---
title: "Sampling distribution of mean; standard error"
subject: "Probability and Statistics"
chapter: 4
key: "semester_5/probability_and_statistics:sampling-distribution-mean-se-finite"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/sampling-distribution-mean-se-finite
site: BCT Engineering
---

# Sampling distribution of mean; standard error

Probability and Statistics · Chapter 4: Sampling Distribution

- Asked in: 2072 Chaitra, 5 marks; 2071 Chaitra, 3+1 marks (asked 2 times)
- Video: [Mean and Standard Error of a Sampling Distribution of Sample Means](https://www.youtube.com/watch?v=E5wWzyJi2rU) — Math and Stats Help
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/sampling-distribution-mean-se-finite
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-4

## Question

What do you mean by sampling distribution of a sample mean and its standard error? Explain with example. What would be the variance of sampling distribution of mean, if sample is taken from finite population?

## Short answer

The sampling distribution of the sample mean is the probability distribution of means from all possible samples of size n. Its standard error is sigma/√n, and for finite population sampling without replacement the variance is (sigma²/n)((N−n)/(N−1)). Example: population 2,4,6, n=2 gives SE 1.1547.

## Answer

**Sampling distribution of the sample mean:** if all possible samples of size *n* are drawn from a population and the mean x̄ of each is calculated, the probability distribution of these x̄ values is called the sampling distribution of the sample mean.

Its properties (population mean μ, standard deviation σ, sample size n):
- Mean: E(x̄) = μ
- Variance: σ²/n (infinite population or sampling with replacement)
- If the population is normal, or n is large (CLT), x̄ is normal.

**Standard error (SE):** the standard deviation of the sampling distribution of the mean:

SE(x̄) = σ/√n

It measures how much sample means vary from sample to sample; a larger n gives a smaller SE.

#### Example
Population: 2, 4, 6 (μ = 4, σ = 1.6330). Take all samples of size n = 2 with replacement (9 samples):

| Sample | Mean | Sample | Mean | Sample | Mean |
|---|---|---|---|---|---|
| (2,2) | 2 | (4,2) | 3 | (6,2) | 4 |
| (2,4) | 3 | (4,4) | 4 | (6,4) | 5 |
| (2,6) | 4 | (4,6) | 5 | (6,6) | 6 |

Mean of x̄ values = 36/9 = 4 = μ.

Standard deviation of x̄ values = 1.1547, which equals σ/√n = 1.6330/√2 = 1.1547.

#### Variance when sampling from a finite population
If the sample of size *n* is drawn without replacement from a finite population of size N, the variance is reduced by the finite population correction factor:

Var(x̄) = (σ²/n) × ((N − n)/(N − 1))

and SE(x̄) = (σ/√n) √((N − n)/(N − 1)).

The factor is nearly 1 when the sample is a small part of the population (n/N < 0.05), so it is then ignored.

## Check yourself

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

1. The sampling distribution of the sample mean is:
   A. the distribution of x̄ over all possible samples of size n
   B. the distribution of the population values
   C. the distribution of one sample
   D. the distribution of σ
   Answer: A. It is the probability distribution of x̄ computed from every possible sample. (Key point: Definition)

2. The standard error of x̄ is:
   A. σ^2/n
   B. σ/√n
   C. σ√n
   D. σ/n
   Answer: B. SE is the standard deviation of x̄, equal to σ/√n. (Key point: Standard error)

3. Sampling without replacement from a finite population gives Var(x̄) =
   A. (σ^2/n)(N − n)/(N − 1)
   B. (σ^2/n)(N − 1)/(N − n)
   C. σ^2/N
   D. σ^2 n/N
   Answer: A. The finite population correction (N − n)/(N − 1) reduces the variance. (Key point: Finite population correction)

4. The correction factor is ignored when:
   A. n/N is very small (below 0.05)
   B. n is greater than N/2
   C. σ is unknown
   D. the population is normal
   Answer: A. If the sample is a small part of the population the factor is nearly 1. (Key point: When correction is ignored)

## Related in Probability and Statistics

- [Sampling distribution: population 5,6,9,12](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apopulation-5-6-9-12-samples) — Probability and Statistics, chapter 4
- [Sampling distribution: population 4,8,12,16,20](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apopulation-4-8-12-16-20) — Probability and Statistics, chapter 4
- [Standard error; population 12,19,13,16](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Astandard-error-population-12-19-13) — Probability and Statistics, chapter 4
- [Sampling distribution: population 2,4,6,8](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apopulation-2-4-6-8-samples) — Probability and Statistics, chapter 4
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- [Sampling distribution: population 2,8,14,20](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apopulation-2-8-14-20-samples) — Probability and Statistics, chapter 4, asked 2×

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