The biggest problem is that an averaged shifted histogram has positive dependence in adjacent bins, so a test derived on an independence assumption (aside the negative dependence induced by the total count being conditioned on, which is adjusted for) won't have the right distribution for its test statistic.
It's possible to adapt a test for such dependence, but the vanilla version of the test will be wrong.
[If you want to test for normality, doing it from a histogram isn't a particularly good way to do it. A Shapiro-Wilk or Shapiro-Francia test, an Anderson-Darling test, or perhaps a Smooth test of the kind discussed in Rayner and Best's book Smooth Tests of Goodness of Fit would be better. The nice thing about a Shapiro-Francia test is it's just based on the correlation in a normal scores plot (Q-Q plot for normality), which gives a visual assessment of the non-normality]
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Edit - looking at your QQ plot - the data are very far from normal. No reasonable test would fail to reject normality at that sample size. A Lilliefors test or an Anderson-Darling or a Shapiro-Wlik or a smooth test with a standard number of terms ($k=4$ or $k=6$) will all reject easily... you don't even need to test that.
To clarify the answer by user777, there are two packages that I and my collaborators have developed specifically for rigorously fitting power-law distributions. Both can be found here. One is for when your data are integer or real values, while the other (which is the one that user777 linked to) is for binned data, i.e., when you only know the number of the measurements within each of several contiguous ranges.
In each case, the packages have four parts:
- fit the power-law model to your data,
- estimate the uncertainty in your parameter estimates,
- estimate the p-value for your fitted power law, and
- compare your power-law model to alternative heavy-tail models.
These methods are described exhaustively in two references, both of which are freely available on the arxiv pre-print server (just search for their titles). The approach they use to fit the model is maximum likelihood, which is far more accurate than classic "curve fitting" approaches on scatter plots.
[integer and continuous quantities] A Clauset, C R Shalizi, and MEJ Newman. "Power-law distributions in empirical data." SIAM Review 51, 661-703 (2009).
[binned quantities] Y Virkar and A Clauset. "Power-law distributions in binned empirical data." Annals of Applied Statistics 8, 89-119 (2014).
In your case, I'm not entirely sure what you mean by each data point carrying an uncertainty. Do you mean a classic measurement uncertainty, like the kind you have when you measure the length or weight of an object (and is thus normally distributed)? If the variance is modest and the number of data points large, then you can get a pretty good estimate of the power-law parameter using our methods even if you ignore the uncertainty (because the estimator takes the logarithm of each value, and normally distributed fluctuations become highly compressed under the log). If you don't have much data, or if the uncertainty is really large, then I would recommend choosing a reasonable binning scheme (powers of 2 or something) and applying the binned-data methods.
Best Answer
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.