Normal Probability Calculator for Sampling Distributions (2024)

This normal probability calculator for sampling distributions finds the probability that your sample mean lies within a specific range.

It calculates the normal distribution probability with the sample size (n), a mean values range (defined by X₁ and X₂), the population mean (μ), and the standard deviation (σ).

Keep reading to learn more about:

• What is the sampling distribution of the mean?
• How to find the standard deviation of the sampling distribution.
• How to calculate probabilities for sampling distributions.
• How to use our normal probability calculator for sampling distributions.

🔎 If you need a calculator that makes the same, but for sample proportions, check our sampling distribution of the sample proportion calculator. If you're interested in the opposite problem: finding a range of possible population values given a probability level, look at our sampling error calculator.

Usually, we use samples to estimate population parameters like a population mean height. The most common example is using the sample mean to estimate the population mean.

If you take different samples from a population, you'll probably get different mean values each time. Therefore, the sample mean is also a random variable we can describe with some distribution. This distribution is known as the sampling distribution of the sample mean, which we will name the sampling distribution for simplicity.

If the original population follows a normal distribution, the sampling distribution will do the same, and if not, the sampling distribution will approximate a normal distribution. The central limit theorem describes the degree to which it occurs.

A common task is to find the probability that the mean of a sample falls within a specific range. We can do it using the same tools for calculating normal distributions (using the z-score). The only difference is that the standard deviation of the sampling distribution ($σ_{\bar X}$σXˉ​) is equal to the population standard deviation divided by the square root of the sample size:

🙋 If you're interested in the $σ_{\bar X}$σXˉ​ term, you can learn more about it in our standard deviation of the sample mean calculator.

The average height of the American women (including all race and Hispanic-origin groups) aged 20 and over is approximately 161.3 cm, with a standard deviation of about 7.1 cm. Let's suppose you randomly sample 7 American women. What is the probability that the average height falls below 160 cm?

To know the answer, follow these steps:

1. Input the population parameters in the sampling distribution calculator (μ = 161.3, σ = 7.1)
2. Select left-tailed, in this case.
3. Input the sample data (n = 7, X = 160).
4. Your result is ready. It should be 0.314039. Therefore, the probability that the average height of those women falls below 160 cm is about 31.4%.

Alternatively, we can calculate this probability using the z-score formula:

1. Define your population mean (μ), standard deviation (σ), sample size, and range of possible sample means.
2. Input those values in the z-score formula z_score = (X̄ - μ)/(σ/√n).
3. Considering if your probability is left, right, or two-tailed, use the z-score value to find your probability.
4. Alternatively, you can use our normal probability calculator for sampling distributions.

Depending on the information you possess, there are two ways:

• If you know the population standard deviation (σ), divide it by the square root of the sample size: σ_X̄ = σ/√n.
• If you don't have σ, estimate it with the sample standard deviation (s): σ_X̄ = s/√n.