Proportion

1. Definition

• A proportion is a part-to-whole ratio that expresses the fraction of the population or sample with a certain characteristic.
• In statistics, it is usually the fraction of successes (or “yes” outcomes) out of the total number of observations.

Mathematically: $\hat{p} = \frac{x}{n}$

Where:

• $x$ = number of successes (e.g., people who answered “yes”)
• $n$ = total sample size
• $\hat{p}$​ = sample proportion (estimate of true population proportion $p$)

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2. Population Proportion vs Sample Proportion

• Population proportion (p): The true proportion in the entire population (usually unknown).
• Sample proportion ($\hat{p}$​): Calculated from sample data, used to estimate $p$.

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3. Examples

Example 1 – Voting Poll

• Out of 1,000 voters surveyed, 540 support Candidate A.

$\hat{p} = \frac{540}{1000} = 0.54$

Sample proportion = 54%.

If the true population proportion is $p = 0.55$, then $\hat{p}$​ is close to the truth.

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Example 2 – Quality Control

• Factory inspects 200 lightbulbs, finds 10 defective.

$\hat{p} = \frac{10}{200} = 0.05$

5% defective rate.

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4. Proportions in Inference

• Used in:
  • Confidence intervals for population proportion:
    • $CI = \hat{p} \pm Z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
  • Hypothesis tests:
    • One-sample proportion test → compare $\hat{p}$ to a hypothesized $p_0$​.
    • Two-proportion z-test → compare $\hat{p}_1$​ and $\hat{p}_2$​.

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5. Relation to Probability

• A proportion in a sample estimates a probability in the population.
• Example: If 30% of surveyed users clicked an ad, we estimate the probability of a random user clicking ≈ 0.30.

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6. Key Takeaways

• A proportion is simply the fraction of observations with a certain property.
• Sample proportion ($\hat{p}$) estimates the true population proportion (p).
• It’s central to survey analysis, A/B testing, medical studies, and quality control.

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In short:
A proportion is the ratio of successes to total observations ($\hat{p} = x/n$). It summarizes categorical data (like “yes/no” outcomes) and is used to estimate population probabilities, build confidence intervals, and perform hypothesis tests.