🎲 Proportion

Proportion#

The fraction of a sample or population that has a given attribute.

Important

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What it is#

A proportion is a part-to-whole ratio — the fraction of a sample or population with a given characteristic, usually the fraction of successes (“yes” outcomes). The sample proportion estimates the true population proportion \(p\):

\[\hat{p} = \frac{x}{n},\]

with \(x\) the number of successes and \(n\) the sample size.

Population vs sample#

\(p\) is the true proportion in the whole population (fixed, usually unknown); \(\hat{p}\) is computed from a sample and estimates \(p\).

Examples#

In a poll, 540 of 1,000 voters back candidate A → \(\hat{p} = 0.54\) (true \(p\) might be 0.55). In quality control, 10 defective bulbs out of 200 → \(\hat{p} = 0.05\), a 5% defect rate.

In inference#

Proportions drive categorical inference: a confidence interval

\[\hat{p} \pm z\, \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}},\]

and hypothesis tests — a one-sample proportion test compares \(\hat{p}\) to a hypothesised \(p_0\), and a two-proportion z-test compares \(\hat{p}_1\) and \(\hat{p}_2\) (the workhorse of A/B testing).

Proportion ≈ probability#

A sample proportion estimates a population probability: if 30% of surveyed users clicked an ad, the probability a random user clicks is \(\approx 0.30\). This is why proportions sit at the centre of surveys, A/B tests, medical studies and quality control — they turn yes/no data into estimable probabilities.

Theme: Probability & Statistics Foundations · All terminology

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Mind map — connected ideas

True Conversion Rate · Standard Error (SE) · Probability · True Population Parameter · Frequentist · A/B Testing

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More in Probability & Statistics Foundations

Beta Distribution · Confidence Level · Correlation · Critical Value · Cumulative Distribution Function (CDF) · Frequentist · IID (Independent and Identically Distributed) · Likelihood · Margin of Error (MoE) · Mean · Median · Normal Distribution · Outlier · Population Proportion