A distribution shows how often different values occur in a data set, and its shape tells you a lot before you do any calculations. Symmetric distributions are balanced around the center, while skewed distributions have a long tail on one side. Recognizing the shape helps you choose the best measure of center and describe typical values clearly.
This matters in science, economics, medicine, and any field where data can be pulled by extreme values.
In a symmetric distribution, the mean and median are usually close together because the data balance evenly on both sides. In a right-skewed distribution, a few large values pull the mean to the right, so the mean is usually greater than the median. In a left-skewed distribution, a few small values pull the mean to the left, so the mean is usually less than the median.
The median is often a better measure of a typical value when a distribution is strongly skewed.
Key Facts
- Symmetric distribution: mean ≈ median ≈ mode when the curve is single-peaked and balanced.
- Right-skewed distribution: mean > median because the long tail points toward larger values.
- Left-skewed distribution: mean < median because the long tail points toward smaller values.
- Mean formula: mean = (sum of all data values) / n.
- Median: the middle value when data are ordered, or the average of the two middle values if n is even.
- Skewness describes asymmetry: positive skew means right-skewed, and negative skew means left-skewed.
Vocabulary
- Distribution
- A distribution is the pattern showing how data values are spread across possible values.
- Symmetric distribution
- A symmetric distribution has roughly the same shape on both sides of its center.
- Right-skewed distribution
- A right-skewed distribution has a long tail extending toward larger values.
- Left-skewed distribution
- A left-skewed distribution has a long tail extending toward smaller values.
- Median
- The median is the middle value of an ordered data set and is resistant to extreme values.
Common Mistakes to Avoid
- Calling the tall side of the graph the direction of skew is wrong because skew is named for the long tail, not the peak.
- Assuming the mean is always the best center is wrong because extreme values can pull the mean away from a typical value in skewed data.
- Saying mean equals median in every bell-shaped graph is wrong because small asymmetries or outliers can separate them.
- Ignoring the scale of the horizontal axis is wrong because stretched or compressed axes can make a distribution look more or less skewed than it really is.
Practice Questions
- 1 For the data set 4, 5, 5, 6, 6, 7, 7, 8, 30, find the mean and median, then decide whether the distribution is likely right-skewed or left-skewed.
- 2 A left-skewed data set has a median of 72. Which is more likely for its mean: 65 or 80? Explain using the position of the tail.
- 3 A town reports household income using the mean, but most households earn less than the reported mean. Explain what distribution shape is likely present and why the median may be more informative.