ML Mean Median Mode

ML Central Tendency: Mean, Median, & Mode Calculations

Central tendency metrics describe the center of a numerical data distribution. Selecting the correct metric depends on data distribution skewness and the presence of outliers.

Metric Characteristics

flowchart TD
    A["Central Tendency Metrics"] --> B["Mean (Arithmetic Average)"]
    A --> C["Median (Middle Value)"]
    A --> D["Mode (Most Frequent Value)"]
    B --> E["Sensitive to Outliers!"]
    C --> F["Robust Against Outliers!"]
    D --> G["Ideal for Categorical Data"]

Mathematical Definitions

• Mean ($\mu$): $rac{1}{N} \sum_{i=1}^{N} x_i$
• Median: Middle value of sorted array (or mean of two middle values if $N$ is even).
• Mode: Data value(s) appearing with highest frequency.

Practical Code Example

import numpy as np
from scipy import stats
from typing import Dict, Any

def calculate_central_tendency(data: np.ndarray) -> Dict[str, Any]:
    mean_val = float(np.mean(data))
    median_val = float(np.median(data))
    mode_result = stats.mode(data, keepdims=True)
    mode_val = float(mode_result.mode[0])

    return {
        "mean": mean_val,
        "median": median_val,
        "mode": mode_val
    }

if __name__ == "__main__":
    # Data with a heavy outlier (1000)
    incomes = np.array([45000, 48000, 50000, 52000, 55000, 1000000])
    metrics = calculate_central_tendency(incomes)

    print("--- Income Distribution Metrics ---")
    print(f"Mean Income   : ${metrics['mean']:,.2f} (Distorted by outlier)")
    print(f"Median Income : ${metrics['median']:,.2f} (Robust representative)")
    print(f"Mode Income   : ${metrics['mode']:,.2f}")

Best Practices & Gotchas

• Outlier Sensitivity: Highly skewed distributions (like income or real estate prices) should use Median rather than Mean.
• Multimodal Distributions: Scipy's stats.mode() returns the smallest modal value if multiple modes exist.
• Empty Datasets: Ensure input arrays are non-empty before calculating metrics.

Self-Check Challenge

Calculate the mean and median of [10, 20, 30, 40, 500] using np.mean() and np.median(). Explain the difference.