer as a decimal (2 places).
Quick check
Heights are normal with mean 170 and standard deviation 8. Approximately what fraction exceed 186? Answer as a decimal (3 places).
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