Skewed data and skewness

Skewed data is data whose distribution is not symmetric about its mean: one tail of the distribution is longer than the other. The skewness measures this asymmetry (0 for a symmetric distribution such as the normal one).

Positive (right) skew

• The tail is longer on the right; a few very large values pull the mean up.
• Relation: mode < median < mean.
• Examples: income, house prices, number of website visits per user.

Negative (left) skew

• The tail is longer on the left; a few very small values pull the mean down.
• Relation: mean < median < mode.
• Examples: marks in an easy exam (most students score high), age at death in developed countries.

 Positive skew          Symmetric          Negative skew
 |\                       /\                       /|
 | \___                  /  \                  ___/ |
 |     \____ tail      _/    \_          tail ____/  |
   mode<median<mean    mean=median=mode   mean<median<mode

Why real-life data are skewed

• Natural lower limit with no upper limit (income, price, time cannot be below zero but can be very large).
• A few extreme cases (rich people, big transactions, viral pages) exist beside many ordinary ones: "rich get richer" and power-law effects.
• Natural growth processes are multiplicative.
• Ceiling or floor effects in tests, and outliers or errors.

Skewed data affect the mean and many algorithms; they are often handled by log or square-root transformation, and by using median instead of mean.