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.

## Calculation of normal probability for sampling distributions

Many real-life phenomena follow a normal distribution. For example, the **American men's height** follows that distribution with a mean of approximately 176.3 cm and a standard deviation of about 7.6 cm. In the following plot, you can see the distribution graph of those heights.

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:

$\footnotesize σ_{\bar X}=\frac{σ}{\sqrt{n}}$σXˉ=nσ

Then, the formula to calculate the z-score is:

$\footnotesize z= \frac{X-μ}{σ/ \sqrt{n}}$z=σ/nX−μ

With the z-score value, you can calculate the probability using available tables or, even better and faster, using our p-value calculator. Read on to look at an example of how to do it.

🙋 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.

## How this sampling distribution calculator works: an example

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:

**Input the population parameters**in the sampling distribution calculator (μ = 161.3, σ = 7.1)- Select
**left-tailed**, in this case. - Input the
**sample data**(n = 7, X = 160). - 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:

$\footnotesize\begin{align*}z_{score}&=\frac{X-μ}{σ/\sqrt{n}}\\\\&= \frac{160-161.3}{ 7.1/ \sqrt{7}}=-0.484433\end{align*}$zscore=σ/nX−μ=7.1/7160−161.3=−0.484433

$\footnotesize\begin{align*}P(\bar X<170)&=P(z_{score}<−0.484433)\\&=0.314039\end{align*}$P(Xˉ<170)=P(zscore<−0.484433)=0.314039

## How to find the mean of the sampling distribution?

If you know the **population mean**, you know the mean of the sampling distribution, as they're both the same. If you don't, you can assume your **sample mean** as the mean of the sampling distribution.

## FAQ

### What is the sampling distribution of the mean?

The sampling distribution of the mean describes the distribution of possible means you could obtain from infinitely sampling from a given population.

### How to calculate probability in sampling distribution?

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

### How to find the standard deviation of the sampling distribution?

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.

### What is the probability of getting a sample mean greater than the population mean?

The probability of getting a sample mean greater than μ (population mean) is **50%**, as long as your sampling distribution follows a normal distribution (this occurs if the population distribution is normal or the sample size is large).