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 |
| z-score/zed skɔː/ | 标准分 | biāo zhǔn fēn |
State the normal assumption
- A normal distribution 正态分布 is a continuous symmetric bell-shaped model, defined by mean and positive standard deviation.
- Those summaries alone do not prove that actual scores, heights or errors are normal. Inspect the data and the proposed modelling assumptions before using normal areas.
How many numbers are needed to describe a normal distribution completely? · 完全描述一个正态分布需要几个数?
The mean fixes where it sits and the standard deviation fixes how wide it is. · 平均数决定它的位置,标准差决定它的宽度。
Separate approximate rules from table areas
- Under a normal model, about 68%, 95% and 99.7% fall within one, two and three standard deviations of the mean respectively.
- These rounded empirical-rule values are estimates. With a cumulative table, use its values consistently: $\Phi(2)=0.9772$ gives an upper tail of 0.0228 rather than the rough 0.025.
Under a normal model, approximately what percentage lies within two standard deviations of the mean, using the rounded empirical rule? · 在正态分布模型下，使用简化经验法则估算均值左右两个标准差范围内大约包含百分之多少的数据？
The rounded empirical rule gives about 95%. A cumulative normal table gives a more precise area, so do not treat this rounded value as exact. · 简化经验法则给出约95%。累积正态分布表提供更精确的面积，故不应将此近似值视为精确值。
Standardise a bound and select its area
- A z-score 标准分 is $z=(x-\mu)/\sigma$. Its sign distinguishes a value below or above the model mean.
- Use $1-\Phi(z)$ for an upper tail and subtract two cumulative values for an interval. Equal z-scores compare relative positions; they do not establish equal test content or learner ability.
A test has mean 62 and standard deviation 8. What is the z-score of 78? · 某测试均值为62，标准差为8。分数78对应的z分数是多少？
(78 − 62) ÷ 8 = 2, so the score is two standard deviations above the mean. · (78 − 62) ÷ 8 = 2,即高出平均数两个标准差。
A stated normal score model. With mean 62 and standard deviation 8, threshold 78 has $z=(78-62)/8=2$. The approximate empirical rule gives an upper tail near 2.5%; the supplied table gives $P(X>78)=1-0.9772=0.0228$, or 2.28%. Label which method and precision are requested.
For the stated normal model with mean 62 and standard deviation 8, use the rounded 95% empirical rule to estimate the percentage above 78. · 针对均值62、标准差8的正态分布模型，使用简化95%经验法则估算高于78分的百分比。
78 is two standard deviations above the mean. The rounded rule leaves 5% outside the central interval, split symmetrically into about 2.5% in each tail. A table gives about 2.28% instead. · 78位于均值上方两个标准差处。简化法则显示中心区间外占5%，对称分布于两尾，每尾约2.5%。查表得约为2.28%。
Put a normal-distribution question in the order that avoids a tail error. · 把正态分布题按"避免尾部方向出错"的顺序排列。
The sketch is what tells you whether the answer should be small or large before you trust the arithmetic. · 在相信算术之前,是草图告诉你答案应该很小还是很大。
An expected count need not be observed exactly. If a model probability is 0.0228 in 500 trials, expected count is $E=Np=11.4$. Any actual count is an integer and can vary. Sheet 2.8 keeps model probabilities, expected counts and observed findings separate.
The empirical rule assumes a normal model. It is not a rule for every data set with a mean and standard deviation. A strictly positive real quantity may sometimes be approximated by a normal model over a relevant range, but its impossible negative tail still needs consideration.
The empirical rule can be applied to any data set. · 经验法则可以用在任何数据集上。
It assumes a normal shape. Income and other skewed data break it, and the answer looks confident and is wrong. · 它以正态为前提。收入等偏态数据会破坏它,而算出的答案看似笃定,实则错误。