How to Detect and Handle Skewed Distributions in EDA

In Exploratory Data Analysis (EDA), detecting and handling skewed distributions is crucial for ensuring the accuracy and validity of any subsequent statistical analysis or machine learning models. Skewed distributions can affect the results, as most algorithms assume a normal or symmetric distribution of the data. Here’s a breakdown of how to detect and handle skewed distributions:

1. Understanding Skewed Distributions

Before diving into detection and handling, it’s important to understand what skewness is:

• Positive Skew (Right Skew): When the right tail (larger values) is longer than the left, resulting in a distribution where most values are clustered on the lower side, and few large values exist.

• Negative Skew (Left Skew): When the left tail (smaller values) is longer than the right, indicating that the data has more values on the higher side but with some extreme low values.

2. Detecting Skewed Distributions

A. Visual Inspection

Several visual tools can help identify skewness in a dataset:

• Histogram: This is a simple way to observe the frequency of values across intervals. A skewed distribution will not appear symmetric and will have a pronounced tail on one side.

• Boxplot: A boxplot can show the distribution’s symmetry. In a skewed dataset, the box will be shifted to one side, and the whiskers (lines extending from the box) will be uneven. The longer whisker will indicate the direction of skewness.

• Density Plot: This is similar to a histogram but smoothens the data into a continuous curve. If the data is skewed, the curve will show an asymmetrical distribution.

B. Statistical Methods

For a more quantitative approach, the skewness of a dataset can be calculated:

• Skewness Coefficient (Pearson’s or Fisher’s method): A skewness value close to 0 indicates a symmetric distribution. Positive values indicate right skewness, while negative values suggest left skewness.

  • Skewness ≈ 0 (symmetrical distribution)

  • Skewness > 0 (right/positive skew)

  • Skewness < 0 (left/negative skew)

  For example, the skewness formula is:

  $$text{Skewness} = frac{n}{(n-1)(n-2)} sum left(frac{x_i – mu}{sigma}right)^3$$

  Where:

  • $n$ is the number of data points,

  • $x_i$ is each data point,

  • $mu$ is the mean of the dataset,

  • $sigma$ is the standard deviation.

• Kurtosis: While kurtosis is a measure of the tail’s thickness, it can sometimes help identify extreme skewness when combined with skewness metrics.

C. Statistical Tests

Skewness can also be tested using statistical methods such as:

• D’Agostino’s K-squared Test: Tests the null hypothesis that the data is normally distributed. If the result is significant, it suggests skewness.

• Shapiro-Wilk Test: A normality test where a significant p-value indicates that the distribution is not normal, which may also be indicative of skewness.

3. Handling Skewed Distributions

Once a skewed distribution is detected, the next step is to decide on how to handle it. The handling method depends on the type of analysis you are performing, but some common strategies include:

A. Transformations

Applying mathematical transformations to the data can help reduce skewness and make the distribution more symmetric, especially for algorithms that assume normality.

• Log Transformation: If the distribution is right-skewed, applying a log transformation can compress large values and reduce the skewness.

  $$text{New Value} = log(text{Original Value})$$

  This is useful for variables such as income or price, which often follow a log-normal distribution.

• Square Root Transformation: If the data has a moderate right skew, a square root transformation may be helpful. This is effective when the range of values is limited but still stretches to the right.

• Box-Cox Transformation: This is a family of power transformations that can handle both positive and negative skewness. The transformation is flexible and can be adjusted to minimize skewness.

• Reciprocal Transformation: Taking the reciprocal of a variable ($frac{1}{x}$) can also help with reducing right skewness, but it works best when values are positive and the data has a significant right tail.

• Logit Transformation (for proportions): For data that is constrained between 0 and 1 (e.g., probabilities), the logit transformation can handle skewness:

  $$text{Logit}(p) = logleft(frac{p}{1-p}right)$$

  where $p$ is the probability.

B. Removing Outliers

Skewness is often caused by extreme values or outliers. Identifying and removing these outliers may help correct the skewed distribution:

• Z-scores or IQR (Interquartile Range) methods can be used to identify outliers. Any data point beyond 3 standard deviations from the mean (for Z-scores) or beyond 1.5 times the IQR (for IQR-based methods) can be considered an outlier.

C. Non-Parametric Methods

In some cases, it may be best to work with non-parametric methods, which do not assume any specific distribution. Techniques like decision trees, random forests, and k-nearest neighbors (KNN) can handle skewed data effectively without needing to address the skewness explicitly.

D. Data Partitioning

If there are extreme outliers that are legitimate but cause skewness, another approach could be to split the data into more manageable subsets. For example, separate data into categories based on ranges (e.g., low, medium, high) and treat each range individually.

E. Bootstrapping

Bootstrapping is a technique where you repeatedly sample from your dataset with replacement to create multiple simulated samples. This can help mitigate the impact of skewed data when performing statistical analysis or building machine learning models.

F. Use Robust Models

Certain machine learning models are robust to skewness and can handle such data directly:

• Tree-based models (e.g., Random Forest, Gradient Boosting) are less sensitive to skewed distributions because they work with decision thresholds that don’t assume normality.

• Neural Networks may also handle skewed data well, depending on the architecture.

4. Conclusion

Detecting and handling skewed distributions in EDA is essential for ensuring that your data is well-prepared for statistical analysis or machine learning modeling. Identifying skewness using both visual and statistical methods is a key first step. Once detected, strategies such as data transformations, outlier removal, and the use of non-parametric methods can help make the data more suitable for analysis, improving the performance and interpretability of your models.