# Statistical test for non uniform counts

I have data on counts for about 1000 categories in a sample. I want to get an estimate if the counts are somewhat uniformly distributed across 1000 categories, or most of the counts are cominf from only few categories and rest are either 0 or minimal.

Any idea what could be a good statistical test, and visualization method? if these were not 1000 but 5 categories, pie-chart would be perfect to get a visual feel.

Also, if I have different samples and want to find relative non-uniform property one sample vs the other, what could be a good way.

Thanks

I want to elaborate my question. Suppose i have only 10 categories instead of 1000, i want to see if the counts look like

Blockquote

5,5,5,5,5,5,5,5,5,5 or

Blockquote

0,0,0,0,0,50,0,0,0,0.

Blockquote

It is possible that all 50 counts are assigned to a single category rather than distributed across 10. I want to reject this sample only accept if they are reasonably distributed across all 10 categories. What could be a good statistical test, and visualization way for this?

• How many data points do you have in total? – Stephan Kolassa Apr 1 '20 at 22:08
• I have a 1000 data points, corresponding to 1000 categories. – paulG Apr 2 '20 at 1:21
• Hm. I may have been unclear. What is your total count, i.e., the total number of samples? For instance, in your example, this total count would be 50. (This is important because the $\chi^2$ test Ertxiem proposes needs a sufficiently large number of expected counts per bin.) – Stephan Kolassa Apr 2 '20 at 5:58
• Lets say I have only one sample, total counts is variable but would be well above 5000. I want to find if this 5000 belongs to only few categories or distributed across all 1000 categories in that sample. – paulG Apr 2 '20 at 10:58

If the data sample is large enough (I would say at least 10 times the number of categories), then you may apply a chi-square test of homogeneity for an uniform distribution.

Regarding the graphical methods, consider a bar chart showing a subset of categories, for instance, the top 5 and the bottom 5 categories.

Ertxiem's answer is precisely what you want. Calculate a $$\chi^2$$ test with $$1000-1=999$$ degrees of freedom.

I personally am a big fan of simulating the null hypothesis a couple of times and plotting the results of such simulations, to get a feeling for the randomness that the null hypothesis would imply - and then comparing these plots to the actual data you have. In the present case, I would simulate distributing $$n=5,000$$ items into $$k=1,000$$ bins, tabulating the numbers of items in the bins and plot the top 5 and the bottom 5 bins, as Ertxiem proposes. You can do this, say, 20 times and arrange the resulting histograms in a $$4\times 5$$ matrix:

For instance, we see that the fullest bin typically has about 14-15 items in it. Now, you can insert your actual histogram at a random position in this matrix. Does it "stand out"? For instance because the largest bin contains not 14-15, but 20 items? If so, then it is sufficiently far away from the null distribution that you can safely say there is something there. You can do a fun little exercise with this, by showing your 19 null distribution and one observation plots to random colleagues and ask them to identify the "special" case. If people can do so consistently, then there is something there.

This will also work if your bin counts are "too uniform". We wouldn't expect all bins to contain only 3-7 items, so if your fuller bins are too empty (and your emptier bins too full), then this illustrates a different departure from uniformity.

(Note: I didn't come up with this kind of "visual significance test". See Buja et al., "Statistical Inference for Exploratory Data Analysis and Model Diagnostics" (2009, Philosophical Transactions: Mathematical, Physical and Engineering Sciences))

R code:

n_sims <- 20
set.seed(1)
n_items <- 5000
n_bins <- 1000
y_max <- 20 # set through trial and error

opar <- par(mfrow=c(4,5),las=2,mai=c(.1,.5,.1,.1))
for ( ii in 1:n_sims ) {
sim <- factor(sample(1:n_bins,n_items,replace=TRUE),levels=1:n_bins)
barplot(c(sort(table(sim),decreasing=TRUE)[1:5],
NA,NA,
rev(sort(table(sim),decreasing=FALSE)[1:5])),
xaxt="n",lwd=2,col="gray",ylim=c(0,y_max))
text(7.2,1,"...",cex=2,font=2)
}
par(opar)