What is the sampling distribution of the sample mean?+
The sampling distribution of the sample mean is the probability distribution of all possible sample means from repeated samples of size n from a population. By the Central Limit Theorem, it is approximately normal with mean equal to the population mean and standard deviation (the standard error) equal to sigma divided by the square root of n. It is the foundation of confidence intervals and hypothesis tests for means.
What is the standard error of the mean formula?+
The standard error is SE = sigma divided by the square root of n, where sigma is the population standard deviation and n is the sample size. For example, with sigma = 15 and n = 25, SE = 15 / 5 = 3. The standard error is always smaller than the population SD (for n greater than 1) and measures how much variation to expect in sample means from sample to sample.
How do I calculate P(X-bar less than x) for a sampling distribution?+
Compute SE = sigma / sqrt(n), then Z = (x minus mu) / SE, then P(X-bar less than x) = Phi(Z) using the standard normal CDF. For mu = 100, sigma = 15, n = 25, SE = 3, x = 105: Z = (105 minus 100) / 3 = 1.667, and P(X-bar less than 105) = Phi(1.667) approximately 95.25%.
What is the Central Limit Theorem and when does it apply?+
The Central Limit Theorem states that for large enough samples, the sampling distribution of the sample mean is approximately normal regardless of the population distribution. The common rule of thumb is n at least 30 for most populations. For symmetric distributions n of 10 to 15 may suffice. If the population is itself normal, the result is exact for any n.
What is the difference between standard deviation and standard error?+
Standard deviation sigma measures the spread of individual data points in the population. Standard error SE = sigma / sqrt(n) measures the spread of sample means across many samples of size n. The standard error is always smaller than sigma for n greater than 1. Using sigma instead of SE when computing sample mean probabilities overstates variability and gives incorrect probabilities.
How does sample size affect the probability that a sample mean is extreme?+
Larger samples produce smaller standard errors, so the sampling distribution is narrower and extreme sample means become less probable. With sigma = 15 and mu = 100: at n = 25, P(X-bar greater than 105) is about 4.75%, but at n = 100 (SE = 1.5) it drops to about 0.04%, and at n = 400 (SE = 0.75) it is essentially zero. This illustrates why large samples give more reliable estimates of the population mean.
Can I use this calculator when I only have the sample standard deviation?+
This calculator uses the population standard deviation sigma. If you only know the sample standard deviation s, the exact distribution is the t-distribution with n minus 1 degrees of freedom, not the standard normal. For large samples (n at least 30), substituting s for sigma gives a good approximation. For small samples use a t-distribution calculator for accurate results.
How do I find the probability between two sample mean values?+
Switch to Between Two Values mode. Enter the population mean, standard deviation, sample size, and lower bound a and upper bound b. The calculator computes Z1 = (a minus mu) / SE and Z2 = (b minus mu) / SE, then P(a less than X-bar less than b) = Phi(Z2) minus Phi(Z1). For a symmetric interval of mu plus or minus 1 SE, the probability is about 68.27%.
What does the Z-score mean in a sampling distribution context?+
The Z-score Z = (x-bar minus mu) / SE tells you how many standard errors the sample mean lies from the population mean. A Z of 2 means the sample mean is 2 standard errors above the population mean, which happens about 2.28% of the time under the right-tail area. Z-scores allow use of the standard normal table regardless of the original data units.
What is the 68-95-99.7 rule for sampling distributions?+
About 68% of sample means fall within 1 SE of the population mean, about 95% within 2 SE (more precisely 1.96 SE), and about 99.7% within 3 SE. These are the same percentages as for any normal distribution, but now applied to sample means. Use the Between Two Values mode with bounds set at mu minus k times SE and mu plus k times SE to verify these percentages for any k.
How is this calculator different from a normal distribution calculator?+
A standard normal distribution calculator works with individual observations from a single distribution. This calculator is specifically designed for the distribution of sample means, so it automatically converts the input standard deviation sigma into the standard error SE = sigma / sqrt(n) before computing probabilities. This distinction is critical: using sigma directly instead of SE gives completely wrong probabilities for sample means.
What inputs does this calculator require?+
You need the population mean mu, the population standard deviation sigma (must be positive), and the sample size n (must be a positive integer). For Single Tail mode you also enter the target sample mean value x and choose the direction. For Between Two Values mode you provide lower bound a and upper bound b, where a must be strictly less than b. The calculator validates all inputs before computing.