Analyze sample means from population inputs and targets. Compare one tail, two tail, and ranges. See clear outputs, examples, formulas, and downloads in seconds.
| μ | σ | n | Type | Target | Probability | Percent |
|---|---|---|---|---|---|---|
| 100 | 15 | 36 | Less | 104 | 0.945201 | 94.520% |
| 50 | 12 | 64 | Greater | 52 | 0.091211 | 9.121% |
| 80 | 20 | 49 | Between | 77 to 84 | 0.772384 | 77.238% |
| 200 | 30 | 100 | Outside | 194 to 206 | 0.045500 | 4.550% |
The central limit theorem says that the sample mean tends to follow an approximately normal distribution when the sample size becomes reasonably large. This page turns that theorem into a practical probability tool. You enter the population mean, the population standard deviation, and the sample size. Then you choose whether you want a left-tail probability, a right-tail probability, a between-range probability, or an outside-range probability.
The calculator uses the sampling distribution of the sample mean. Its center is the population mean. Its spread is the standard error, which equals the population standard deviation divided by the square root of the sample size. If you sample without replacement from a limited population, the finite population correction can reduce the standard error further. That adjustment is included here as an option.
You also get z scores, a confidence-based central interval, and a plot of the sampling distribution. These outputs help you interpret how unusual a target sample mean is under the assumed population model. The graph supports quick decision making when you need a visual view of tails or ranges.
This calculator is useful in statistics classes, quality checks, forecasting, operations, research planning, and exam practice. It is especially helpful when you want a direct link between raw population inputs and the probability of observing a sample average in a chosen region.
Sampling distribution of the sample mean:
X̄ approximately follows a normal model with mean μ and standard error σ / √n.
Finite population correction:
Adjusted SE = (σ / √n) × √((N - n) / (N - 1)), when sampling without replacement from a finite population.
Z score:
z = (x̄ - μ) / Adjusted SE
Tail probabilities:
Left tail: P(X̄ ≤ a) = Φ(z)
Right tail: P(X̄ ≥ a) = 1 - Φ(z)
Between: P(a ≤ X̄ ≤ b) = Φ(z₂) - Φ(z₁)
Outside: P(X̄ ≤ a or X̄ ≥ b) = 1 - [Φ(z₂) - Φ(z₁)]
Central interval:
μ ± zα/2 × Adjusted SE
It estimates probabilities for the sample mean using the central limit theorem. You can evaluate left tails, right tails, inside ranges, outside ranges, and central intervals.
Use it when you study sample means and the sample size is reasonably large, or when the population is already close to normal.
No, not always. Large samples often make the sample mean approximately normal. Small samples need more caution unless the original population is already close to normal.
Standard error measures the spread of the sample mean distribution. It equals the population standard deviation divided by the square root of the sample size.
Use it when sampling without replacement from a finite population, especially when the sample is a noticeable fraction of the full population.
Outside probability measures the chance that the sample mean falls below the lower bound or above the upper bound, not inside the interval.
It shows the symmetric range around the sampling mean that contains the chosen confidence percentage. This helps interpret spread and typical sample-mean variation.
Yes. The result block supports CSV export and PDF download after a successful calculation.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.