Skewness and Kurtosis Calculator

Analyze the symmetry and tailedness of your data distribution.

Understanding Skewness and Kurtosis

💡 Skewness and Kurtosis are fundamental statistical measures used to describe the shape of a data distribution. They provide insights into the symmetry and peakedness of data, beyond basic measures like mean and variance.

What is Skewness?

Skewness measures the asymmetry of a distribution:

  • Positive Skew: The tail on the right side of the distribution is longer/more pronounced.
  • Negative Skew: The tail on the left side is longer/more pronounced.
  • Zero Skew: The distribution is symmetric, like a normal distribution.

The formula for skewness (\(g_1\)) is:

\[ g_1 = \frac{n}{(n-1)(n-2)} \sum_{i=1}^{n} \left(\frac{x_i - \bar{x}}{s}\right)^3 \] Where:
  • \(n\): Number of data points
  • \(\bar{x}\): Mean of the data
  • \(s\): Standard deviation
  • \(x_i\): Individual data points

What is Kurtosis?

Kurtosis measures the "tailedness" of a distribution:

  • Mesokurtic (K=3): A normal distribution.
  • Leptokurtic (K>3): More extreme outliers (heavier tails).
  • Platykurtic (K<3): Fewer outliers (lighter tails).

The formulas for kurtosis are:

Fisher's Kurtosis (Excess Kurtosis)

Fisher's kurtosis, also known as excess kurtosis, measures how much the tails deviate from a normal distribution:

\[ g_2 = \frac{n(n+1)}{(n-1)(n-2)(n-3)} \sum_{i=1}^{n} \left(\frac{x_i - \bar{x}}{s}\right)^4 - \frac{3(n-1)^2}{(n-2)(n-3)} \] Where the terms are the same as in skewness.

Pearson's Kurtosis

Pearson's kurtosis includes the baseline kurtosis of a normal distribution (3) and is calculated by adding 3 to Fisher's kurtosis:

\[ K = g_2 + 3 \] Where:
  • \(g_2\): Fisher's kurtosis (excess kurtosis)
  • \(K\): Pearson's kurtosis

Key Concepts

  • Symmetry: Skewness reveals the symmetry (or lack thereof) of the data distribution.
  • Peakedness: Kurtosis highlights how heavy or light the tails of the distribution are compared to a normal distribution.
  • Higher Moments: Both skewness and kurtosis are part of the higher moments of a dataset (third and fourth moments).

Note: While skewness and kurtosis provide useful insights, interpreting them should be done alongside visualizations and other statistical measures.

Real-Life Applications

Skewness and Kurtosis are widely used in various fields:

  • Finance: Measuring the risk of extreme events in stock returns.
  • Healthcare: Analyzing test results or patient characteristics.
  • Social Sciences: Studying income distributions or survey data.

Python Implementation

Python Function for Skewness and Kurtosis
from scipy.stats import skew, kurtosis

# Example data
data = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

# Calculate skewness
data_skewness = skew(data)

# Calculate Fisher's kurtosis (excess kurtosis is the default)
fisher_kurtosis = kurtosis(data, fisher=True)

# Calculate Pearson's kurtosis
pearson_kurtosis = fisher_kurtosis + 3

# Print results
print(f"Skewness: {data_skewness:.4f}")
print(f"Fisher's Kurtosis (Excess): {fisher_kurtosis:.4f}")
print(f"Pearson's Kurtosis: {pearson_kurtosis:.4f}")

R Implementation

R Function for Skewness and Kurtosis
# Load the required library
library(moments)

# Example data
data <- c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)

# Calculate skewness
data_skewness <- skewness(data)

# Calculate Pearson's kurtosis (default from moments::kurtosis())
pearson_kurtosis <- kurtosis(data)

# Calculate Fisher's kurtosis (excess kurtosis)
fisher_kurtosis <- pearson_kurtosis - 3

# Print results
cat(sprintf("Skewness: %.4f\n", data_skewness))
cat(sprintf("Fisher's Kurtosis (Excess): %.4f\n", fisher_kurtosis))
cat(sprintf("Pearson's Kurtosis: %.4f\n", pearson_kurtosis))

JavaScript Implementation

JavaScript Function for Skewness and Kurtosis
function calculateMean(data) {
    return data.reduce((a, b) => a + b, 0) / data.length;
}


function calculateSD(data, mean = null) {
    const m = mean === null ? calculateMean(data) : mean;
    const variance = data.reduce((a, b) => a + Math.pow(b - m, 2), 0) / (data.length - 1);
    return Math.sqrt(variance);
}

function calculateSkewness(data) {
    // Input validation
    if (!Array.isArray(data)) {
        throw new Error('Input must be an array');
    }
    if (data.length < 3) {
        throw new Error('Skewness requires at least 3 data points');
    }
    if (!data.every(x => typeof x === 'number' && !isNaN(x))) {
        throw new Error('All elements must be valid numbers');
    }

    const n = data.length;
    const mean = calculateMean(data);
    const sd = calculateSD(data, mean);

    // Check for division by zero
    if (sd === 0) {
        return 0; // All values are identical
    }

    const skewnessFactor = (n / ((n - 1) * (n - 2)));
    return skewnessFactor * data.reduce((sum, x) => sum + Math.pow((x - mean) / sd, 3), 0);
}

function calculateKurtosis(data, fisher = true) {
    // Input validation
    if (!Array.isArray(data)) {
        throw new Error('Input must be an array');
    }
    if (data.length < 4) {
        throw new Error('Kurtosis requires at least 4 data points');
    }
    if (!data.every(x => typeof x === 'number' && !isNaN(x))) {
        throw new Error('All elements must be valid numbers');
    }

    const n = data.length;
    const mean = calculateMean(data);
    const sd = calculateSD(data, mean);

    // Check for division by zero
    if (sd === 0) {
        return fisher ? -3 : 0; // All values are identical
    }

    const kurtosisFactor = (n * (n + 1)) / ((n - 1) * (n - 2) * (n - 3));
    const fourthMoment = data.reduce((sum, x) => sum + Math.pow((x - mean) / sd, 4), 0);
    const correction = (3 * Math.pow(n - 1, 2)) / ((n - 2) * (n - 3));

    const excessKurtosis = kurtosisFactor * fourthMoment - correction;
    return fisher ? excessKurtosis : excessKurtosis + 3;
}

// Example usage with error handling
try {
    const data = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10];

    const skewness = calculateSkewness(data);
    const fisherKurtosis = calculateKurtosis(data, true);
    const pearsonKurtosis = calculateKurtosis(data, false);

    console.log(`Skewness: ${skewness.toFixed(10)}`);
    console.log(`Fisher's Kurtosis (Excess): ${fisherKurtosis.toFixed(10)}`);
    console.log(`Pearson's Kurtosis: ${pearsonKurtosis.toFixed(10)}`);
} catch (error) {
    console.error('Error:', error.message);
}

Note: While the formulas for skewness and kurtosis are standardized, the calculated values may differ slightly between JavaScript, Python, and R due to differences in numerical precision and rounding methods used by each language.

Further Reading

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Senior Advisor, Data Science | [email protected] | + posts

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.

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