---
title: "Sampling distribution of proportion; parameter vs statistic"
subject: "Probability and Statistics"
chapter: 4
key: "semester_5/probability_and_statistics:parameter-vs-statistic-sampling-proportion"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/parameter-vs-statistic-sampling-proportion
site: BCT Engineering
---

# Sampling distribution of proportion; parameter vs statistic

Probability and Statistics · Chapter 4: Sampling Distribution

- Asked in: 2068 Chaitra, 1+3 marks
- Video: [Statistic vs Parameter & Population vs Sample](https://www.youtube.com/watch?v=Mb9BuEkbaHQ) — The Organic Chemistry Tutor
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/parameter-vs-statistic-sampling-proportion
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-4

## Question

What do you mean by the sampling distribution of sample proportion? Distinguish between parameter and statistics.

## Short answer

The sampling distribution of sample proportion p̂ is the distribution of p̂ = X/n from all samples of size n, with mean p and SE √(pq/n). A parameter is a fixed population measure (e.g., μ, p), while a statistic is a sample-based variable (e.g., x̄, p̂) used to estimate it.

## Answer

The **sampling distribution of the sample proportion** is the probability distribution of the values of p̂ = X/n obtained from all possible random samples of size *n* drawn from a population in which the proportion of "successes" is *p*.

For a population proportion *p* and q = 1 − p:

- Mean of p̂: E(p̂) = p
- Standard error: σ(p̂) = √(pq/n) (multiply by √((N−n)/(N−1)) for a finite population when n/N > 0.05)
- Shape: if *np* ≥ 5 and *nq* ≥ 5 (large *n*), p̂ is approximately normal, so Z = (p̂ − p)/√(pq/n) ~ N(0, 1).

For example, if 30% of the components in a lot are defective (p = 0.3) and samples of n = 100 are taken, p̂ has mean 0.3 and standard error √(0.3×0.7/100) = 0.0458.

#### Parameter and statistic

| Basis | Parameter | Statistic |
|---|---|---|
| Meaning | Numerical measure describing the **population** | Numerical measure computed from a **sample** |
| Value | Fixed and usually unknown | Known, changes from sample to sample |
| Nature | Constant | Random variable with its own sampling distribution |
| Symbols | μ, σ, σ², p, ρ | x̄, s, s², p̂, r |
| Use | What we wish to know | Used to estimate or test the parameter |

Example: the mean life μ of all bulbs made by a factory is a parameter; the mean x̄ of 50 tested bulbs is a statistic used to estimate μ.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The sampling distribution of the sample proportion p̂ is:
   A. The distribution of p in the population
   B. The distribution of p̂ = X/n over all possible samples of size n
   C. The distribution of X in one sample
   D. The distribution of n over samples
   Answer: B. It is the probability distribution of p̂ values from all possible random samples of size n. (Key point: Sampling distribution of proportion)

2. For p = 0.3 and n = 100, the standard error of p̂ is:
   A. 0.3
   B. 0.21
   C. 0.0021
   D. 0.0458
   Answer: D. SE = √(pq/n) = √(0.3 × 0.7/100) = 0.0458. (Key point: Standard error of p̂)

3. When is p̂ approximately normal?
   A. Always, for any n
   B. When np ≥ 5 and nq ≥ 5
   C. Only when p = 0.5
   D. When n < 30
   Answer: B. Both np and nq should be at least 5 for a normal approximation. (Key point: Normal approximation)

4. Which statement correctly distinguishes parameter and statistic?
   A. A parameter is computed from a sample and varies
   B. A statistic describes the population and is fixed
   C. A parameter describes the population and is fixed; a statistic is computed from a sample and varies
   D. They are the same except for symbols
   Answer: C. Parameters (μ, σ, p) are fixed population measures; statistics (x̄, s, p̂) are random variables computed from samples. (Key point: Parameter vs statistic)

## Related in Probability and Statistics

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- [Population, parameter, statistic; samples of 3,7,11,15](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apopulation-parameter-statistic-3-7-11) — Probability and Statistics, chapter 4
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