2 clarification edited Oct 19 '17 at 22:13 eSurfsnake 70122 silver badges1111 bronze badges One of two things: 1) make fixed histogram bucket sizes and then count the number of points you get that occur in thateach bucket. Caveat In other words, break up the range of $$x$$ into n equal intervals, and then the count for each interval is the number of times your CDF has a 'step' up in that interval, for each interval. Caveat: you will need to normalize, when done, so that all buckets add to 100% probability. 2) Just take the differences between each pair of CDF points (thus the change in height between them), divide by $$\delta x_i$$ to get the slope of the CDF at that point along the $$x$$ axis, and use a linelines of that slopethose slopes to connect the points of a PDF plot. Essentially, you are taking and using the numerical approximation to the derivative to the CDF, which is the PDF. Warning: you will need to think through very carefully if how you do this does not, accidentally, shift the distribution up or down by something like $$\delta x_i/2$$ at each point. In other words, centering each segment will be important to get right. If you have a good number of points, method 1 will be a lot less error-prone - e.g., with 1000 points you can probably get a good discrete histogram representation to something like a normal distribution with 20-50 buckets which you can do numerical statistics on easily (mean, moments).   Since that is usually what you want, it does the job. I seesense your desire to do something that looks more like a continuous function, which method 2 would get, but I would warn you away from that, unless you have a small number of data points. You will find that: (1) it is going to be hard to represent somehow;somehow (i.e., on a spreadsheet or as a data structure); (2) it will be hard to work with even a good representation, and (3) it will take a lot of thought to get right. I do a lot of numerical methods with unknown distributions and method one is surprisingly accurate most of the time (again, with enough points). One of two things: 1) make fixed histogram bucket sizes and then count the number of points you get that occur in that bucket. Caveat: you will need to normalize so that all buckets add to 100% probability. 2) Just take the differences between each pair of CDF points (thus the change in height between them), divide by $$\delta x_i$$ to get the slope of the CDF at that point along the $$x$$ axis, and use a line of that slope to connect the points. Essentially, taking and using the numerical approximation to the derivative to the CDF, which is the PDF. Warning: you will need to think through very carefully if how you do this does not, accidentally, shift the distribution up or down by something like $$\delta x_i/2$$ at each point. In other words, centering each segment will be important to get right. If you have a good number of points, method 1 will be a lot less error-prone - e.g., with 1000 points you can probably get a good discrete histogram representation to something like a normal distribution with 20-50 buckets which you can do numerical statistics on easily (mean, moments).   I see your desire to do something that looks more like a continuous function, which method 2 would get, but (1) it is going to be hard to represent somehow; (2) it will be hard to work with, and (3) it will take a lot of thought to get right. I do a lot of numerical methods with unknown distributions and method one is surprisingly accurate most of the time (again, with enough points). One of two things: 1) make fixed histogram bucket sizes and then count the number of points you get that occur in each bucket. In other words, break up the range of $$x$$ into n equal intervals, and then the count for each interval is the number of times your CDF has a 'step' up in that interval, for each interval. Caveat: you will need to normalize, when done, so that all buckets add to 100% probability. 2) Just take the differences between each pair of CDF points (thus the change in height between them), divide by $$\delta x_i$$ to get the slope of the CDF at that point along the $$x$$ axis, and use lines of those slopes to connect the points of a PDF plot. Essentially, you are taking and using the numerical approximation to the derivative to the CDF, which is the PDF. Warning: you will need to think through very carefully if how you do this does not, accidentally, shift the distribution up or down by something like $$\delta x_i/2$$ at each point. In other words, centering each segment will be important to get right. If you have a good number of points, method 1 will be a lot less error-prone - e.g., with 1000 points you can probably get a good discrete histogram representation to something like a normal distribution with 20-50 buckets which you can do numerical statistics on easily (mean, moments). Since that is usually what you want, it does the job. I sense your desire to do something that looks more like a continuous function, which method 2 would get, but I would warn you away from that, unless you have a small number of data points. You will find that: (1) it is going to be hard to represent somehow (i.e., on a spreadsheet or as a data structure); (2) it will be hard to work with even a good representation, and (3) it will take a lot of thought to get right. I do a lot of numerical methods with unknown distributions and method one is surprisingly accurate most of the time (again, with enough points). 1 answered Oct 19 '17 at 21:56 eSurfsnake 70122 silver badges1111 bronze badges One of two things: 1) make fixed histogram bucket sizes and then count the number of points you get that occur in that bucket. Caveat: you will need to normalize so that all buckets add to 100% probability. 2) Just take the differences between each pair of CDF points (thus the change in height between them), divide by $$\delta x_i$$ to get the slope of the CDF at that point along the $$x$$ axis, and use a line of that slope to connect the points. Essentially, taking and using the numerical approximation to the derivative to the CDF, which is the PDF. Warning: you will need to think through very carefully if how you do this does not, accidentally, shift the distribution up or down by something like $$\delta x_i/2$$ at each point. In other words, centering each segment will be important to get right. If you have a good number of points, method 1 will be a lot less error-prone - e.g., with 1000 points you can probably get a good discrete histogram representation to something like a normal distribution with 20-50 buckets which you can do numerical statistics on easily (mean, moments). I see your desire to do something that looks more like a continuous function, which method 2 would get, but (1) it is going to be hard to represent somehow; (2) it will be hard to work with, and (3) it will take a lot of thought to get right. I do a lot of numerical methods with unknown distributions and method one is surprisingly accurate most of the time (again, with enough points).