Separate concentration from normal approximation

Try

An average of many measurements often looks steadier than a single measurement. But “steadier” and “approximately normal” are different claims. Imagine a strongly skewed population of waiting times. Increasing sample size does not transform the individual waiting times into bell-shaped data, even when the distribution of their average becomes approximately normal.

Predict

Predict whether a running average must get closer to the population mean after every additional observation. A new extreme value can temporarily move it away. Then predict which quantity the central limit theorem standardizes: the original measurements or the sample mean.

Experiment

Assume independent identically distributed measurements with mean10 and SD6, both in minutes. For n=100, the sample mean has expectation10 and SD6/√100=0.6. An observed mean11.2 is two standard errors above10. If a normal approximation is justified, a supplied standard-normal upper-tail area at2, about0.0228, approximates the chance of a mean at least11.2 under this model.

Live experiment: precision is not the same as lack of bias

In this synthetic population, 40% have a specified characteristic. Half belong to a subgroup with 70%, and the other half to a subgroup with 10%. Predict what happens if every sample is drawn only from the first subgroup, then increase the sample size.

Sampling method Uniform draws from the entire populationDraw only from the 70% subgroup Independent draws per sample 125102040 Reset experiment

Every model draw is with replacement, independently and uniformly within the chosen sampling group. Filled bars give exact-model probabilities. You can separately generate seeded trials below and compare their empirical frequencies with these probabilities.

Expected sample proportion: 0.4. Bias relative to the population: 0.0000.

Standard deviation of the sample proportion: √(0.4 × (1 − 0.4) / 5) ≈ 0.2191.

Probability of a result from 0.3 through 0.5, inclusive: ≈ 34.56%.

The expected sample proportion is0.4 for every sample size. Individual samples still vary. The probability inside a fixed interval need not increase at every small change of n because only the fractions k/n are possible.

Inspect exact-model probabilities

Successes k	Proportion k/n	Probability
0	0.0000	7.78%
1	0.2000	25.92%
2	0.4000	34.56%
3	0.6000	23.04%
4	0.8000	7.68%
5	1.0000	1.02%

Run a reproducible sampling investigation

Define a success as drawing an item with the characteristic. Each trial draws n items and records the proportion. Predict how often that proportion will fall from 0.3 through 0.5, then run 100 trials. Compare the simulated fraction with the exact probability above.

Simulation seed Run 100 simulated samples

The simulation uses a deterministic pseudorandom generator to approximate independent model draws. It is not a new real-world sample and cannot repair an unrealistic sampling model. Changing settings hides mismatched simulated results. Reset or navigation clears the temporary run.

Probabilities are calculated from the binomial model and displayed with rounding; a displayed0.00% may be small but nonzero. Sampling without replacement needs a different model when the sampling fraction is material. Settings are temporary and do not count as assessment evidence.

Use the widget’s representative mode to compare exact proportion distributions at small and larger n. Notice discrete bars rather than a continuous density. For p=0.4 and n=40, expected successes16 and failures24 support a common normal-approximation check, but the displayed masses remain exact. Run the seeded sampling simulation at both sample sizes. Its empirical frequencies need not equal the exact bars; distinguish the number of observations per sample from the number of repeated samples. Switch to biased selection and explain why concentration can reinforce the wrong population target.

Understand

A law of large numbers says, under appropriate assumptions, the sample average concentrates near its expected target as n grows. In an independent identically distributed finite-variance setting, the varianceσ²/n shrinks. More precisely, for any fixed positive tolerance ε, the probability that x̄ differs from μ by more than ε tends to zero. This is convergence in probability, not a guarantee of monotonic improvement in a realized sequence.

The classical independent identically distributed central limit theorem assumes a finite positive variance. It says the standardized mean (x̄−μ)/(σ/√n) approaches the standard normal distribution as n grows. The centering and scaling matter: the unstandardized mean instead becomes more concentrated near μ. The theorem describes repeated samples, not the shape of one sample’s raw observations.

If the original population is normal and sampling is independent, its mean is exactly normal for every positive n. For nonnormal populations, normality is an approximation whose quality depends on skew, tails, sample size and the probability region being approximated. A rule such as n≥30 is a heuristic, not a universal guarantee. Rare-tail probabilities can require more care than central probabilities.

Dependence changes the standard-error calculation and can invalidate the familiar theorem’s assumptions. Infinite-variance models also require different theory. Repeating the same observation one hundred times does not provide the information of one hundred independent observations. Random selection, time structure and experimental design therefore remain essential.

For a binomial proportion, checking both np and n(1−p) helps diagnose whether the normal shape is plausible; values of at least10 are a common introductory guideline. Exact binomial calculations remain available when appropriate. The choice of p for checking a proposed null differs from using an estimated p in other inferential settings, which the next sequence will treat explicitly.

Apply

Write separate one-sentence interpretations of LLN and CLT without using the word “normal” in the first. Then explain why a biased sampling process can satisfy both theorems relative to its own distribution while failing to estimate the intended population. Theorems do not repair a mismatch between the statistical target and the collection mechanism.

Continue

The final foundations lesson audits readiness for inference. A valid approximation is one ingredient, not permission to skip design, parameter definitions or contextual interpretation.