Normal distribution N(μ, σ²): the bell curve and exam probabilities

The bell curve

A variable $X$ follows $\mathcal{N}(\mu, \sigma^2)$ if its density is bell-shaped, centered at $\mu$ (mean), with “width” related to $\sigma$ (standard deviation).

The 68-95-99.7 rule (memorize!)

For any $X \sim \mathcal{N}(\mu, \sigma^2)$:

Interval	Probability
$[\mu - \sigma, \mu + \sigma]$	≈ 68%
$[\mu - 2\sigma, \mu + 2\sigma]$	≈ 95%
$[\mu - 3\sigma, \mu + 3\sigma]$	≈ 99.7%

Concrete example

Adult heights follow $\mathcal{N}(170, 8^2)$ (mean 170 cm, sd 8 cm).

• About 68% of people are between 162 and 178 cm.
• About 95% between 154 and 186 cm.
• About 99.7% between 146 and 194 cm.

Calculator

For $P(a \leq X \leq b)$ with $X \sim \mathcal{N}(\mu, \sigma^2)$:

• Casio: Menu STAT → DIST → NORM → Ncd: between $a$ and $b$, $\mu$, $\sigma$.
• TI-83/84: 2nd → DISTR → normalcdf(a, b, μ, σ).

For $P(X \leq c)$: use $-10^{99}$ as lower bound.

Standard normal

Special case: $\mu = 0$, $\sigma = 1$, written $\mathcal{N}(0, 1)$.

Any $X \sim \mathcal{N}(\mu, \sigma^2)$ standardizes with:

$Z = \dfrac{X - \mu}{\sigma} \sim \mathcal{N}(0, 1)$

Lets you use the z-table if no calculator.

Common mistakes

1. Mixing $\sigma$ and $\sigma^2$: notation $\mathcal{N}(\mu, \sigma^2)$ uses variance, but 68-95-99.7 uses $\sigma$.
2. Forgetting it’s continuous: $P(X = k) = 0$. Always use intervals.
3. Thinking 95% = exactly 2σ: it’s 1.96σ exactly. The “2” is a shortcut.

?À toi de jouer

X ~ N(50, 4²). What's P(46 ≤ X ≤ 54)?

A.≈ 68%B.≈ 95%C.≈ 50%D.≈ 34%

Valider