I want to plot a histogram of some timing data. The timing data, represented by a continuous variable t, is binned as follows:
t=0
0<t<=1
1<t<=2
2<t<=3
3<t<=4
I have frequency data for each bin. To plot this as a histogram, I understand that I ought to use frequency density; that is, the frequency divided by the bin width. But my first bin has zero width! How can one cope with this?
For data that are analytically derived, where some positive percentage of the data occur at a single exact value and others may be found throughout some interval(s) on the real line, a cumulative distribution function (CDF) is one way to clearly graph the data.
If this actually is a probability distribution of a random variable $X$, the CDF is given by $F(t) = P(X \leq t)$. For the situation described in the question, where only values $t \geq 0$ can occur, you would have $F(t) = 0$ for all $t < 0$, then $F(t) = P_0$ for $t = 0$, where $P_0$ is the fraction of data that fall at $t = 0$ exactly, and $F(t)$ is increasing for all $t > 0$ where the probability density at $t$ is positive, $F(t)$ constant anywhere else.
This also works for data that are not random but that act like a probability distribution, in this example a certain percentage at one exact value, a certain percentage distributed in the interval $(0,1]$, a certain percentage in the interval $(1,2]$, and so forth. If all you had available (or all you wanted to determine) was the frequencies for each of these bins and for the value $t=0$, you could interpolate a straight line segment from $(0,P_0)$ to $(1,P_0 + P_1)$ where $P_1$ was the fraction of data falling in the interval $(0,1]$.