The Normal Distribution · 正态分布
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| normal distribution/ˈnɔːml ˌdɪstrɪˈbjuːʃn/ | 正态分布 | zhèng tài fēn bù |
| empirical rule/emˈpɪrɪkl ruːl/ | 经验法则 | jīng yàn fǎ zé |
| standard normal/ˈstændəd ˈnɔːml/ | 标准正态 | biāo zhǔn zhèng tài |
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.
著名的钟形曲线
- 许多真实变量——身高、测量误差、考试分数——遵循钟形模式。
- 正态分布是理想化的钟形曲线,关于其中心对称。
- 它完全由两个数描述:平均数 $\mu$(中心)和标准差 $\sigma$(分散)。
- 因为如此多数据落入这种形状,正态模型是统计学的主力。
A normal distribution is completely described by its... · 正态分布完全由其...描述
Two numbers: $\mu$ (center) and $\sigma$ (spread). · 两个数：$\mu$ (中心) 和 $\sigma$ (离散度)。
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.
经验法则(68–95–99.7)
- 对正态分布,经验法则告诉你有多少数据落在平均数几个标准差内:
- 约 $68\%$ 在平均数 $1$ 个标准差内。
- 约 $95\%$ 在 $2$ 个标准差内;约 $99.7\%$ 在 $3$ 个标准差内。
- 这三个数让你不用计算器就能估计比例。
The normal bell curve · 正态钟形曲线
Shade a normal curve by z-score — about 68% lies within 1 SD, 95% within 2, 99.7% within 3. · 通过 z 分数着色正态曲线——约 68% 位于 1 SD 内，95% 位于 2 SD 内，99.7% 位于 3 SD 内。
About what percent of a normal distribution lies within $1$ standard deviation of the mean? · 正态分布中约有多少百分比位于均值 $1$ 个标准差之内？
The empirical rule: about $68\%$. · 经验法则：约 $68\%$。
About what percent lies within $2$ standard deviations of the mean? · 约有多少百分比位于均值 $2$ 个标准差之内？
About $95\%$ within $2$ SDs. · 约 $95\%$ 位于 $2$ 个 SD 内。
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 $x$."
标准分与标准正态
- 把任何值转成标准分 $z=\frac{x-\mu}{\sigma}$——它离平均数的距离,以标准差计。
- 标准正态分布的平均数为 $0$、标准差为 $1$;每条正态曲线标准化后都变成它。
- 用 $z$ 表(或技术工具)把标准分变成比例或百分位数。
- 这就是你如何求"有百分之几低于 $x$"。
For · 支持 $\mu=170$, $\sigma=8$, find the $z$-score of $x=186$. · 对于 $\mu=170$, $\sigma=8$，找到 $z$的得分 $x=186$.
$z=(186-170)/8=2$.
The standard normal distribution has mean $0$ and standard deviation . · 标准正态分布的均值为 $0$，标准差为。
Standardizing gives mean $0$, SD $1$. · 标准化后得到均值 $0$，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.
正态模型合理吗?
- 并非一切都正态——假设之前先检查。
- 大致对称、钟形、单峰的图支持正态模型。
- 强烈偏斜、多峰或严重离群值意味着正态模型拟合不佳。
- 先评估数据的形状;经验法则和 $z$ 表只在它近似正态时适用。
A normal model is a poor fit when the data are... · 当数据为...时，正态模型拟合不佳
Skew or multiple peaks break the normal assumption. · 偏态或多峰破坏正态假设。
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.
经验法则和 $z$ 表比例仅适用于(近似)正态分布——别把它们用在强烈偏斜的数据上。而经验法则是 68–95–99.7(对应 $1$、$2$、$3$ 个标准差),按此顺序;搞混这些百分比是常见失误。读表前永远用 $z=\frac{x-\mu}{\sigma}$ 标准化。
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.
身高服从正态分布,$\mu=170$ cm、$\sigma=8$ cm。有百分之几在 $162$ 到 $178$ cm 之间?
- $162=170-8$、$178=170+8$——恰好在平均数下方和上方 $1$ 个标准差。
- 由经验法则,约 $68\%$ 的身高落在 $1$ 个标准差内。
- 所以大约 $68\%$ 在 $162$ 到 $178$ cm 之间。
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.
正态分布是由平均数 $\mu$ 和标准差 $\sigma$ 确定的对称钟形曲线。经验法则(68–95–99.7)给出 $1$/$2$/$3$ 个标准差内的比例。用标准分 $z=\frac{x-\mu}{\sigma}$ 标准化,并用标准正态求精确比例——但仅当正态模型合理拟合数据时。