Skewed Distribution
In short: Skewed distribution is a data shape with one long tail: a right-skewed one stretches toward large values and pulls the mean above the median, and a left-skewed one does the opposite.
AP StatisticsStatisticsAP Statistics Unit 1
What is Skewed Distribution?
A skewed distribution is lopsided. Most values bunch up on one side, and a thin tail trails away on the other. The tail’s direction names the skew: tail to the right means right-skewed (positively skewed), tail to the left means left-skewed.
Skew changes which summary to trust. Extreme values in the tail drag the mean toward them, while the median stays near the bulk of the data. In right-skewed data, the mean is greater than the median; in left-skewed data, the mean is less than the median. For lopsided shapes, the median and IQR describe a typical value and spread better than the mean and standard deviation.
How to spot it
- A histogram has a tall cluster at one end and a long, thin tail stretching to the other.
- The variable is bounded on one side but not the other, as with waiting times, incomes or house prices.
- The mean and median are given and they differ noticeably.
- A question asks which measure of center best describes a typical value.
The method
- Look at where the peak sits and which side has the long tail; name the skew after the tail.
- Compare mean and median: mean above median points to right skew, below points to left skew.
- Choose median and IQR as the summary when the shape is strongly skewed or has outliers.
- Describe the shape in context, mentioning the skew, center and spread together.
Skewed Distribution: solved example
Problem
Nine patients at a walk-in clinic waited 2, 3, 3, 4, 5, 5, 6, 8 and 24 minutes. Describe the shape and say which center to report.
Solution
The median is the 5th value: 5 minutes. The sum is 60, so the mean is \(\frac{60}{9} \approx 6.67\) minutes.
The mean exceeds the median, and the single wait of 24 minutes stretches the tail to the right, so the distribution is skewed right.
Report the median of 5 minutes as the typical wait, because the 24-minute wait inflates the mean.
Try it
Question 1
On an easy exam, the median score is 88 and the mean is 79. Which way is the distribution skewed?
Show the answer
The mean is below the median, pulled down by a tail of low scores: skewed left.
Question 2
A city’s home prices are strongly right-skewed. Why do news reports quote the median price?
Show the answer
A few luxury homes pull the mean up, so the median represents a typical home better.
Skewed Distribution vs. Outlier
Skew describes the overall lopsided shape of the whole distribution. An outlier is one unusual value far from the others. A single outlier can cause apparent skew in a small data set, but a strongly skewed distribution has a long, gradual tail with many values in it.
Keep going
Lessons
Worksheets and quizzes
- Graphs of Quantitative Data and Shape – Worksheet
- Graphs of Quantitative Data and Shape – Quiz
- Mean, Median, Standard Deviation and Outliers – Worksheet
- Mean, Median, Standard Deviation and Outliers – Quiz
Related words
Whole dictionary: A–Z · AP Statistics · Statistics