The median of a skewed density curve is the point that divides the area under the curve in half, meaning 50% of the data falls below it and 50% falls above it.
When a density curve is skewed, the median and the mean are not the same. Skewness indicates that the data is not symmetrically distributed.
Understanding Skewness
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Left Skew (Negative Skew): In a left-skewed distribution, the tail is longer on the left side. The mean is typically less than the median because the extreme values in the left tail pull the mean in that direction. The median remains at the center of the data's distribution.
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Right Skew (Positive Skew): In a right-skewed distribution, the tail is longer on the right side. The mean is typically greater than the median because the extreme values in the right tail pull the mean in that direction. Again, the median marks the center of the data.
Locating the Median
The median's location is best described as the point where the area to the left of it under the density curve equals the area to the right of it. Because the density curve represents probabilities, this means the area on each side of the median is 0.5 (or 50%). It's visually where you can "cut" the distribution in half by area.
Example
Imagine a dataset representing income. Income data is often right-skewed. This means that a few individuals earn significantly more than the majority. In this scenario, the median income might be $60,000, while the mean income is $80,000. The median is lower because it's not as influenced by the very high incomes at the far right of the distribution.
Key Differences: Mean vs. Median
The crucial distinction between the mean and median in skewed distributions lies in their sensitivity to extreme values.
| Feature | Mean | Median |
|---|---|---|
| Definition | Average of all data points | Middle value when data is sorted |
| Skewed Data | Significantly affected by outliers | Less affected by outliers |
| Location | Pulled towards the tail of the distribution | Remains closer to the "center" of the data mass |
In summary, the median of a skewed density curve represents the midpoint of the data, dividing the distribution into two equal areas, and it's less sensitive to extreme values than the mean.
