You know two measures of center from the mean as a fair share and the median. They answer different questions, and in skewed data they can be far apart.
Key idea
The mean $\bar{x} = \frac{\sum x}{n}$ is the balance point of the data. The median is the middle value of the sorted data. The median is resistant: a few extreme values barely move it. The mean is not: it is pulled toward a long tail or an outlier.
Why it works
The mean uses the size of every value. If one value grows by 30, the total grows by 30 and the mean grows by $30/n$, no matter how strange that value is. The median only uses the order of the values: making the largest value even larger does not change which value is in the middle. That is why a few huge incomes raise a town's mean income but leave its median alone.
Worked example: Mean and median with an outlier
Seven students' commute times, in minutes: $8, 10, 10, 12, 15, 17, 47$. Find the mean and the median.
- Mean: $$8 + 10 + 10 + 12 + 15 + 17 + 47 = 119$$ and $119 \div 7 = 17$.
- Median: the values are already sorted; the 4th of 7 values is $12$.
- Only one student (17 min) is at or above the mean, because the 47 drags the mean up.
Answer: Mean 17 minutes, median 12 minutes.
Worked example: Fixing a typo
The 47 was a typo for 19. What happens to the mean and the median?
- The total drops by $47 - 19 = 28$: new total $91$, new mean $91 \div 7 = 13$.
- Sorted data: $8, 10, 10, 12, 15, 17, 19$. The middle value is still $12$.
- The mean moved 4 minutes; the median did not move at all.
Answer: Mean 13, median 12.
Common mistake
Expecting the mean and median to match. They match only for symmetric data. In data skewed right the mean is usually larger than the median, and in data skewed left it is usually smaller. Check by finding both: a big gap is a clue that the data are skewed or have an outlier.
- Symmetric data: mean and median are close; either is fine.
- Skewed data or outliers: report the median as the typical value (and say why).
- Need the total, such as total rainfall: the mean times n gives it, the median does not.