The Normal Distribution
Introduced| English |
|---|
| normal distribution/ˈnɔːml ˌdɪstrɪˈbjuːʃn/ |
| empirical rule/emˈpɪrɪkl ruːl/ |
| standard normal/ˈstændəd ˈnɔːml/ |
The famous bell curve
- Many real variables — heights, measurement errors, test scores — follow a bell-shaped pattern.
- The normal distribution 正态分布 is the idealized bell curve, symmetric about its center.
- It's completely described by two numbers: the mean $\mu$ (center) and standard deviation $\sigma$ (spread).
- Because so much falls into this shape, the normal model is the workhorse of statistics.
A normal distribution is completely described by its...
Two numbers: $\mu$ (center) and $\sigma$ (spread).
The empirical rule (68–95–99.7)
- For a normal distribution, the empirical rule 经验法则 tells you how much data falls within a few SDs of the mean:
- About $68\%$ within $1$ standard deviation of the mean.
- About $95\%$ within $2$ SDs; about $99.7\%$ within $3$ SDs.
- These three numbers let you estimate proportions without a calculator.
The normal bell curve
Shade a normal curve by z-score — about 68% lies within 1 SD, 95% within 2, 99.7% within 3.
About what percent of a normal distribution lies within $1$ standard deviation of the mean?
The empirical rule: about $68\%$.
About what percent lies within $2$ standard deviations of the mean?
About $95\%$ within $2$ SDs.
z-scores and the standard normal
- Convert any value to a $z$-score $z=\frac{x-\mu}{\sigma}$ — its distance from the mean in SDs.
- The standard normal 标准正态 distribution has mean $0$ and SD $1$; every normal curve becomes it after standardizing.
- Use a $z$-table (or technology) to turn a $z$-score into a proportion or percentile.
- This is how you find "what percent scored below $x$."
For $\mu=170$, $\sigma=8$, find the $z$-score of $x=186$.
$z=(186-170)/8=2$.
The standard normal distribution has mean $0$ and standard deviation ____.
Standardizing gives mean $0$, SD $1$.
Is a normal model reasonable?
- Not everything is normal — check before assuming it.
- A roughly symmetric, bell-shaped, single-peaked graph supports a normal model.
- Strong skew, multiple peaks, or heavy outliers mean a normal model is a poor fit.
- Assess the data's shape first; the empirical rule and $z$-tables only apply if it's approximately normal.
A normal model is a poor fit when the data are...
Skew or multiple peaks break the normal assumption.
The empirical rule and $z$-table proportions apply only to (approximately) normal distributions — don't use them on strongly skewed data. And the empirical rule is 68–95–99.7 (for $1$, $2$, $3$ SDs), in that order; mixing up the percentages is a common slip. Always standardize with $z=\frac{x-\mu}{\sigma}$ before reading a table.
Heights are normal with $\mu=170$ cm, $\sigma=8$ cm. What percent are between $162$ and $178$ cm?
- $162=170-8$ and $178=170+8$ — exactly $1$ SD below and above the mean.
- By the empirical rule, about $68\%$ of heights fall within $1$ SD.
- So roughly $68\%$ are between $162$ and $178$ cm.
The normal distribution is a symmetric bell curve set by its mean $\mu$ and standard deviation $\sigma$. The empirical rule (68–95–99.7) gives proportions within $1$/$2$/$3$ SDs. Standardize with a $z$-score $z=\frac{x-\mu}{\sigma}$ and use the standard normal for exact proportions — but only when a normal model reasonably fits the data.