What Is Sample Mean Probability?
The central limit theorem says that sample means are approximately normal, centred on the population mean, with a spread called the standard error: σ ÷ √n. That lets you find the probability of any range of sample means with a z-score.
Pick below, above or between, then enter μ, σ and n and the sample mean value or values. The calculator returns the standard error, the z-scores, the probability and a shaded sampling distribution. An optional population size adds the finite population correction.
Sample means vary far less than single observations, so the probabilities are very different. A value 4 above the mean may be common for one observation but rare for the average of 36.
How to Use the Sample Mean Probability Calculator
- Choose below, above or between.
- Enter μ, σ and n, then the sample mean value or values.
- Read the probability and the shaded sampling distribution.
Sample Mean Probability Formula
Standard error
z for a sample mean
Finite population correction
Sample Mean Probability Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| μ | Population mean | |
| σ | Population standard deviation | |
| n | Sample size | |
| σ/√n | Standard error of the mean | |
| z | Standard score of the sample mean | (x̄ − μ) / SE |
Sample Mean Probability Worked Example
With the example values loaded in the calculator above, the steps are:
- The sample mean has mean μ = 100 and standard error σ ÷ √n = 15 ÷ √36 = 2.5.
- z = (104 − 100) ÷ 2.5 = 1.6.
- Probability = P(x̄ > 104) = 0.0547993.
Result: P(x̄ > 104) = 0.0547993; Standard error of the mean = 2.5.
Worked example
A population has μ = 100 and σ = 15. For samples of n = 36 the standard error is 15 ÷ 6 = 2.5. What is P(x̄ > 104)?
z = (104 − 100) ÷ 2.5 = 1.6 and P(Z > 1.6) = 0.0548. About 5.5% of samples of 36 will have a mean above 104.
Single value versus sample mean
A single value above 104 has probability P(Z > 0.267) = 0.395, which is far larger. Averages are much steadier than individual values, because the standard error shrinks as n grows.
Where Sample Mean Probability Is Used
Statistics coursework
Solve central limit theorem probability problems.
Quality control
Find how likely an average measurement is to drift past a limit.
Survey planning
See how a larger n narrows the likely range of the sample mean.
Common Sample Mean Probability Mistakes to Avoid
- Use σ of the population, not of the sample, for this calculation.
- Use the finite population correction only if the sample is a large share of a small population.
Sample Mean Probability Terms and Definitions
- Standard error of the mean
- The standard deviation of the sampling distribution of the sample mean.
Sample Mean Probability FAQs
How do you find the probability of a sample mean?
Compute the standard error σ ÷ √n, convert the value to a z-score with (x̄ − μ) ÷ SE, and read the normal probability.
Why is the standard error smaller than the standard deviation?
Averaging cancels out some of the variation, so means vary less than single values, by a factor of √n.
When is the central limit theorem valid?
For large samples (about 30 or more) from almost any population. For small samples, the population should be nearly normal.
Do I need the population standard deviation?
Yes, this calculation uses σ. If you only have the sample standard deviation, a t-based method is more appropriate.
What does P(x̄ > a) mean?
The proportion of all possible samples of size n whose mean is above a.
When do I use the finite population correction?
When you sample more than about 5% of a finite population without replacement.