What distribution family would best fit this graph?

Sorry if this type of question is not kosher. I'm new around here, so please forgive me.

Anyway, I have a dataset that describes the probability that users will like certain articles from my corpus (with 5 articles chosen randomly and graphed below. Along the x axis are individual users, and along the y-axis their score. The users are sorted by score.)

I'm looking to fit a distribution to each line, but am unsure of which one to choose that would likely best these lines. I was thinking of using an exponential distribution? But there's a lot of information that exponential probably would not capture here. Does anyone have a good idea?

EDIT: I will update my title to include the name of a reasonable distribution so that this question will be more searchable and others would possibly learn from it in the future.

• After some research, I think it's similar to a sigmoid curve, except flipped? Sep 13 '14 at 0:30
• Probably, a sigmoid curve isn't a distribution, though. You may have an empirical cumulative distribution function (ECDF). Sep 13 '14 at 0:51

To many statisticians, information like this would usually be thought of as an empirical CDF:

with the scores on the x-axis, and the proportion of people scoring it less than or equal to a given x-value on the y-axis (that is, your x-axis ranks become scaled to proportions of people scoring no more than the person with that rank, and then becomes the new y-axis).

But it's sometimes difficult to identify good approximating distributions from the ECDF; it might be easier to look initially at some other displays (histograms with plenty of bins, kernel density estimates, Q-Q plots or other distributional plots.

You don't say what the range of possible scores is, but I presume they're bounded. If so, a scaled beta distribution might be a reasonable starting model:

As you see, with appropriate choices for the 4 parameters (I didn't try to fit these, just tried some values roughly in the right region), they can look quite similar to at least 4 of the 5 distributions you have there. However, they're probably not suitable in general. Mixture distributions might get you further along.

On the other hand you have a lot of data, so the empirical distribution probably has all you need to answer all kinds of questions about the data.

• thank you so much @Glen_b. I'm so sorry I wasn't more specific, but I'm mainly looking for a parameterized model, which empirical cdf doesn't look to be? I'll start with Beta distribution. I'll keep the question open for a bit longer and then accept your answer. Thanks again! Sep 13 '14 at 1:01
• @Glen_b I've been turning my head sideways for about 5 minutes trying to see how Alex's plot is an ECDF, and I'm not seeing it. The x axis is user ID's and the y axis looks like a logit transform (except sorted backwards). It seems like "probability of like" is a decreasing function of user ID. I think the resemblance to an ECDF is incidental. Sep 13 '14 at 1:26
• @ssdecontrol Unless I missed something, the numbers on OP's x-axis appears to be (effectively) ranks rather than IDs. If you take all the individual scores in a sample in order, then a plot of their scores against the values $1,2,..., n$ is (up to interchanging y and x, possible axis flipping and some scaling, possibly including a known shift) the same as the ECDF. The rank of a user corresponds to the number of users who score at least as extreme as they do, so a user's ranking can be scaled to the proportion $\leq$ than that user's score. Sep 13 '14 at 1:37
• Alex - what do you need from a parameterized model you can't get from the sample? Incidentally a beta won't fit your green one there. Looks like a second mode near 0.8 Sep 13 '14 at 1:41
• @Glen_b Got it. With regard to the OP wanting a parameterized distribution, interpreting CDFs isn't that easy. I think a kernel density might be more natural. Maybe a triangular one to be more "empirical." And as regards bounds on the scores, I still suspect the scores are unbounded transformations of the probabilities, although that is up to the OP to clarify. Sep 13 '14 at 2:18