The Central Limit Theorem
Introduced| English | Meaning |
|---|---|
| Central Limit Theorem/ˈsentrəl ˈlɪmɪt ˈθɪərəm/ | A theorem stating that, under suitable conditions, the sampling distribution of a mean approaches a normal distribution as sample size increases. |
| population distribution/ˌpɒpjʊˈleɪʃn ˌdɪstrɪˈbjuːʃn/ | population distribution |
The most important theorem
- The Central Limit Theorem (CLT) 中心极限定理 is the engine behind most inference.
- It says: the sampling distribution of the sample mean $\bar{x}$ is approximately normal for a large sample.
- This holds no matter what the population's own shape is.
- A near-magical result — normality emerges from averaging.
Works even for weird populations
- The population distribution 总体分布 can be skewed, bimodal, or bizarre.
- Yet the distribution of $\bar{x}$ over many samples still turns out roughly normal.
- Averaging "smooths out" the population's odd shape.
- Only a large enough sample is needed — not a normal population.
The n ≥ 30 guideline
- A common rule of thumb: $n \ge 30$ is usually enough for the CLT to kick in.
- For a roughly symmetric population, smaller $n$ can already be fine.
- For a very skewed population, you may want $n$ even larger.
- The guideline is practical, not an exact law.
Bigger n, better normality
- The larger the sample size, the more nearly normal the sampling distribution of $\bar{x}$.
- Larger $n$ also makes the sampling distribution narrower (less spread).
- So big samples give estimates that are both more normal and more precise.
- This is why sample size matters so much in study design.
The CLT is about the sampling distribution of $\bar{x}$, not the data itself. A large sample does not make the population or the raw data normal — it makes the distribution of the sample mean approximately normal. If the population is already normal, $\bar{x}$ is exactly normal for any $n$; the CLT only matters when it isn't.
Incomes are strongly right-skewed (a few very high earners).
- One sample's incomes look skewed — the CLT says nothing about that.
- But average the incomes of $n = 50$ people, and repeat: those $\bar{x}$ values form a roughly normal distribution.
- So we can use a normal model for $\bar{x}$ even though incomes aren't normal.
Raw observations and repeated sample means
$E(\bar X)=\mu,\quad \mathrm{SD}(\bar X)=\sigma/\sqrt n$
The population stays the same while the distribution of sample means changes. These are illustrative independent models; a large n does not make raw data normal.
The Central Limit Theorem says the sampling distribution of $\bar{x}$ is approximately normal for a large sample size, even when the population distribution isn't normal. A common guideline is $n \ge 30$; a larger sample makes the sampling distribution both more normal and narrower.
Write from a sample of 40
- A population is strongly skewed.
- A random sample of size 40 is used to study the mean. The usual guideline quoted in the lesson is n at least 30.
Does this sample size meet the guideline for the mean?
The central limit theorem applies to the sample mean as n grows.
Does n equal to 40 meet the guideline?
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40 is above the lesson guideline of 30.
The Central Limit Theorem says that for a large sample, the sampling distribution of x-bar is...
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The CLT delivers approximate normality of x-bar for large n.
The CLT requires the population itself to be normal.
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It works even for skewed populations — that's the point.
What sample size is the common rule-of-thumb minimum for applying the CLT?
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n ≥ 30 is the usual guideline.
As the sample size increases, the sampling distribution of x-bar becomes...
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Bigger n → more normal and less spread.
The CLT makes the raw data values normal, not just the distribution of x-bar.
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It's about the sampling distribution of the mean, not the raw data.