# Normal Probability Calculator for Sampling Distributions

Created by Luis Hoyos
Reviewed by Steven Wooding
Last updated: Jun 05, 2023

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).

• 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. Distribution of American men's heights. You can note that Barack Obama's height is slightly above the mean. At the same time, Danny Devito and Shaquille O'neal are at the very extremes of the percentile, as they're extremely short and high, respectively. Source: Introductory Statistics Explained Edition 1.10CC, by Jeremy Balka

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 ($σ_{\bar X}$) is equal to the population standard deviation divided by the square root of the sample size:

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

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

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

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 $σ_{\bar X}$ 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?

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:

\footnotesize \begin{align*} z_{score}&=\frac{X-μ}{σ/\sqrt{n}}\\\\&= \frac{160-161.3}{ 7.1/ \sqrt{7}}=-0.484433 \end{align*}
\footnotesize \begin{align*} P(\bar X<170)&=P(z_{score}<−0.484433)\\&=0.314039 \end{align*}

## 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?

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 zscore = (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.

### 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 = σ/√n.
• If you don't have σ, estimate it with the sample standard deviation (s): σ = 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).

Luis Hoyos
As an example, you can see the distribution of means of a normal distribution (μ = 100, σ = 5). Please fill the calculator with your own values.
Population mean (μ)
Population standard deviation (σ)
Sample size (n)
What probability do you want?
X₁ < X̄ < X₂
X₁
X₂
Normal probability:
P(X₁ < X̄ < X₂)
0.734529
Standard deviation of the mean (σ/√n)
Z-score of X₁
Z-score of X₂
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