Normal Probability Calculator for Sampling Distributions

Find the probability that a sample mean falls above, below, or between values. Enter the population mean, standard deviation, and sample size to get the standard error, z-score, and normal probability.

Quick Facts

Method
Standard error SE = σ/√n, then z = (x − μ)/SE and P = Φ(z)
Valid when the population is normal, or by the Central Limit Theorem when n ≥ 30.

Your Results

Calculated
Probability
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Area under the sampling distribution
Standard error (σ/√n)
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Std. dev. of the sample mean
Z-score
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Standardized sample mean
Percent
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Probability as a percentage

Ready

Enter μ, σ, n and a target sample mean, then calculate.

Understanding the sampling distribution of the sample mean

When you draw a random sample and compute its mean, that sample mean is itself a random variable. Take a different sample and you get a slightly different mean. The sampling distribution of the sample mean is the distribution of all those possible sample means. This calculator finds the probability that a single sample mean, X̄, lands above, below, or between values you specify.

Two facts make the calculation tractable. First, the mean of the sampling distribution equals the population mean: E(X̄) = μ. Second, its spread — the standard error of the mean — is smaller than the population spread by a factor of √n:

SE = σ / √n

Larger samples produce sample means that cluster more tightly around μ, which is why bigger surveys yield more precise estimates.

The formula this calculator uses

To find a probability, the calculator standardizes your target sample mean into a z-score using the standard error, then reads the area from the standard normal curve:

  • Standard error: SE = σ / √n
  • Z-score: z = (x − μ) / SE = (x − μ) / (σ/√n)
  • Below: P(X̄ < x) = Φ(z)
  • Above: P(X̄ > x) = 1 − Φ(z)
  • Between: P(x₁ < X̄ < x₂) = Φ(z₂) − Φ(z₁)

Here Φ is the cumulative distribution function (CDF) of the standard normal distribution — the area under the bell curve to the left of z. The calculator evaluates Φ with the Abramowitz & Stegun 7.1.26 approximation, accurate to better than 1.5×10⁻⁷.

Why you can use the normal curve: the Central Limit Theorem

The z-score approach assumes X̄ is normally distributed. This holds exactly when the underlying population is normal. When the population is not normal, the Central Limit Theorem (CLT) guarantees that the sampling distribution of X̄ becomes approximately normal as n grows. A common rule of thumb is n ≥ 30, though heavily skewed populations may need more. Note that this calculator describes the distribution of the sample mean, not of individual observations — for a probability about a single value, set n = 1, which makes SE = σ.

Common reference points

  • z = 1.645: 95% of sample means fall below this; 5% above (one-tailed).
  • z = 1.96: the middle 95% of sample means fall within ±1.96 standard errors of μ.
  • z = 2.576: the middle 99% fall within ±2.576 standard errors.
  • The 68–95–99.7 rule: about 68%, 95%, and 99.7% of sample means fall within 1, 2, and 3 standard errors of μ.

Worked example

Suppose IQ scores have μ = 100 and σ = 15. For a sample of n = 25 people, the standard error is 15/√25 = 15/5 = 3. What is the chance the sample's average IQ exceeds 106? The z-score is (106 − 100)/3 = 2.0, and P(X̄ > 106) = 1 − Φ(2.0) ≈ 1 − 0.9772 = 0.0228, or about 2.28%.

Frequently Asked Questions

What is the standard error of the mean?
The standard error of the mean, SE = σ/√n, is the standard deviation of the sampling distribution of the sample mean. It tells you how much a sample mean typically differs from the true population mean. Because it divides by √n, quadrupling the sample size halves the standard error, and larger samples give more precise estimates of μ.
How is this different from a normal probability for a single value?
For a single observation you use σ directly: z = (x − μ)/σ. For a sample mean you use the standard error σ/√n instead: z = (x − μ)/(σ/√n). Because the standard error is smaller, sample means cluster more tightly around μ than individual values do. Set n = 1 in this calculator to reproduce the single-value case.
Do I need to know the population standard deviation?
This calculator assumes σ is known, which lets you use the normal (z) distribution. If σ is unknown and you estimate it from the sample using s, the correct sampling distribution is the Student's t-distribution with n − 1 degrees of freedom rather than the normal. For large n the two are nearly identical, but for small samples with an estimated standard deviation, use a t-based tool.
When does the normal approximation fail?
It can be inaccurate for small samples drawn from strongly skewed or heavy-tailed populations, because the Central Limit Theorem has not yet "kicked in." It also assumes independent observations and, for finite populations sampled without replacement, that the sample is a small fraction of the population (otherwise apply a finite population correction factor √((N−n)/(N−1))).

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