Mean vs Median: Balancing Point of Population Density Graphs

[MeatyDumplings]

in population density graphs, the median is known to be the line where it splits the graph into two equal sections. And the mean, according to the textbook, is the 'balancing point" of the graph. What exactly is this "balancing point"? what are its properties and how do i calculate the coordinate/location?
thanks in advance


sorry if this is in the wrong section, the Statistics section says not to post homework probelms there...

 

[Gokul43201]

What's on the x-axis?

The mean is the flat line drawn such that the (rectangular) area to the left of it equals the area to the left of the density curve.

 

[MeatyDumplings]

the density cirve looks like this, some line intersecting the x-axis at (x, 0) is the mean

http://img241.imageshack.us/img241/8884/untitledtf0.png

 

[Gokul43201]

What do the axes represent?

 

[HallsofIvy]

Assuming that x (taking on values from 0 to b) is some random variable and y (taking on values of 0 to a) is the probability density, you can think of "probability density" as if it were a real "weight density". The "balancing point" would be where the total weight (integral of density) is the same on both sides. If the total weight were one, that would mean that the weight on each side of the balancing point would be 1/2. You are looking for x such that
$$\int_0^x f(t)dt= \int_x^b f(t)dt= \frac{1}{2}$$.