1. What is Kurtosis?

Kurtosis is a statistical measure that describes the shape of a distribution's tails in relation to its overall shape. It measures the "tailedness" of the probability distribution of a real-valued random variable.

2. How Does the Calculator Work?

The calculator uses the kurtosis formula for ungrouped data:

Where:

• \( x_i \) — Individual data points
• \( \mu \) — Mean of the data
• \( \sigma \) — Standard deviation of the data
• \( n \) — Number of data points

Explanation: This formula calculates the fourth standardized moment about the mean, providing a measure of the distribution's tail heaviness compared to a normal distribution.

3. Importance of Kurtosis

Details: Kurtosis helps identify whether data are heavy-tailed or light-tailed relative to a normal distribution. High kurtosis indicates heavy tails and more outliers, while low kurtosis indicates light tails and fewer outliers.

4. Using the Calculator

Tips: Enter numerical data separated by commas. The calculator will compute the kurtosis along with mean, standard deviation, and sample size. Ensure data contains at least 4 values for meaningful results.

5. Frequently Asked Questions (FAQ)

Q1: What does kurtosis value indicate?
A: For a normal distribution, kurtosis is 3. Values >3 indicate leptokurtic (heavy-tailed), values <3 indicate platykurtic (light-tailed).

Q2: What is excess kurtosis?
A: Excess kurtosis = kurtosis - 3. This centers the normal distribution at 0, making interpretation easier.

Q3: When is kurtosis most useful?
A: In risk management, finance, and quality control where tail risk and outlier detection are important.

Q4: What are limitations of kurtosis?
A: Sensitive to sample size, can be influenced by extreme values, and doesn't distinguish between different tail shapes.

Q5: How does kurtosis relate to skewness?
A: Skewness measures asymmetry, kurtosis measures tail heaviness. Both are important for understanding distribution shape.