Use this calculator to visualize the Central Limit Theorem by providing values for the population mean, standard deviation, and sample size. The calculator will simulate a distribution of sample means.

**Sample Mean (x̄):**

**Sample Standard Deviation (s):**

## Central Limit Theorem Explained

The **Central Limit Theorem (CLT)** states that the distribution of the sample means approaches a normal distribution, regardless of the population's original distribution, as the sample size becomes large enough.

### Key Points

**Population Mean (μ)**: The average of all values in the population.**Sample Mean (x̄)**: The average of the values in a random sample taken from the population.**Sample Size (n)**: The number of observations in a sample.**Standard Error (SE)**: The standard deviation of the sampling distribution, calculated as \( \text{SE} = \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the population standard deviation.

### CLT Formula

The mean of the sampling distribution of the sample mean is equal to the population mean:

The standard deviation of the sampling distribution (also called the standard error) is given by:

### Conditions for Applying the Central Limit Theorem

The Central Limit Theorem applies under the following conditions:

**Random Sampling:**The data must be obtained through a random process. This ensures that the sample is representative of the population.**Independent Observations:**The sampled observations must be independent of each other. This is generally satisfied if the sample size is small relative to the population (usually less than 10% of the population).**Sample Size:**The sample size should be sufficiently large. A common rule of thumb is that the sample size should be at least 30. For highly skewed distributions, larger sample sizes may be required.**Population Distribution:**If the population distribution is normal, the CLT holds even for small sample sizes. If the population distribution is not normal, the sample size must be large enough for the sample means to approximate a normal distribution.

### Why Is CLT Important?

- It allows for statistical inference, making it possible to apply hypothesis testing and confidence intervals.
- It justifies the use of the normal distribution in many practical applications, even when the underlying population distribution is unknown or not normal.

### References

## Implementations

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.