Enter the population mean, standard deviation, sample size, and target sample mean to calculate the probability of obtaining that sample mean in a normal distribution.

**Probability of Obtaining the Sample Mean:**

## Understanding Sampling Distribution

In a normal distribution, the sampling distribution of the sample mean describes the probabilities of different possible values for the sample mean. The formula for calculating the probability of a given sample mean is:

where:

- \(\overline{X}\) = Target sample mean
- \(\mu\) = Population mean
- \(\sigma\) = Population standard deviation
- \(n\) = Sample size

### Real-Life Example: Quality Control in Manufacturing

Suppose a manufacturer produces components with an average length of 100 mm and a standard deviation of 15 mm. If a random sample of 30 components is taken, what is the probability that the sample mean length will be 102 mm or greater?

Here’s how the calculation works:

**Step 1:**Set the values: population mean (μ) = 100, population standard deviation (σ) = 15, sample size (n) = 30, and target sample mean (\(\overline{X}\)) = 102.**Step 2:**Calculate the standard error of the mean:$$ \text{Standard Error} = \frac{\sigma}{\sqrt{n}} = \frac{15}{\sqrt{30}} \approx 2.74 $$**Step 3:**Find the z-score:$$ Z = \frac{102 - 100}{2.74} \approx 0.73 $$**Step 4:**Calculate the probability based on the z-score to find the likelihood of obtaining this sample mean or greater.

Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.