---
title: "Define sampling distribution of mean"
subject: "Probability and Statistics"
chapter: 4
key: "semester_5/probability_and_statistics:sampling-distribution-of-mean-define"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/sampling-distribution-of-mean-define
site: BCT Engineering
---

# Define sampling distribution of mean

Probability and Statistics · Chapter 4: Sampling Distribution

- Asked in: 2070 Asar, 4 marks
- Video: [The Sampling Distribution of the Sample Mean (fast version)](https://www.youtube.com/watch?v=0zqNGDVNKgA) — jbstatistics
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/sampling-distribution-of-mean-define
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-4

## Question

Define sampling distribution of mean.

## Short answer

The sampling distribution of the mean is the probability distribution of sample means from all possible random samples of size n from a population. Its mean equals the population mean μ, and its standard error is σ/√n (or adjusted for finite population without replacement).

## Answer

**Sampling distribution of the mean:** the probability distribution of the sample mean x̄ obtained from all possible random samples of the same size *n* drawn from a population. A sample statistic varies from one sample to another, and this distribution describes how.

**Properties** (population mean μ, standard deviation σ):
1. Mean of x̄: μ(x̄) = μ (x̄ is an unbiased estimator of μ).
2. Standard error (SD of x̄): σ(x̄) = σ/√n for infinite population or sampling with replacement; σ(x̄) = (σ/√n)√((N − n)/(N − 1)) for sampling without replacement from a finite population of size N.
3. Shape: if the population is normal, x̄ is normal for any n; for other populations, x̄ is approximately normal when n is large (central limit theorem).
4. As n increases, the spread σ/√n decreases.

**Example:** population 2, 4, 6, with samples of size 2 taken with replacement give 9 samples and sample means 2, 3, 4, 3, 4, 5, 4, 5, 6.

| x̄ | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|
| P(x̄) | 1/9 | 2/9 | 3/9 | 2/9 | 1/9 |

Mean of x̄ = 4 = μ, and SE = √(4/3) = 1.1547 = σ/√2 with σ = 1.633.

## Check yourself

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

1. The sampling distribution of the mean is:
   A. The distribution of the values in the population
   B. The distribution of values within one sample
   C. The distribution of the sample size
   D. The probability distribution of x̄ over all possible samples of the same size n
   Answer: D. It describes how x̄ varies from sample to sample for all possible samples of size n. (Key point: Definition)

2. What is the mean of the sampling distribution of x̄?
   A. μ, so x̄ is an unbiased estimator of μ
   B. σ/√n
   C. μ/n
   D. σ
   Answer: A. The mean of x̄ equals the population mean μ. (Key point: Mean of x̄)

3. As the sample size n increases, the standard error σ/√n:
   A. Increases
   B. Stays the same
   C. Decreases
   D. Becomes equal to σ
   Answer: C. A larger n makes σ/√n smaller, so sample means cluster more tightly around μ. (Key point: Effect of sample size)

4. When is the sampling distribution of x̄ normal?
   A. Only when n < 30
   B. Never, it is always skewed
   C. Only when σ is unknown
   D. When the population is normal (any n) or when n is large (CLT)
   Answer: D. Normal populations give normal x̄ for any n; other populations give approximately normal x̄ for large n. (Key point: Shape of the distribution)

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