Normal Sampling Distribution Calculator

Calculator Inputs

This tool models the sampling distribution of the sample mean. It supports lower tails, upper tails, middle intervals, outside ranges, percentile cutoffs, confidence-based central intervals, and finite population correction.

Population mean (μ)

Population standard deviation (σ)

Sample size (n)

Analysis mode

First cutoff / lower bound

Second cutoff / upper bound

Percentile probability (%)

Confidence level (%)

Population size (N) for FPC

Apply finite population correction

Result cards appear above this form after submission, directly below the page header.

Formula used

Mean of the sampling distribution
μ_x̄ = μ

Standard error of the sample mean
σ_x̄ = σ / √n

Finite population correction
σ_x̄,FPC = (σ / √n) × √((N − n) / (N − 1))

Z score for a sample mean cutoff
z = (x̄ − μ_x̄) / σ_x̄

Probability relationships
P(X̄ ≤ x) = Φ(z)
P(X̄ ≥ x) = 1 − Φ(z)
P(a ≤ X̄ ≤ b) = Φ(z_b) − Φ(z_a)

Central interval and percentile cutoff
Central interval = μ ± z_α/2σ_x̄
Percentile cutoff = μ + z_pσ_x̄

Here, Φ(z) is the standard normal cumulative distribution function. The calculator assumes a normal sampling distribution for the sample mean, which is exact for normal populations and often reasonable for large samples by the central limit theorem.

How to use this calculator

Enter the population mean, population standard deviation, and planned sample size.
Select the analysis mode that matches your probability or cutoff question.
Fill the needed cutoff, percentile, or confidence fields for that mode.
Use finite population correction only when sampling without replacement from a limited population.
Click the calculate button to view the result table, interpretation, and graph above the form.
Use the export buttons to save the generated result table as CSV or PDF.

Example data table

Item	Example value	Explanation
Population mean (μ)	50	Expected average of the full population.
Population standard deviation (σ)	12	Known variability of individual values.
Sample size (n)	36	Number of observations per sample.
Standard error	2.000000	Computed as 12 / √36.
Mode	Between values	Find probability between two sample-mean cutoffs.
Lower cutoff	48	First sample-mean threshold.
Upper cutoff	53	Second sample-mean threshold.
Lower Z score	-1.000000	(48 − 50) / 2
Upper Z score	1.500000	(53 − 50) / 2
Probability	0.774538	P(48 ≤ X̄ ≤ 53)

FAQs

1. What does this calculator estimate?

It estimates probabilities, cutoffs, and intervals for the sampling distribution of the sample mean. It focuses on X̄ rather than single observations.

2. When is the normal sampling model appropriate?

It is exact when the population itself is normal. It is also often reliable for large samples because the central limit theorem makes sample means approximately normal.

3. Why does sample size matter so much?

Larger samples reduce the standard error because σ/√n gets smaller. That makes the sampling distribution narrower and pushes probabilities closer to the population mean.

4. What is the difference between σ and standard error?

σ measures the spread of individual population values. Standard error measures the spread of sample means across repeated samples of the same size.

5. When should I apply finite population correction?

Apply it when sampling without replacement from a limited population and the sample is not tiny relative to that population. It reduces the standard error.

6. What does percentile mode return?

Percentile mode returns the sample-mean cutoff associated with a chosen cumulative probability. For example, the 95th percentile gives the value below which 95% of sample means fall.

7. Can I use this for individual raw observations?

Not directly. This page models the distribution of sample means. For single observations, use the original population distribution instead of the sampling distribution.

8. Why might my plotted values look slightly rounded?

The displayed table and graph use rounded points for readability. The underlying calculations remain precise enough for practical statistical analysis and reporting.