Use this calculator to compute probabilities, percentiles, and visualize the normal distribution. Input the population mean, standard deviation, and the specific point(s) to calculate left, right, between, or outside probabilities.

**Calculated Probability: **

## Understanding the Normal Distribution

The **normal distribution**, also known as the Gaussian distribution, is one of the most important probability distributions in statistics. It describes how the values of a variable are distributed — most values cluster around a central region, and values become less frequent as they move further from the center.

### Key Characteristics of the Normal Distribution

**Symmetry**: The normal distribution is perfectly symmetrical around its mean. This means that the left and right halves of the distribution are mirror images.**Bell-shaped Curve**: The distribution forms a bell shape, with the peak at the mean (μ) and tails extending indefinitely in both directions.**Mean, Median, Mode**: In a normal distribution, the mean, median, and mode are equal and all located at the center of the distribution.**68-95-99.7 Rule**: Also known as the empirical rule, about 68% of data falls within one standard deviation (σ) of the mean, 95% within two, and 99.7% within three standard deviations.

### Conditions for Using the Normal Distribution

While the normal distribution applies to many real-world scenarios, it is important to check the following conditions:

**Independence**: The data points should be independent of each other. One data point should not influence the next.**Sample Size**: When applying the Central Limit Theorem (CLT), ensure your sample size is sufficiently large (generally, n ≥ 30) to assume a normal distribution of sample means, regardless of the population distribution.**No Extreme Skewness**: The normal distribution is symmetric. If your data has extreme skewness or outliers, it may not fit a normal distribution well, even with a large sample size.

### Caveats and Assumptions

- Small sample sizes or data with significant skewness may require a transformation or alternative distributions (e.g., t-distribution) for accurate analysis.
- The
**Central Limit Theorem (CLT)**applies to sample means, meaning that with a large enough sample size, the distribution of the sample means will approximate a normal distribution, even if the population distribution is not normal. - Assuming normality is a common approach in many statistical models, but it is always important to check the fit before making inferences.

### Further Reading

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.