Difficulty: Intermediate | Prerequisites: Basic probability (Chapter 5–6), normal distribution, z-scores.
Big picture: Up to this point, you have been working with individual data points drawn from a population. This chapter shifts the focus to what happens when you repeatedly take samples and compute their means. The sampling distribution of the sample mean is the bridge between descriptive statistics and inferential statistics. It is what allows you to make claims about an entire population using only a sample. If you skipped the normal distribution material, go back and review it first.
When you take random samples from a population and compute the mean of each sample, those means form their own distribution called a sampling distribution. This distribution has the same centre as the population but is narrower (less spread), and the spread shrinks as sample size grows. If the population is normal, the sampling distribution of the mean is also normal.
Statistic
Any quantity computed from values in a sample. Examples include the sample mean (x̄) and the sample standard deviation (s). In simple terms, a statistic is a number you calculate from the data you actually collected.
Parameter
A numerical summary of an entire population, such as the population mean (μ) or the population standard deviation (σ). In simple terms, parameters describe the whole group, but you rarely know them exactly.
Sampling distribution
The probability distribution of a statistic (such as x̄) obtained from all possible samples of a given size drawn from a population. Think of it as: "If I repeated this sampling process thousands of times and plotted all the results, what pattern would I see?"
Standard error (σ_x̄)
The standard deviation of the sampling distribution of the sample mean. It measures how much sample means vary from sample to sample. Formally: σ_x̄ = σ / √n. In simple terms, this tells you how "jumpy" your sample average is likely to be.
Population distribution
The probability distribution of individual values in the entire population, described by P(population distribution).
Parameters describe populations: μ (mean), σ (standard deviation).
Statistics describe samples: x̄ (sample mean), s (sample standard deviation).
Probability links population to sample; inferential statistics works backwards from sample to population.
Take all possible random samples of size n from a population.
Compute x̄ for each sample.
The distribution of all those x̄ values is the sampling distribution of the sample mean.
Two key properties of this distribution:
Its mean equals the population mean.
Its standard deviation (the standard error) is smaller than the population standard deviation.
Formula: μ_x̄ = μ_X
The expected value of the sample mean equals the population mean.
Derivation:
x̄ = Σxᵢ / n
E(x̄) = E(Σxᵢ / n) = (1/n) · ΣE(xᵢ) = (1/n) · nμ = μ
This means the sample mean is an unbiased estimator of the population mean. On average, your sample mean hits the target.
Formula: σ_x̄ = σ_X / √n
Derivation:
σ_x̄ = √(Var(x̄))
Var(x̄) = Var(Σxᵢ / n) = (1/n²) · ΣVar(xᵢ) = (1/n²) · nσ² = σ²/n
Therefore σ_x̄ = σ / √n
What this means in practice:
As sample size n increases, the standard error decreases.
Larger samples give more precise estimates of the population mean.
Example: if σ = 10, then for n = 4, σ_x̄ = 10/√4 = 5; for n = 100, σ_x̄ = 10/√100 = 1.
If the population itself is normally distributed, X ~ N(μ, σ²), then the sampling distribution of x̄ is also exactly normal:
x̄ ~ N(μ, σ²/n)
This holds for any sample size, no matter how small.
You can then convert to a z-score: z = (x̄ − μ) / (σ / √n)
Quantity | Formula |
|---|---|
Sample mean | x̄ = Σxᵢ / n |
Mean of sampling distribution | μ_x̄ = μ |
Standard error | σ_x̄ = σ / √n |
z-score for sample mean | z = (x̄ − μ) / (σ / √n) |
Visual concept: As n increases from 2 to 4 to 10, the sampling distribution becomes progressively taller and narrower, always centred on μ. The population distribution is the widest; each sampling distribution sits inside it.
Setup: μ = 1.5 min, σ = 0.35 min, n = 5
(a) Mean of the sampling distribution:
μ_x̄ = μ = 1.5 min
(b) Standard error:
σ_x̄ = 0.35 / √5 = 0.1565 min
(c) P(x̄ ≤ 2.0):
z = (2.0 − 1.5) / 0.1565 = 3.19
P(z ≤ 3.19) = 0.9993
(d) P(μ − 0.3 ≤ x̄ ≤ μ + 0.3):
This is P(1.2 ≤ x̄ ≤ 1.8).
z₁ = (1.2 − 1.5) / 0.1565 = −1.92
z₂ = (1.8 − 1.5) / 0.1565 = 1.92
P(−1.92 ≤ z ≤ 1.92) = 0.9726 − 0.0274 = 0.9452
The standard error formula is the reason political polls report a "margin of error." A poll of 1,000 people has a smaller standard error (and thus tighter margin) than a poll of 100. This same principle drives quality control in manufacturing, where companies test batches of products and use the sample mean to decide whether the production line is within spec.
"A larger sample changes the population distribution." It does not. The population is fixed. A larger sample only narrows the sampling distribution (reduces the standard error).
"The sampling distribution and the population distribution are the same thing." They are not. The population distribution describes individual values; the sampling distribution describes sample means.
"You need a normal population to use these formulas." The mean and standard error formulas (μ_x̄ = μ, σ_x̄ = σ/√n) hold regardless of the population's shape. Normality of the sampling distribution requires either a normal population or a large sample size (CLT, covered in Part 2).
⚠️ You will be asked to compute σ_x̄ = σ / √n. Do not confuse σ (population SD) with σ_x̄ (standard error).
⚠️ When converting x̄ to a z-score, the denominator is σ/√n, not σ. Using plain σ is the single most common error on this chapter's exam questions.
⚠️ Know the difference between a parameter and a statistic. Exam questions often ask you to classify a given number as one or the other.
⚠️ If the population is normal, the sampling distribution is exactly normal for any n. You do not need the Central Limit Theorem in that case.
True or False: The mean of the sampling distribution of x̄ equals the population mean.
True or False: The standard error increases as sample size increases.
Fill in the blank: σ_x̄ = σ / ___
True or False: If the population distribution is normal, the sampling distribution of x̄ is normal regardless of sample size.
True or False: The μ_x̄ and σ_x̄ of a sampling distribution depend on the shape of the population distribution.
Answers: 1. True. 2. False (it decreases). 3. √n. 4. True. 5. False (μ_x̄ = μ and σ_x̄ = σ/√n always, regardless of shape).
Q: A population has μ = 50 and σ = 12. A random sample of n = 36 is taken. What are μ_x̄ and σ_x̄?
A: μ_x̄ = 50. σ_x̄ = 12/√36 = 12/6 = 2.
Q: If σ = 0.35 and n = 5, what is the standard error of the sample mean?
A: σ_x̄ = 0.35/√5 ≈ 0.1565.
Q: What is the difference between a parameter and a statistic?
A: A parameter describes a population (μ, σ). A statistic describes a sample (x̄, s). Parameters are usually unknown; statistics are computed from collected data.
Q: A population is normally distributed with μ = 100 and σ = 10. For a sample of 25, find P(x̄ < 95).
A: σ_x̄ = 10/√25 = 2. z = (95 − 100)/2 = −2.5. P(z < −2.5) = 0.0062.
This material connects directly to confidence intervals and hypothesis testing (Chapters 8–9), where you will use the standard error to build intervals around your sample mean and decide whether a population mean is plausible. The z-score conversion here is the same one you will use in those chapters. Understanding the sampling distribution is also the foundation for understanding why larger clinical trials and larger survey samples produce more reliable results.
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