The Normal Distribution, Revisited
Introduced| English |
|---|
| normal distribution/ˈnɔːml ˌdɪstrɪˈbjuːʃn/ |
| z-score/zed skɔː/ |
| standard normal/ˈstændəd ˈnɔːml/ |
| statistical inference/stəˈtɪstɪkl ˈɪnfərəns/ |
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.
The standard normal, shaded by z
Areas under the curve give probabilities; z marks the cutoff.
A statistic is normal with mean 50 and SD 4. Find the z-score of the value 58.
z = (58 − 50)/4 = 2.
The standard normal distribution has...
By definition, mean 0 and SD 1.
The critical value z* for the middle 95% of the standard normal is about... (two decimals)
z* = 1.96 leaves 2.5% in each tail.
The z-score for a sample statistic uses the standard deviation of its sampling distribution.
Not the spread of one sample's raw data — the sampling distribution's SD.
Under a normal model, a statistic with a large |z| is...
Large |z| = far from center = surprising evidence.