The chapter explains how the central limit theorem (CLT) enables strong inferences from limited data. Wheelan makes up an outlandish story to demonstrate how sample means signal a population’s identity. A bus full of marathoners goes missing in a city that is also hosting a sausage festival ferrying visitors by bus. One errant bus is found, but since its passengers’ average weight is over 220 pounds—an unlikely mean for runners—it is most likely not the lost marathon bus.

The CLT holds that means of large, random samples from any population are distributed normally around the true population mean, regardless of the population’s shape. This allows four core inferences: predicting sample characteristics from known populations, inferring population parameters from samples (as in polling), testing whether a sample likely came from a given population, and assessing whether two samples likely share a common source.

For example, to survey US household income, even with a skewed distribution, repeated samples of 1,000 households produce a normal distribution of sample means around the true mean. With at least about 30 observations, the CLT’s normal approximation typically applies. The standard error captures how sample means vary: It shrinks as sample size grows and expands with greater population dispersion. Consequently, about 68% of sample means fall within one standard error of the population mean, 95% within two, and 99.7% within three.

The bus example applies these tools. If the mean of marathoner population weight is 162 pounds, with a standard deviation of 36, then the standard error for a 62-person sample is roughly 4.6. A bus full of people weighing an average of 194 pounds is over three standard errors from the mean, making it extremely unlikely to be made up of runners. A colleague notes that in real life, logic also matters (buses rarely go missing, so any missing bus is most likely the right one). However, the scenario clarifies how the CLT quantifies uncertainty.

In college, Wheelan spent most of a semester blowing off one of his classes. Then, when other time commitments shifted, he decided to really learn the material for the final. His dramatic improvement on the exam prompted his professor to question whether he had cheated. This anecdote illustrates statistical inference: weighing how likely observed outcomes are under competing explanations. Statistics rarely proves anything with certainty; rather, it gauges probabilities. For example, rolling 10 consecutive sixes while gambling is astronomically unlikely by chance, making cheating plausible, yet rare events do occur (e.g., an individual has been struck by lightning multiple times).

Inference starts with a null hypothesis (meaning that a given treatment has no effect on results) and an alternative hypothesis. Researchers reject the null when observed results would be very unlikely if the null hypothesis were true. In medical clinical trials, a null hypothesis might posit that a new drug matches a placebo; if the treatment group then shows starkly lower infection rates compared to the control group, the study is justified in rejecting the null. This can help catch cheating. For instance, in the Atlanta school scandal, wrong-to-right erasures were 20-50 standard deviations above norms, meaning they were so improbable that investigators concluded cheating had occurred.

A common threshold for rejecting the null is .05 (5%). In the bus-weight scenario with a population mean of 162 pounds and a standard error of 4.6, a bus population averaging 136 pounds falls well below the 95% range, yielding a p-value below .0001 and justifying rejection of the null.

Wheelan notes that statistical significance does not imply causation or meaningful magnitude. A study in the Archives of General Psychiatry found larger average brain volumes among children with autism (p = .002), with nonoverlapping 95% confidence intervals. The correlation does not clarify the causal relationship, but does invite further study. Conversely, a 2011 paper claiming evidence of extrasensory perception met the .05 threshold but failed the standard of extraordinary evidence for extraordinary claims. Its non-reproducible results were the result of luck. 

Choosing a significance level involves trade-offs between Type I errors (false positives) and Type II errors (false negatives), with different domains—spam filtering, cancer screening, counterterrorism—prioritizing errors differently.

A late-2011 New York Times/CBS poll reported widespread public anxiety, including low approval of Congress and polarized views on President Obama. Such findings rely on polling, which applies the central limit theorem: a large, representative sample mirrors the population. The “margin of error” is the 95% confidence interval around a sample proportion; ± 3% sampling error means that in 95 of 100 similar polls, results fall within three points of the true value.

Because many polling questions estimate proportions, the relevant standard error depends on both the proportion and sample size; larger samples yield narrower margins. An exit poll of 500 voters showing 53% for one candidate has a margin of ± 2% and an approximate confidence level of 68%. Raising confidence to 95% widens the band, sometimes preventing a clear call. Increasing the sample to 2,000 tightens the ± 2% margin to 95% confidence, allowing more decisive inference.

Methodology matters. Representative sampling avoids self-selection biases common in call-in or online polls: Professional pollsters use random digit dialing of numbers that include cell phones, randomly select one adult per household, and make repeated callbacks to reduce nonresponse bias. Question wording also shapes results. Gallup finds over 60% of respondents support the death penalty for murder, but this number drops significantly when the question offers life without parole as an alternative. Pollsters use split-sample tests to detect such effects and avoid leading language.

Finally, respondents may misreport details about their lives or opinions, particularly on sensitive topics. For example, checking voting records shows that many of those polled overreport their voting frequency. This is why pollsters screen for likely voters instead of simply voting-age adults. Interestingly, the National Opinion Research Center’s 1995 “Sex Study,” designed to inform HIV/AIDS research, achieved a nearly 80% response and found less sexual activity than commonly assumed. 

Ultimately, the challenges of polling are twofold: first, drawing a proper sample, and second, eliciting accurate responses. The statistics calculations are straightforward, but execution and interpretation are critical.