Central Limit Theorem

As explained in Causal Inference for Data Science

The central limit theorem states that, under suitable independence and finite-variance conditions, a normalized sample mean approaches a normal distribution as sample size grows.

The central limit theorem explains why a sample mean becomes approximately normal as the sample grows, even when the observations’ original distribution is not normal. It requires identically distributed, independent observations with a common finite variance. The sample mean approaches the population mean at rate 1/√n, and its standard deviation is σ/√n. section 8.1.2 section 8.2.2

The result matters because it connects large-sample behavior to practical statistical estimation:

• It provides a benchmark for estimator speed: root-n convergence is achieved by the sample mean and cannot be improved by another unbiased estimator. section 8.1.2
• It supports analytical confidence intervals by approximating the estimator’s distribution with a normal distribution when the sample is large. section 8.2.2
• It describes an asymptotic approximation, so independence, a common distribution, finite variance, and a sufficiently large sample remain important. section 8.1.2 section 8.2.2

From a sample mean to an interval

Suppose x₁, …, xₙ are independently observed. Their sample average and sample standard deviation can be combined with a standard-normal critical value to calculate a confidence interval. This analytical route is faster than bootstrapping when an appropriate formula is available, though such formulas are not available for every problem. section 8.2.2

Figure 8.4 Convergence rate for sequences 1/n and 1/√n. We can see that 1/n converges faster to 0 than 1/√n. The convergence rate of an algorithm tells us how fast this algorithm converges to its objective. In statistics and machine learning, the faster an estimate converges, the less data we need.

Compare 1/n with 1/√n to see why convergence rate matters for the amount of data needed by an estimator. section 8.1.2

How Causal Inference for Data Science covers it

How deep it goes

Defines assumptions, normal approximation, standard deviation, and convergence rate, while deferring the technical proof.

Examples

Uses independently observed samples, sample averages, standard deviations, normal critical values, and convergence-rate comparisons.

What it assumes

Assumes familiarity with sample means, expectation, variance, independence, normal distributions, and confidence intervals.

What you can do after

Apply it via normal approximations to assess sample-mean uncertainty and construct analytical confidence intervals.

How it connects

Connects sample means, normal distributions, confidence intervals, root-n consistency, efficiency, and Double Machine Learning.

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Also explained in

• 

  liveBookmanning.com
  Grokking StatisticsA theorem stating that, under suitable conditions, the distribution of sample means approaches a normal distribution as sample size increases, regardless of the population distribution's shape.
• 

  liveBookmanning.com
  Experimentation for EngineersA theorem stating that suitably normalized sums or averages of many observations tend toward a normal distribution.

Compare how 3 books explain Central Limit Theorem →

Related concepts

• normal distributionproduces · A normal distribution is a continuous probability distribution with a symmetric, bell-shaped density determined by its mean and variance.
• sampling distributionproduces · A sampling distribution is the probability distribution of a statistic across repeated samples drawn from a specified population or process.
• meanuses · A mean is a measure of central tendency, calculated as the arithmetic average of values or, for a random variable, as the probability-weighted average of its possible values.
• student's t-distributioncontrasts with · Student’s t-distribution is a family of continuous probability distributions with heavier tails than the normal distribution, used in inference when variability is estimated from data.

This explainer was generated by Manning's knowledge graph from the text of Causal Inference for Data Science. The section links point to the parts of the book it draws on.