Normal Probability Calculator for Sampling Distributions
Enter the population mean, population standard deviation, and sample size. Then choose a probability type: the probability that the sample mean falls between two values, below a value, or above a value. The calculator applies the Central Limit Theorem to find the standard error, converts your bounds to z-scores, and reads the probability from the standard normal distribution. Results update instantly as you type, with a full worked solution showing every step.
Formula
Worked example
Population with mu = 50, sigma = 10, sample size n = 30. Standard error = 10 / sqrt(30) = 1.8257. For P(45 < X-bar < 55): z1 = (45 - 50) / 1.8257 = -2.739, z2 = (55 - 50) / 1.8257 = 2.739. P = Phi(2.739) - Phi(-2.739) = 0.9969 - 0.0031 = 0.9938, or about 99.38%.
What is the sampling distribution of the mean?
When you draw a random sample of size n from a population with mean mu and standard deviation sigma, the sample mean X-bar is itself a random variable. If you repeated the sampling process many times, the collection of sample means would form its own distribution, called the sampling distribution of the mean. The Central Limit Theorem (CLT) tells us that this distribution is approximately normal for large enough n, regardless of the shape of the original population. Specifically, X-bar follows N(mu, sigma-squared / n), meaning it is centered on the population mean and has a spread equal to sigma divided by the square root of n. That spread is called the standard error of the mean (SE), and it gets smaller as n increases, which is why larger samples give more precise estimates.
How to use this calculator
Enter the population mean (mu) and population standard deviation (sigma). Then set the sample size n (the number of observations in each sample). Choose a probability type: between two values, left-tailed (below a value), or right-tailed (above a value). For a between calculation, enter both X1 (lower bound) and X2 (upper bound); for left- or right-tailed, enter only X1. The calculator computes the standard error, converts your bounds to z-scores using z = (X - mu) / SE, and evaluates the standard normal CDF to return the probability. The "Show your work" panel shows every arithmetic step with your actual numbers.
The Central Limit Theorem and when it applies
The CLT approximation is excellent when n is at least 30, and it becomes exact when the population is itself normally distributed regardless of sample size. For smaller samples from non-normal populations the approximation deteriorates, so exercise caution when n < 30 and you do not know the population shape. The standard error formula SE = sigma / sqrt(n) assumes the sample is a small fraction of the population (or that sampling is with replacement). If your sample is more than about 5% of a finite population, apply a finite population correction factor.
Interpreting z-scores and the standard normal table
A z-score measures how many standard errors a sample mean is from the population mean. A z-score of 2 means the sample mean is two standard errors above mu, which corresponds to roughly the 97.7th percentile of sampling means. The cumulative distribution function Phi(z) gives the probability of observing a z-score at or below z, which is the area to the left of z under the standard normal curve. To find P(X1 < X-bar < X2), you subtract Phi(z1) from Phi(z2). To find a left-tailed probability P(X-bar < X1) you use Phi(z1) directly. For a right-tailed probability P(X-bar > X1) you compute 1 - Phi(z1).
Common z-score probability reference
| Z-score range | Probability (%) | Interpretation |
|---|---|---|
| μ ± 1σ | 68.27% | 1 standard deviation |
| μ ± 1.645σ | 90.00% | 90% confidence interval |
| μ ± 1.960σ | 95.00% | 95% confidence interval (standard) |
| μ ± 2σ | 95.45% | 2 standard deviations |
| μ ± 2.576σ | 99.00% | 99% confidence interval |
| μ ± 3σ | 99.73% | 3 standard deviations (six-sigma) |
| μ ± 4σ | 99.994% | 4 standard deviations |
Two-tailed probabilities for the most frequently cited z-score ranges. Derived from the standard normal distribution.
Frequently asked questions
What is the difference between population standard deviation and standard error?
Population standard deviation (sigma) measures how much individual data points vary from the population mean. Standard error (SE = sigma / sqrt(n)) measures how much the sample mean varies from the population mean across repeated samples. Because SE shrinks with larger n, the sample mean becomes a more precise estimate as you collect more data. SE is never larger than sigma, and for n = 1 the two are equal.
How large does the sample size need to be for the normal approximation to work?
A commonly cited rule of thumb is n >= 30. At this threshold the Central Limit Theorem approximation is reliable for most real-world distributions. If the population is already normally distributed, any sample size works exactly. If the population is highly skewed, you may need n > 50 or more for a good approximation. For n < 30 from a non-normal population, consider using t-distribution methods instead.
What is a left-tailed versus a right-tailed probability?
A left-tailed probability P(X-bar < X1) is the chance the sample mean falls below the cutoff X1. It equals the area under the sampling distribution curve to the left of X1. A right-tailed probability P(X-bar > X1) is the complementary area to the right. A two-tailed (between) probability P(X1 < X-bar < X2) is the area between the two cutoffs, calculated by subtracting the two cumulative probabilities.
Can I use this calculator for proportions?
This calculator is designed for the sample mean of a continuous or count variable. For proportions (where each observation is a 0 or 1), the population mean mu equals the proportion p and sigma = sqrt(p * (1 - p)). You can substitute those values here, but a dedicated sampling distribution of proportions calculator handles that setup more naturally and can apply the continuity correction.
Why does increasing sample size not change the probability for a fixed X range?
Increasing n reduces the standard error, which makes the sampling distribution narrower and taller. If your X bounds stay the same distance from mu in absolute terms, a narrower distribution actually changes the probability substantially: bounds that span many standard errors become more likely (the distribution is more concentrated near mu), while bounds that span only a fraction of a standard error become less likely. Try increasing n and watch the probability change.
What does a probability of 0.9938 mean in this context?
It means that in repeated random sampling, about 99.38% of all samples of size n would produce a sample mean that falls in your specified range. Equivalently, if you drew one sample, there is a 99.38% chance its mean would be in that range. It does NOT mean that 99.38% of the individual data points lie in the range; that is a different calculation using the original population distribution, not the sampling distribution.