Transforming Skewed Distributions

Skewness is a measure of how symmetrical or asymmetrical a data distribution is. A distribution is asymmetrical if its left and right sides are not mirror images. Skewness helps us understand the shape of the data and whether it deviates from a normal (bell-shaped) distribution.

Types of Skewness

1. Zero Skew (Symmetrical Distribution):
• The left and right sides of the peak are mirror images.
• Examples: Normal distribution, uniform distribution, and some bimodal distributions.
• Key point: In zero skew, the mean and median are approximately equal.
2. Right Skew (Positive Skew):
• The right tail (the longer part of the curve) extends further than the left.
• Common in data with extreme high values, such as income or sales data.
• Key point: In right skew, the mean > median.
3. Left Skew (Negative Skew):
• The left tail extends further than the right.
• Common in data where most values are high with a few very low values, such as test scores.
• Key point: In left skew, the mean < median.

How to Check Skewness

1. Visual Check:
• Plot a histogram to observe the shape of the distribution.
• If the data is symmetrical, it has zero skew. If it leans right or left, it's skewed.
2. Mathematical Check:
• Use Pearson’s median skewness formula

• A skewness close to 0 indicates symmetry, while higher positive or negative values show skew.

Handling Skewed Data

1. Do Nothing:
• Mild skewness often doesn't significantly affect statistical tests like linear regression.
2. Choose a Different Model:
• Use models that don't assume normality, such as non-parametric tests or generalized linear models.
3. Transform the Data:
• Apply a mathematical transformation to reduce skewness and make the distribution closer to normal.

Transformations for Skewness

Note: Reflection reverses the direction of the data. The reflection is calculated as K+1−xK + 1 - xK+1−x, where KKK is the largest observation.

Handling Skewed Data

1. Do Nothing:
• Mild skewness often doesn't significantly affect statistical tests like linear regression.
2. Choose a Different Model:
• Use models that don't assume normality, such as non-parametric tests or generalized linear models.
3. Transform the Data:
• Apply a mathematical transformation to reduce skewness and make the distribution closer to normal.

Transformations for Skewness

Note: Reflection reverses the direction of the data. The reflection is calculated as K+1−xK + 1 - xK+1−x, where KKK is the largest observation.