The median on a density curve is located at the equal-areas point with 50% of the “mass” on either side.
Understanding the Median on a Density Curve
A density curve represents the distribution of a continuous variable. The total area under the curve is equal to 1, representing 100% of the data or probability.
- What is the "Mass"? In the context of a density curve, "mass" refers to the area under the curve. The area under the curve between two points represents the proportion of observations that fall within that range.
- The Equal-Areas Point: The median is the point on the horizontal axis (the variable's value) that divides the total area under the curve exactly in half. This means that the area under the curve to the left of the median is 0.5 (or 50%), and the area under the curve to the right of the median is also 0.5 (or 50%).
Think of it like cutting a shape (the area under the curve) into two equal pieces by a vertical line. The point where that line crosses the horizontal axis is the median.
Median vs. Mean
The reference also contrasts the median with the mean:
- Median: The equal-areas point (50% of the area on either side).
- Mean: The balancing point of the curve, as if the curve were a solid object and you were trying to find the point where it would balance perfectly.
The relationship between the mean and median depends on the shape of the density curve:
-
Symmetric Density Curve:
- The mean and median are equal.
- The balancing point and the equal-areas point are the same.
- Examples: Normal distribution (bell curve).
-
Skewed Density Curve:
- The mean is pulled away from the median in the direction of the long tail.
- Skewed Right (Positive Skew): The long tail is on the right. The mean is greater than the median (Mean > Median). The balancing point is pulled towards the higher values in the tail.
- Skewed Left (Negative Skew): The long tail is on the left. The mean is less than the median (Mean < Median). The balancing point is pulled towards the lower values in the tail.
Here is a summary of the key locations:
| Measure | Location on Density Curve | Relationship to Skewness |
|---|---|---|
| Median | The point dividing the area under the curve into two equal halves (50% on each side). | Less affected by extreme values in the tails. |
| Mean | The balancing point of the curve. | Pulled in the direction of the long tail. |
In essence, the median's position is solely determined by where 50% of the probability mass (area) lies, making it a robust measure of the center that is not overly influenced by outliers or the shape of the distribution's tails.
