The mean and median are two common ways to describe the center of a dataset. Both try to answer what a typical value looks like, but they do it in different ways. The mean uses every number in the dataset, while the median depends on the order of the values.
Choosing the right measure matters because a single very large or very small value can change the story the data tells.
The mean is the balancing point of the data, so values far from the center pull it in their direction. The median is the middle value, so it is more resistant to outliers and skewed distributions. For roughly symmetric data with no extreme values, the mean and median are often close and either may work well.
For skewed data, such as incomes or house prices, the median often better represents a typical value.
Key Facts
- Mean = sum of all values divided by number of values, x̄ = (x1 + x2 + ... + xn) / n
- Median = middle value after the data are ordered from least to greatest.
- If n is odd, the median is the value in position (n + 1) / 2.
- If n is even, the median is the average of the two middle values.
- Outliers strongly affect the mean but usually have little effect on the median.
- In a right-skewed distribution, mean > median; in a left-skewed distribution, mean < median.
Vocabulary
- Mean
- The mean is the arithmetic average found by adding all data values and dividing by the number of values.
- Median
- The median is the middle value of an ordered dataset, or the average of the two middle values when there are an even number of values.
- Outlier
- An outlier is a data value that is unusually far from the rest of the dataset.
- Skew
- Skew describes a distribution that has a longer tail on one side than the other.
- Distribution
- A distribution shows how data values are spread across possible values.
Common Mistakes to Avoid
- Forgetting to order the data before finding the median. The median is based on position in the sorted list, not the original order.
- Using the mean for a dataset with extreme outliers without checking their effect. Outliers can pull the mean away from the value most data points are near.
- Assuming the mean and median always give the same typical value. They are close for symmetric data but can be very different for skewed data.
- Finding the median of an even-sized dataset by choosing only one middle value. For an even number of values, the median is the average of the two middle values.
Practice Questions
- 1 Find the mean and median of the dataset: 4, 6, 7, 8, 10.
- 2 Find the mean and median of the dataset: 25, 28, 30, 32, 35, 120. Which measure is more affected by the outlier?
- 3 A neighborhood report lists home prices, and one mansion is much more expensive than all other homes. Should the report use the mean or median to describe a typical home price? Explain your reasoning.