The normal distribution
The normal distribution
- The normal distribution models a continuous variable with a symmetric bell shape: $X \sim N(\mu, \sigma^2)$.
- To use the tables, standardize to $Z \sim N(0, 1)$:
$$Z = \frac{X - \mu}{\sigma}$$
- Then $P(X < x) = P\!\left(Z < \dfrac{x - \mu}{\sigma}\right)$, read from the $\Phi$ table.
Practice
X ~ N(50, 4) means μ = 50, σ = 2. Standardize x = 56: what is Z = (x − μ)/σ?
Z = (56 − 50)/2 = 6/2 = 3.
Practice
The normal distribution curve is:
It is the symmetric bell-shaped curve centred on the mean μ.
Approximating the binomial
- For large $n$, the normal distribution approximates the binomial.
- Because you replace a discrete variable by a continuous one, apply a continuity correction (adjust by 0.5).
Practice
When the normal distribution approximates the binomial, a continuity correction of 0.5 is applied.
Replacing a discrete variable with a continuous one needs a ±0.5 continuity correction.
You've got it
Key idea
- $X \sim N(\mu, \sigma^2)$; standardize with $Z = \dfrac{X - \mu}{\sigma}$
- read probabilities as areas from the $\Phi$ table for $Z \sim N(0,1)$
- the normal approximates the binomial for large $n$ (use a continuity correction)