What is a CDF?
First of all, let's start with a histogram:
A histogram is bar graph which shows us what proportion of our data are in a
particular data range (sometimes called 'bins'). For example, we could easily
get a sense of how our observations are distributed in terms of age, if we type:
histogram age, bin(50)
You can see that many of our observations are at younger ages (below 40).
With continuous variables (of which age is the closest approximation to this
that we
have in our data set), we can get a smoother histogram, for which the area
represented by the histogram sums to 1.
This smoother histogram is known as a PDF. Most of us will have seen the
familiar bell-shaped curve of the normal PDF, from one or other introductory
statistics class. Recall that each point on the PDF shows us how much
probability mass there is at that value in the distribution. If we cumulatively
add up all the area under the PDF, we get the CDF, as represented below:
A point on the CDF may be interpreted as follows: looking at the xline
plotted at 40 years of age, we can tell that about 80% of our sample is at or
below 40 years.
Notice 2 properties of this CDF:
1. it's Y-values (or probability values) always lie between 0 and 1, and the total area under the CDF sums to 1
2. the function is monotonic - or, continually increasing.
This is because we are keeping a running sum of the area under the PDF to
generate the graph, and area can never be negative, so the CDF cannot turn down.
The increasing function is not linear though; the Y-value decreases faster and
faster as age decreases towards 0, and increases more and more slowly as age
increases to 60+.
These 2 properties characterize the normal distribution, and all other
distributions of continuous variables. The important thing to remember about
CDF's is that they are functions which map X-values into probability
numbers, and so are bounded between [0,1].