From Data to Probability Densities without Histograms

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Abstract
When one deals with data drawn from continuous variables, a histogram is often inadequate to display their probability density. It deals inefficiently with statistical noise, and binsizes are free parameters. In contrast to that, the empirical cumulative distribution function (obtained after sorting the data) is parameter free. But it is a step function, so that its differentiation does not give a smooth probability density. Based on Fourier series expansion and Kolmogorov tests, we introduce a simple method, which overcomes this problem. Error bars on the estimated probability density are calculated using a jackknife method. We give several examples and provide computer code reproducing them. You may want to look at the corresponding figures 4 to 9 first.

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