What Is Normal Sampling Distribution?
The sampling distribution of the sample mean shows how x̄ varies from sample to sample. When the population is normal, or the sample is large enough for the central limit theorem, it is normal with mean μ and standard error σ divided by the square root of n.
Enter μ, σ and n, then choose P(x̄ below a), P(x̄ above a) or P(a below x̄ below b). The calculator standardises to a z score and reads the probability from the normal curve.
Bigger samples give a smaller standard error, so the sample mean lands closer to μ.
How to Use the Normal Sampling Distribution Calculator
- Enter μ, σ and n.
- Pick less than, greater than or between and enter the value or values.
- Read the standard error, the z score and the probability.
Normal Sampling Distribution Formula
Standard error
z score of a sample mean
Normal Sampling Distribution Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| μ | Population mean | |
| σ | Population standard deviation | |
| n | Sample size | |
| SE | Standard error of the mean | σ / √n |
Normal Sampling Distribution Worked Example
With the example values loaded in the calculator above, the steps are:
- The sampling distribution of x̄ has mean μ = 100 and standard error σ ÷ √n = 15 ÷ √36 = 2.5.
- Standardise: z = (x̄ − μ) ÷ SE = (103 − 100) ÷ 2.5 = 1.2.
- P(x̄ > 103) = 0.11507.
Result: P(x̄ > 103) = 0.11507; Standard error (σ ÷ √n) = 2.5.
Worked example
Let μ = 100, σ = 15 and n = 36. The standard error is 15 ÷ 6 = 2.5.
For P(x̄ > 103), z = (103 − 100) ÷ 2.5 = 1.2, so the probability is 1 − 0.8849 = 0.1151.
The central limit theorem
For large samples the sample mean is close to normal even if the population is not. Heavy skew needs a larger n before this works well.
Where Normal Sampling Distribution Is Used
Statistics homework
Find probabilities for a sample mean.
Quality control
Estimate how likely a batch average is to drift past a limit.
Understanding the CLT
See how the spread of x̄ shrinks as n grows.
Common Normal Sampling Distribution Mistakes to Avoid
- Do not confuse the standard error with the standard deviation of individual values.
- Doubling n shrinks the standard error by a factor of √2, not 2.
Normal Sampling Distribution Terms and Definitions
- Standard error
- The standard deviation of a sample statistic, here of x̄.
- Sampling distribution
- The distribution of a statistic over all possible samples.
Normal Sampling Distribution FAQs
What is the sampling distribution of the sample mean?
The distribution of x̄ across repeated samples. It has mean μ and standard deviation σ ÷ √n.
How do you find the standard error?
Divide the population standard deviation by the square root of the sample size.
When is it normal?
When the population is normal, or when n is large enough for the central limit theorem to apply.
What if σ is unknown?
Use the sample standard deviation and a t distribution.