Statistical error associated with a bin count

Question

I have a collection of data points. I have computed the histogram of this data to create the empirical distribution. How can estimated the error in the value at each bin. Based on the the total number of data points and the counts at each bin.

The data comes from chemical random reactions, between two states. The total number of events may vary according to the lifetime of chemicals used, but considerable big >>5000 data points. The number of bins used is 100 and of homogeneous width. The intention of estimate the error is to propagate this error trough an equation that depends on the distribution of the process and therefore estimate the error of the quantity computed from this histogram.

Answer 1

Suppose the $i$th histogram bin contains $f_i = 34$ of $n =200$ observations. Then the population proportion $p$ in the $i$th bin is estimated by $\hat p = f/n = 0.17.$

A 95% confidence interval for $p$ is $\hat p \pm 1.96\sqrt{\hat p(1-\hat p)/n}.$ In my example this amounts to $(0.118, 0.222).$ That confidence interval may be useful if you are questioning one histogram bin.

The trouble is that this CI has a 5% chance of being incorrect. If you do this procedure for $k$ bins, total chances of being correct greater than $0.05$ and smaller than $\max(0.05k, 1).$

Maybe with more context someone can suggest a more useful Answer: Describe the data, say how many bins in the histogram, say how many observations. Maybe most crucial of all, specifically what to you hope to accomplish by this kind of analysis of the histogram.

Addendum per comment:

Illustration using Poisson data. I have simulated $n = 7000$ counts from a Poisson distribution. With the following results (from R statistical software):

mean(x)
## 200.0743
sum(x > 230)
## 107
sum(x > 220 & x < 225)
## 209

So $\bar X = 200.0743.$ Also, there are 107 of 7000 events exceeding 230 (about 1.53%) and 209 of 7000 events in the interval $[221, 224]$ or about 3.0%. A density histogram of the events is shown below. (A density histogram is scaled so that the total area of all bars is unity.)

You can see that the histogram looks a bit 'raggedy' so these counts from the sample of 7000 events may not match the actual underlying process. But if we use $\bar X$ to estimate the Poisson mean $\lambda,$ we get estimates 1.74% and 3.20%, respectively for the intervals mentioned above directly from the Poisson distribution. [Because this is a simulation, I know that the correct answer is $\lambda = 200,$ so that the estimate $\bar X$ is pretty good.]

1 - ppois(230, 200.0743)
## 0.01737749
diff(ppois(c(220,224), 200.0743))
## 0.03195982

Of course, this analysis using a specific 'named' distribution requires some theoretical basis for choosing that particular distribution. (For $\lambda$ as large as 200, the distribution $\mathsf{Pois}(\lambda)$ is approximately $\mathsf{Norm}(\mu = \lambda, \sigma = \sqrt{\lambda}).)$

Without going into details about 'kernel density estimators' (which you can Google), I show the histogram above with its KDE as a dark green curve.

hist(x, prob=T, br=100, col="skyblue2", main="Event Counts")
lines(density(x), col="darkgreen", lwd=2)