Sampling Distribution Formula

Sampling distribution is the probability distribution of a statistic (such as the sample mean) computed from all possible random samples of the same size drawn from a population.

The Formula

σxˉ=σn

When to use: Imagine you survey 50 random people about their height, compute the average, then repeat with a different group of 50, again and again. Each group gives a slightly different average. The pattern of all those averages forms the sampling distribution. It's like taking the temperature of a city by sending out 100 different thermometers—each reads slightly differently, but together they cluster around the truth.

Quick Example

Population mean height μ=170 cm. Take 1000 random samples of n=40. Each sample mean xˉ differs slightly, but the histogram of all 1000 sample means forms a bell curve centered at 170.

Notation

Xˉ denotes the random variable for the sample mean; its distribution is the sampling distribution.

What This Formula Means

The probability distribution of a statistic (such as the sample mean) computed from all possible random samples of the same size drawn from a population.

Imagine you survey 50 random people about their height, compute the average, then repeat with a different group of 50, again and again. Each group gives a slightly different average. The pattern of all those averages forms the sampling distribution. It's like taking the temperature of a city by sending out 100 different thermometers—each reads slightly differently, but together they cluster around the truth.

Formal View

E(Xˉ)=μ and SD(Xˉ)=σn where μ and σ are the population mean and SD

Worked Examples

Example 1

medium
A population has μ=50 and σ=12. For random samples of size n=36, describe the sampling distribution of Xˉ: find its mean, standard error, and shape.

Answer

Xˉ∼N(μ=50, SE=2). Sample means are normally distributed around 50 with SD=2.

First step

1
Mean of sampling distribution: μXˉ=μ=50

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Example 2

hard
For a population with μ=100, σ=20, and n=25: (a) find P(Xˉ>104), (b) find the value c such that P(Xˉ<c)=0.90.

Example 3

medium
A population has μ=200, σ=40, and we draw samples of size n=64. Find P(Xˉ>210).

Common Mistakes

  • Using the population SD σ as the spread of the mean - the sample mean's spread is the smaller standard error σn.
  • Confusing the data's distribution with the statistic's distribution - raw values and sample means have different spreads.
  • Thinking a bigger sample makes individual data less variable - bigger n shrinks the spread of the MEAN, not of the data.

Why This Formula Matters

The sampling distribution is the hidden engine behind all inference: confidence intervals and hypothesis tests work only because we know how much xˉ varies. Students who confuse the spread of the data with the spread of the mean misjudge every margin of error — the standard error σn is the whole point. Recognizing it by "Am I describing how a statistic (like xˉ) varies across many samples, rather than how raw values vary?" — rather than by familiar numbers — is what lets a student tell it apart from population distribution and central limit theorem and sample (single) in a mixed problem set.

Frequently Asked Questions

What is the Sampling Distribution formula?

The probability distribution of a statistic (such as the sample mean) computed from all possible random samples of the same size drawn from a population.

How do you use the Sampling Distribution formula?

Imagine you survey 50 random people about their height, compute the average, then repeat with a different group of 50, again and again. Each group gives a slightly different average. The pattern of all those averages forms the sampling distribution. It's like taking the temperature of a city by sending out 100 different thermometers—each reads slightly differently, but together they cluster around the truth.

What do the symbols mean in the Sampling Distribution formula?

Xˉ denotes the random variable for the sample mean; its distribution is the sampling distribution.

Why is the Sampling Distribution formula important in Math?

The sampling distribution is the hidden engine behind all inference: confidence intervals and hypothesis tests work only because we know how much xˉ varies. Students who confuse the spread of the data with the spread of the mean misjudge every margin of error — the standard error σn is the whole point. Recognizing it by "Am I describing how a statistic (like xˉ) varies across many samples, rather than how raw values vary?" — rather than by familiar numbers — is what lets a student tell it apart from population distribution and central limit theorem and sample (single) in a mixed problem set.

What do students get wrong about Sampling Distribution?

The procedure for sampling distribution is the easy part; the trap is using the population SD σ as the spread of the mean. Asking "Am I describing how a statistic (like xˉ) varies across many samples, rather than how raw values vary?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Sampling Distribution formula?

Before studying the Sampling Distribution formula, you should understand: normal distribution, mean, standard deviation.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Data Representation, Variability, and Sampling Guide →
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