Key takeaway: A skewed distribution has one long tail stretching toward higher values (right skew) or lower values (left skew), which pulls the mean toward the tail and away from the median.
Grade 12 – AP StatisticsAP Statistics Unit 1Statistics and Data
Definition of Skewed Distribution
A skewed distribution is lopsided: most of the data cluster on one side, while a thin tail trails off in the other direction. The direction of the skew is the direction of the tail, not of the peak.
Incomes, home prices and waiting times are typically right-skewed, since a few large values stretch the tail. Scores on an easy exam are often left-skewed. Because the tail drags the mean, the median is the better measure of a typical value for skewed data.
Formula and notation
Compare the two measures of center:
\[ \text{right skew: } \,\overline{x}\gt\text{median},\qquad \text{left skew: } \,\overline{x}\lt\text{median}. \]
Roughly equal mean and median suggest a symmetric shape. The skew is read from a histogram or dotplot, never from the box alone.
Key facts
- The skew direction names the tail: a long tail to the right is right-skewed.
- The mean follows the tail; the median stays close to the bulk of the data.
- For skewed data, report the median and the interquartile range rather than the mean and standard deviation.
- Strong skew often comes from a natural lower bound, like zero, with no matching upper bound.
Skewed Distribution example
Nine houses on a street sold for the following prices, in thousands of dollars: 212, 225, 231, 238, 244, 251, 259, 268 and 940. Compare the mean and median and describe the shape.
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Median: the 5th value of nine, so the median is \(244\), or \$244,000.
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Total: \(212+225+231+238+244+251+259+268+940=2{,}868\).
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Mean: \(\frac{2868}{9}\approx 318.7\), or about \$318,700.
The mean (about \$318,700) is far above the median (\$244,000), so the data form a right-skewed distribution pulled by the \$940,000 sale.
Common mistake
Watch out
A student looks at a histogram whose tallest bars sit on the right with a tail trailing left and calls it right-skewed because “most of the data are on the right”.
Why it is wrong
Skew is named for the tail. A tail pointing left means the shape is left-skewed, even though the bulk sits on the right. In that shape the mean is below the median, the opposite of the house-price example.
Related terms and lessons
Browse the glossary: A–Z list · AP Statistics vocabulary · Statistics and Data