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.

The Formula

σxˉ=σn\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}

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\mu = 170 cm. Take 1000 random samples of n=40n = 40. Each sample mean xˉ\bar{x} differs slightly, but the histogram of all 1000 sample means forms a bell curve centered at 170.\text{the histogram of all 1000 sample means forms a bell curve centered at } 170.

Notation

Xˉ\bar{X} 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)=μE(\bar{X}) = \mu and SD(Xˉ)=σn\text{SD}(\bar{X}) = \frac{\sigma}{\sqrt{n}} where μ\mu and σ\sigma are the population mean and SD

Worked Examples

Example 1

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

Answer

Xˉ∼N(μ=50, SE=2)\bar{X} \sim N(\mu=50,\ SE=2). Sample means are normally distributed around 50 with SD=2.

First step

1
Mean of sampling distribution: μXˉ=μ=50\mu_{\bar{X}} = \mu = 50

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Example 2

hard
For a population with μ=100\mu=100, σ=20\sigma=20, and n=25n=25: (a) find P(Xˉ>104)P(\bar{X} > 104), (b) find the value cc such that P(Xˉ<c)=0.90P(\bar{X} < c) = 0.90.

Example 3

medium
A population has μ=200\mu = 200, σ=40\sigma = 40, and we draw samples of size n=64n = 64. Find P(Xˉ>210)P(\bar{X} > 210).

Common Mistakes

  • Using the population SD σ\sigma as the spread of the mean - the sample mean's spread is the smaller standard error σn\frac{\sigma}{\sqrt{n}}.
  • 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 nn 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\bar{x} varies. Students who confuse the spread of the data with the spread of the mean misjudge every margin of error — the standard error σn\frac{\sigma}{\sqrt{n}} is the whole point. Recognizing it by "Am I describing how a statistic (like xˉ\bar{x}) 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\bar{X} 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\bar{x} varies. Students who confuse the spread of the data with the spread of the mean misjudge every margin of error — the standard error σn\frac{\sigma}{\sqrt{n}} is the whole point. Recognizing it by "Am I describing how a statistic (like xˉ\bar{x}) 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\sigma as the spread of the mean. Asking "Am I describing how a statistic (like xˉ\bar{x}) 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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