The **MAD (Median Absolute Deviation) rule** is a statistical outlier detection method that uses the median and the median absolute deviation as robust estimators to identify outliers in a dataset. 

It is particularly useful when dealing with **data that may not follow a normal distribution** or when there are potential outliers that can significantly impact the mean and standard deviation.

## How to use MAD rule

1. **Calculate the Median**: Compute the median of the dataset, which is the middle value when the data is sorted;
2. **Calculate the Median Absolute Deviation (MAD)**: For each data point, find the absolute difference between the data point and the median. The MAD is the median of these absolute differences;
3. **Define a Threshold**: Choose a threshold value (usually a constant, e.g., 2 or 3 times the MAD) to determine how far a data point can deviate from the median before being considered an outlier;
4. **Identify Outliers**: Any data point that has an absolute difference from the median greater than the threshold is considered an outlier.
>Note
>
>Mathematically, the absolute difference between two values, `A` and `B`, is denoted as `|A - B|`, where `"|"` represents the absolute value function. This function returns the positive value of the difference between A and B.
## MAD rule implementation
```python
def mad_rule_outlier_detection(data, threshold=3.0):
  
    # Calculate the median
    median = np.median(data)
    
    # Calculate the absolute differences from the median
    abs_diff = np.abs(data - median)
    
    # Calculate the MAD (Median Absolute Deviation)
    mad = np.median(abs_diff)
    
    # Define the threshold for outliers
    outlier_threshold = threshold * mad
    
    # Identify outliers based on the threshold
    outliers = [x for x in data if np.abs(x - median) > outlier_threshold]
    
    return outliers
```

## MAD vs 1.5 IQR rule


What is the main advantage of using MAD for outlier detection?

Anomaly detection is integral to any data scientist's work: high-quality, cleaned, and well-prepared data is the key to success for almost any machine learning problem. 

Let's delve into the concept of data anomalies, their classification, and their impact on decision-making. Throughout this exploration, we will gain a comprehensive understanding of these topics, accompanied by real-life examples to illustrate their significance.

Statistical methods play a crucial role in various domains, including finance, cybersecurity, manufacturing, and healthcare, where detecting anomalies can prevent fraud, ensure data quality, and improve overall system reliability. These methods are particularly valuable when dealing with large datasets and complex systems, as they provide a systematic and quantitative approach to anomaly detection.

Now, let's explore how machine learning methods can effectively address anomalies. We will examine various techniques, including clustering, regularization, dropout, and more, that are employed for this purpose.

Data Anomaly Detection

Median Absolute Deviation

How to use MAD rule

MAD rule implementation

MAD vs 1.5 IQR rule