| English | 意味 |
|---|---|
| normal distribution/ˈnɔːml ˌdɪstrɪˈbjuːʃn/ | 正規分布 |
| z-score/zed skɔː/ | z点 |
| standard normal/ˈstændəd ˈnɔːml/ | 標準正規分布 |
| statistical inference/stəˈtɪstɪkl ˈɪnfərəns/ | 統計的推論 |
The bell curve returns
- Many sampling distributions are well modeled by the normal distribution 正态分布.
- It's the familiar symmetric bell, described by a mean and a standard deviation.
- When a statistic's sampling distribution is normal, we can compute probabilities for it.
- This is why the normal model is the workhorse of inference.
z-scores for a statistic
- A $z$-score 标准分数 says how many standard deviations a value sits from the mean:
-
$$z = \frac{\text{statistic} - \text{mean}}{\text{standard deviation}}$$
- For a sample statistic, the "standard deviation" is that of its sampling distribution.
- The $z$-score converts any normal value onto one common scale.
The standard normal
- The standard normal 标准正态 distribution has mean $0$ and standard deviation $1$.
- Every $z$-score lives on it, so one table (or calculator) handles all normal problems.
- Probabilities: area under the curve left/right of a $z$.
- Critical values: the $z$ that cuts off a chosen tail area (e.g. $z^* = 1.96$ for the middle $95\%$).
Why it matters for inference
- Inference asks: "how surprising is my sample, if a claim were true?"
- The normal model turns that into an area (a probability) under the bell.
- A statistic far out in the tail (large $|z|$) is surprising evidence.
- Every confidence interval and test in Units 6–9 runs on this machinery.
The $z$-score's denominator is the standard deviation of the sampling distribution, not the spread of one sample's raw data. Mixing these up is a classic slip. And remember the empirical rule benchmarks — about $68\%$ within $1$ SD, $95\%$ within $2$, $99.7\%$ within $3$ — for a quick sense of how extreme a $z$ is.
A statistic is normal with mean $50$ and SD $4$. You observe a value of $58$.
- $z = \dfrac{58 - 50}{4} = 2$ — two standard deviations above the mean.
- By the empirical rule, only about $2.5\%$ of values exceed $z = 2$.
- So a value of $58$ is fairly surprising under this model.
The normal distribution models many sampling distributions. A $z$-score $z = \frac{\text{stat} - \text{mean}}{\text{SD}}$ maps a value onto the standard normal (mean $0$, SD $1$), where probabilities are areas and critical values cut off tail areas. This normal machinery powers all statistical inference.
標準正規分布、zによって塗りつぶされる部分
曲線下の面積は確率を示し、zは閾値を示す。
平均50、SD 4の正規分布を持つ統計がある。値58のzスコアを求めよ。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
z = (58 − 50)/4 = 2.
標準正規分布には…
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
定義により、平均0、SD 1である。
中央部95%に対する標準正規分布の臨界値 z* はおおよそ…（小数点以下2桁）
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
z* = 1.96 は両端にそれぞれ2.5%を残す。
サンプル統計子のzスコアには、その標本分布の標準偏差を用いる。
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
単一サンプルの生データのばらつきではなく、標本分布のSDである。
正規モデルにおいて、|z|が大きい統計子とは…
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
大きな |z| ＝ 中心から遠い ＝ 驚異的な証拠。