# Calculating uncertainties for histogram bins of experimental data with known measurement errors

I have a set of experimental data (with each data-point having its own measured uncertainty), and I wish to produce a histogram of it. The x values of the edges of each bin are already defined. The trick is that I need to have uncertainties for the value of each bin, since I am then going to fit a model-histogram to it. (The model is of a physical process, the outcome of which is best described by a histogram. The model will be fit using a non-linear least squares algorithm, and I want to weight each bin based on its uncertainty).

The uncertainties of each histogram bin need to depend on both the known uncertainties associated with each data-point within the bin, and also the number of data-points within the bin. This is where I am stuck - how can I calculate this?

• I think some more details would help - for instance, are you assuming normal (or some other distribution) errors? May 24 '16 at 8:23
• @Silverfish yes indeed - the measured experimental errors are normally distributed, with the SD measured for each (they arise from a photodetector that is known to have this property). Each data point can be assumed to be independent. May 24 '16 at 23:51
• Best thing to do is edit the new information into the question - not everyone reads the comments, and this might also draw some more attention to the question. May 25 '16 at 0:03

It sounds like you want to calculate a standard error for the unobserved count (i.e. counts of values without the error) in each bin.

For each bin you can calculate the probability that a given observation ($x_i^\text{obs}$ with associated standard deviation $\sigma_i$) could have come from any given bin.

So the number of observations actually in some specific bin, say bin $j$, is the sum of a collection of $\text{Bernoulli}(p_i(j))$ random variables, where $p_i$ for a given bin is the proportion of the area under a normal distribution $N(x_i,\sigma_i^2)$ within the bin boundaries of the $j$-th bin.

If the Bernoulli observations are in his would imply the standard error of the total count is

$$\sum_{i=1}^n p_i(j)(1-p_i(j))$$

where

$$p_i(j) = \int_{l_j}^{u_j} \frac{1}{\sqrt{2\pi}\sigma_i} e^{-\frac{(x_i-z)^2}{2\sigma_i^2}}\, dz$$

where $l$ and $u$ represent upper and lower bin boundaries, and so $p_i(j)$ may be written as the differences of two normal cdf values.

Under the assumption that the different observations' contributions to the count in a given bin are independent, the distribution of the unobserved "true" count in a given bin would be distributed as Poisson-binomial, but I don't think we need to use that for anything, and - while we can work out the correlation between bin counts - I don't think we need that if your interest is on the individual per-bin standard errors.

• Awesome, thanks! This makes sense - I'll give it a shot. (And no, I'm not worried about correlation between counts in this case). May 28 '16 at 4:38
• @BdawgN Note that if you have bins in a bounded interval, you need to include two additional bins - below the lower bound and above the upper bound - to get the proportions to sum to 1. May 28 '16 at 4:41
• Yep, sure. A question though - if the SE of the bin count is the sum of the Bernoulli RVs, then the error would increase as the number of counts in the bin increases. This seems counter-intuitive, shouldn't the uncertainty decrease as more data is included? May 28 '16 at 5:23
• @BdawgN The standard error of the estimated count increases as the expectation increases, yes. The standard error of the estimate of the proportion of the total sample in the current bin decreases as you add more data. This is no different from standard errors of means of iid r.v.s decreasing when $n$ increases but standard errors of totals (i.e. of $n\hat\mu$) increasing when $n$ increases. May 28 '16 at 5:27