Sampling Distribution Mean Calculator

Calculator

Enter your parameters. Then compute standard error and probabilities.

White theme Single page

Population mean (μ)

Expected value of the population.

Standard deviation (σ or s)

Use σ if known; otherwise use s.

Sample size (n)

n ≥ 1. Larger n reduces standard error.

Sampling method

Use FPC when sampling without replacement.

Population size (N, optional)

Required only for without replacement.

Probability mode

Adds z-scores and probability estimates.

Value A (x̄A)

Used for tails, interval start, or point.

Value B (x̄B)

Used only for interval probabilities.

Tail / interval type

Ignored for “Standard error only”.

Reset

Formula used

Sampling distribution mean

E[x̄] = μ
SE[x̄] = σ / √n
Var[x̄] = (σ²) / n

If σ is unknown, many users plug in s as an estimate.

Finite population correction (optional)

FPC = √((N − n) / (N − 1))
SE[x̄] = (σ / √n) × FPC

Use FPC when sampling without replacement and n is not tiny compared to N.

Probability using z

z = (x̄ − μ) / SE[x̄]

This page uses a normal CDF approximation. Probabilities assume x̄ is approximately normal (exact if population is normal).

How to use this calculator

Enter μ, σ (or s), and sample size n.
Pick your sampling method. Add N if needed.
Choose “Standard error only” or a probability mode.
Provide A (and B for intervals) then click Compute.
Download CSV or PDF to save your output.

Example data table

Example: μ = 50, σ = 12. Standard error decreases as n grows.

Sample size (n)	SE = σ/√n	Approx. 95% range for x̄ (μ ± 1.96·SE)
9	4	42.16 to 57.84
16	3	44.12 to 55.88
36	2	46.08 to 53.92
64	1.5	47.06 to 52.94
100	1.2	47.65 to 52.35

FAQs

1) What does the sampling distribution of the mean describe?

It describes how sample means vary across repeated samples of size n. Its center is μ, and its spread is the standard error.

2) Why does the standard error shrink when n increases?

Because averaging reduces variability. The formula divides σ by √n, so doubling n does not halve SE, but it still gets smaller.

3) When is the normal assumption for x̄ reasonable?

If the population is normal, x̄ is normal for any n. Otherwise, x̄ becomes approximately normal as n grows, per the CLT.

4) What is finite population correction and when should I use it?

FPC adjusts SE when sampling without replacement from a finite population. It matters most when n is a noticeable fraction of N.

5) Why is P(x̄ = A) not reported as a probability?

For continuous models, the probability of an exact value is effectively zero. This tool can show the density at A instead.

6) Can I use s instead of σ?

Yes. If σ is unknown, many analyses use s as an estimate. Interpret results as approximate, especially for small n.

7) How should I interpret the tail probabilities?

They estimate how likely a sample mean is to fall beyond your threshold under μ and SE. Smaller probabilities indicate rarer outcomes.