정규 분포

The curve that rules the world

• Heights, IQ scores, measurement errors, exam results — all follow the same symmetric bell-shaped curve.
• The normal distribution 正态分布 is the most important distribution in statistics. Master it and you can model almost any continuous variable 连续变量.

The normal distribution

• The normal distribution models a continuous variable with a symmetric bell shape: $X \sim N(\mu, \sigma^2)$.
• The mean $\mu$ determines the centre; the standard deviation $\sigma$ determines the spread.

Example

The 68-95-99.7 rule. About 68% of values lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.

The normal curve is symmetric about the mean; about 68% of the area lies within one standard deviation.

Standardising

• To use the tables, standardize 标准化 to $Z \sim N(0, 1)$:

• Then $P(X < x) = P\!\left(Z < \dfrac{x - \mu}{\sigma}\right)$, read from the $\Phi$ table.

A Galton board: balls fall through rows of pegs and pile up into the bell-shaped normal distribution

Worked example

• $X \sim N(50, 16)$ (so $\mu = 50$, $\sigma = 4$). Find $P(X < 58)$.
• $Z = \dfrac{58 - 50}{4} = 2$.
• $P(Z < 2) = \Phi(2) = 0.9772$.

A Galton board produces the bell-shaped normal distribution

Approximating 近似 the binomial

• For large $n$, the normal distribution approximates the binomial: $B(n, p) \approx N(np, np(1-p))$.
• Because you replace a discrete variable by a continuous one, apply a continuity correction 连续性校正 (adjust by 0.5).

For large n the binomial bars closely follow the smooth normal curve N(np, np(1−p))

The binomial distribution B(10, 0.3); its mean is np = 3

• The normal models a continuous random variable; use standardisation $Z=(X-\mu)/\sigma$, and as an approximation to the binomial.

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