Right Skew (Positive Skew)
Description: In a right-skewed distribution, the tail on the right side of the distribution is longer or fatter than the left side. This indicates that the majority of the data points are concentrated on the left.
Mean and Median: In a right-skewed distribution, the mean is typically greater than the median because the few high values in the long tail can pull the mean to the right.
Example: Income distribution in most countries is a classic example of right skewness. Most people earn near the median income, but a few high earners pull the mean income up, creating a long right tail.
2. Left Skew (Negative Skew)
Description: In a left-skewed distribution, the tail on the left side of the distribution is longer or fatter than the right side. This indicates that the majority of the data points are concentrated on the right.
Mean and Median: In a left-skewed distribution, the mean is typically less than the median because the few low values in the long tail can pull the mean to the left.
Example: The age at retirement might show a left skewness. While the typical retirement age might be around 60-65 years, few people who retire much earlier (like in their 40s) can create a long left tail.
Normalization of Skewed Data
Normalizing skewed data involves transforming the data to approximate a more symmetric, bell-shaped distribution, which can improve the performance and accuracy of many statistical methods and machine learning models. Several techniques can be used to reduce skewness:
1. Log Transformation
Applicable to: Right-skewed distributions.
Method: Apply the logarithm function to each data point. This is effective for reducing positive skewness because it compresses the long tail and stretches out the lower end of the distribution.
Example: If you have data on income that is right-skewed, applying a log transformation can normalize the data.
2. Square Root Transformation
Applicable to: Right-skewed distributions, particularly when data cannot take negative values (e.g., counts, areas).
Method: Take the square root of each data point.
Example: This can be useful in normalizing data where variance increases with the mean (e.g., biological assays).
3. Cube Root Transformation
Applicable to: Both right and left-skewed distributions.
Method: Take the cube root of each data point. This transformation is even milder than a square root and can handle negative values as well.
Example: Useful for certain environmental data sets where variables can have both positive and negative values.
4. Box-Cox Transformation
Applicable to: Right-skewed distributions.
Method: A family of power transformations that includes logarithmic and square root transformations, among others, parameterized by λ (lambda). It tries different values of λ to find the best approximation of a normal distribution.
Example: Often used in regression problems to stabilize variance and make the data more normally distributed.
5. Reflect and Log/Square Root Transformation
Applicable to: Left-skewed distributions.
Method: Reflect the data (i.e., multiply by -1) to make it right-skewed and then apply a log or square root transformation.