The Coefficient of Kurtosis Calculator determines the kurtosis of a dataset, which measures the sharpness of its peak and the thickness of its tails compared to a normal distribution. It helps statisticians and analysts understand the distribution shape and identify whether the data has extreme values.
This calculator belongs to the statistical analysis tools category and is used in fields like finance, quality control, and data science to detect outliers, assess risk, and analyze data patterns.
Formula of Coefficient Of Kurtosis Calculator
Formula for Sample Kurtosis
Kurtosis = (n * Σ(x - x̄)⁴) / ((n-1) * (n-2) * (n-3) * s⁴)
Where:
n = Sample size
x = Individual data point
x̄ = Sample mean
s = Sample standard deviation
Σ = Summation symbol
Formula for Population Kurtosis
Kurtosis = (Σ(x - μ)⁴) / (N * σ⁴)
Where:
N = Population size
μ = Population mean
σ = Population standard deviation
Interpretation of Kurtosis
- Mesokurtic (Kurtosis ≈ 3): Indicates a normal distribution.
- Leptokurtic (Kurtosis > 3): Data has a sharper peak and heavier tails, indicating more outliers.
- Platykurtic (Kurtosis < 3): Data has a flatter peak and lighter tails, with fewer outliers.
Reference Table for Quick Kurtosis Insights
Below is a table to help interpret kurtosis values without detailed calculations:
| Kurtosis Value | Distribution Type | Characteristics |
|---|---|---|
| ≈ 3 | Mesokurtic | Normal distribution |
| > 3 | Leptokurtic | More outliers, sharper peak |
| < 3 | Platykurtic | Fewer outliers, flatter peak |
This table provides a quick reference for users analyzing distribution types.
Example of Coefficient Of Kurtosis Calculator
Let’s calculate kurtosis for a sample dataset: [2, 4, 6, 8, 10].
Step 1: Calculate the sample mean (x̄)
x̄ = (2 + 4 + 6 + 8 + 10) / 5 = 6
Step 2: Calculate the deviations and raise them to the fourth power
(2-6)⁴ = 256
(4-6)⁴ = 16
(6-6)⁴ = 0
(8-6)⁴ = 16
(10-6)⁴ = 256
Σ(x - x̄)⁴ = 256 + 16 + 0 + 16 + 256 = 544
Step 3: Calculate the sample standard deviation (s)
s = sqrt(Σ(x - x̄)² / (n-1)) = sqrt(40 / 4) = sqrt(10)
Step 4: Use the sample kurtosis formula
Kurtosis = (5 * 544) / ((4) * (3) * (2) * (10²)) = 2720 / 2400 ≈ 1.13
Interpretation: The kurtosis value of 1.13 indicates a platykurtic distribution with a flatter peak.
Most Common FAQs
Kurtosis measures the shape of a dataset’s distribution, specifically its peak sharpness and tail thickness. It helps identify outliers and assess whether data follows a normal distribution.
Kurtosis focuses on the peak and tails of a distribution, while skewness measures its asymmetry. Together, they provide a complete picture of a dataset’s shape.
Kurtosis cannot be negative because it involves raising deviations to the fourth power. However, the excess kurtosis (kurtosis - 3) can be negative, indicating a platykurtic distribution.
Kurtosis does not tell you anything about peakedness or flatness. There are infinitely peaked distributions with very low kurtosis, and there are low, flat-topped distributions with very high kurtosis. Examples of such distributions are given on the current Wikipedia page.
Thank you for your comment! You are correct that kurtosis does not directly measure peakedness or flatness, despite common misconceptions. Instead, kurtosis primarily describes the tails of a distribution, indicating the propensity for extreme values or outliers.
The terms leptokurtic, mesokurtic, and platykurtic were historically associated with peakedness but are now better understood in relation to the weight of a distribution’s tails. For example:
Leptokurtic distributions have heavier tails, meaning more extreme outliers.
Platykurtic distributions have lighter tails, indicating fewer extreme values.
The examples you mentioned, such as highly peaked distributions with low kurtosis or flat-topped distributions with high kurtosis, are valid and highlight why interpreting kurtosis strictly as peakedness can be misleading.
I will revise the explanation in the post to reflect this more accurate interpretation of kurtosis. Thank you for pointing this out! Let me know if you have further feedback or suggestions.