Probability and distribution from actual data

Question

Let's say I have some data from a real world system:

data = {0.026666156, 1.27421*^-6, 0.027878597, 0.017035598, 
   0.011036215, 0.038684388, 0.025161479, 0.021353902, 0.027487123, 
   0.021190747, 0.029328752, 0.014554109, 0.040037348, 0.020063044, 
   0.018363514, 0.0116034, 0.02609979, 0.024298555, 0.020568345, 
   0.017299039, 0.005427133, 0.023453297, 0.037443984, 0.031152865, 
   0.020158735, 0.014362383, 0.034687449, 0.01472421, 1.29928*^-6, 
   0.023887209, 0.014531724, 0.03055792, 0.027206875, 0.02227913, 
   0.01618167, 0.032540959, 1.26651*^-6, 0.021810796, 0.038698191, 
   0.014228248, 0.020261827, 0.031292827, 0.01142724, 0.039466342, 
   0.035504951};

If I generate histograms for this data it looks like this:

Column[{
    Histogram[data, {.005}, "Probability", ImageSize -> Large], 
    Histogram[data, {0.005}, "CDF", ImageSize -> Large]
}]

My question is twofold.

1. Is there a way to "fit" this data with a distribution? Or make a pseudo-distribution with this data? I've found d = EmpiricalDistribution[data]; DiscretePlot[CDF[d, x], {x, 0, .05, .0001}], but is this the best way?

2. How do you programmatically calculate the P50 value from this data? (50% confidence that the value will be realized. y = 0.5 on CDF plot)

Answer 1

There are basically four ways you can do this:

1. Discretely, using EmpiricalDistribution, as you suggest in your question
2. Also discretely, but in a bucketed way, using HistogramDistribution
3. Smoothly, using SmoothKernelDistribution or KernelMixtureDistribution.
4. By fitting an assumed distribution, e.g. by using EstimatedDistribution, FindDistributionParameters and related commands.

Which you choose depends on the nature of your data and what you know about it. If you have reason to believe the data are distributed according to some known distribution, choose #4. Since your data are real-valued, it is likely that it comes from a continuous distribution and so choices #3 or #4 seem most appropriate.

Once you have your estimated distribution, you can calculate the P50 value using Quantile.

It is worth noting, though, that particular settings will give different estimates of the quantile. For example there is more than one way to define the kernel used in SmoothKernelDistribution. Using the data in your question:

Quantile[SmoothKernelDistribution[data ,  Automatic, #], 0.5] & /@
 {"Biweight", "Cosine", "Epanechnikov", "Gaussian", 
  "Rectangular", "SemiCircle", "Triangular", "Triweight"}

{0.0222671, 0.0223012, 0.0223522, 0.0223533, 0.0223458, 0.0223374,  0.0222653, 0.0222342}

Of course, in some fields, four decimal places is close enough.