# Particle distribution: how to compute the cumulative distribution?

In powder metallurgy people often use particle distributions such as this one:

I understand this distribution as a probability density function. I integrated this function to obtain the cumulative distribution function:

The value of the CDF is supposed to be one for particle sizes over 50µm but this is not the case. Is there a coefficient to apply to the integration ?

• Without more knowledge about what's on the y axis in your first plot, I'd think dividing by the sum might suffice. – N. Wouda May 31 '17 at 15:03
• The yaxis is the percentage of volume for each size of particle. – snickers Jun 1 '17 at 8:48
• No, the y axis doesn't show a percentage, because the area under the curve is not of the order of 100. It shows proportion given a certain bin width or equivalent. I don't have a better guess than you why the curve is presented as smoothed rather than binned. A rough average for the y variable is 0.05 (units something/$\mu$m) and the horizontal extent is about 50 $\mu$m. So the area under the curve is about 2.5 something. That suggests to me that the data are collected with a bin width of the order of 2.5 $\mu$m. What the machinery does should be documented somewhere! – Nick Cox Mar 19 at 8:41
• The lack of a label on the vertical axis should make us cautious about interpreting the first graphic as a plot of a density: evidently it's something else that is only proportional to a density. – whuber Aug 23 at 14:29