Skewed Distribution — Definition, Formula & Examples
A skewed distribution is a set of data whose graph is lopsided. Most values bunch up on one side, and a long, thin tail stretches out toward the other side.
A distribution is skewed when it is not symmetric about its center. It is positively (right) skewed when its third standardized moment is greater than 0, and negatively (left) skewed when that quantity is less than 0. The direction of skew is named for the side the tail points toward, not the side where the peak sits.
How It Works
Look at where the tail is. If the tail stretches toward large values, the distribution is right-skewed. If it stretches toward small values, it is left-skewed. A quick numerical clue is that extreme values pull the mean toward the tail while the median barely moves. For right skew you usually see mean > median, and for left skew you usually see mean < median. Treat this as a rule of thumb: exceptions exist, especially with small or discrete data sets.
Worked Example
Problem: Seven students report how many hours they spent on homework last week: 2, 3, 3, 4, 5, 6, 12. Use the mean and median to describe the shape of the distribution.
Step 1: Find the mean by adding the values and dividing by 7.
Step 2: The data are already in order. With 7 values, the median is the 4th value.
Step 3: Compare them. The single large value 12 pulls the mean above the median, and the tail points toward the high end.
Answer: The mean (5) is greater than the median (4), and the value 12 forms a long tail on the high side. The distribution is right-skewed (positively skewed).
Visualization
Why It Matters
Skew tells you which measure of center to report. Income, home prices, and wait times are usually right-skewed, so economists and AP Statistics students report the median and IQR instead of the mean and standard deviation. Many tests in a first statistics course also assume roughly symmetric data, so you need to spot strong skew before using them.
Common Mistakes
Mistake: Naming the skew after the side where the peak or the bulk of the data sits. For example, calling a graph with its hump on the left 'left-skewed'.
Correction: Skew is named for the tail. A hump on the left with a long tail to the right is right-skewed. Ask yourself which way the tail points.