1. What is Kurtosis?

Kurtosis is a statistical measure that describes the shape of a probability distribution, specifically how heavily the tails of the distribution differ from the tails of a normal distribution. It measures the "tailedness" of the distribution.

2. How Does the Calculator Work?

The calculator uses the kurtosis formula:

Where:

• \( f_i \) — Frequency of each value
• \( x_i \) — Individual data values
• \( \mu \) — Mean of the data
• \( \sigma \) — Standard deviation of the data
• \( N \) — Total number of observations

Explanation: Kurtosis is calculated as the fourth standardized moment about the mean, normalized by the standard deviation raised to the fourth power.

3. Importance of Kurtosis

Details: Kurtosis helps identify whether data are heavy-tailed or light-tailed relative to a normal distribution. It's crucial in risk management, finance, and quality control for understanding extreme events.

4. Using the Calculator

Tips: Choose between raw data (individual values) or grouped data (values with frequencies). Enter comma-separated values. For grouped data, ensure values and frequencies arrays have the same length.

5. Frequently Asked Questions (FAQ)

Q1: What do different kurtosis values mean?
A: Normal distribution has kurtosis = 3. Values > 3 indicate heavy tails (leptokurtic), values < 3 indicate light tails (platykurtic).

Q2: What is excess kurtosis?
A: Excess kurtosis = kurtosis - 3. It's often used to compare directly with normal distribution (excess kurtosis = 0).

Q3: When is high kurtosis important?
A: High kurtosis indicates more frequent extreme values, important in finance for risk assessment and in quality control for outlier detection.

Q4: Can kurtosis be negative?
A: The formula always produces positive values. Excess kurtosis can be negative when distribution has lighter tails than normal.

Q5: What are limitations of kurtosis?
A: Kurtosis doesn't distinguish between left and right tails, and can be sensitive to outliers in small samples.