I have expected frequencies in each category as shown below:

[![enter image description here](https://i.stack.imgur.com/I2rld.png)](https://i.stack.imgur.com/I2rld.png)
[![enter image description here](https://i.stack.imgur.com/esHnW.png)](https://i.stack.imgur.com/esHnW.png)

So my initial data: category - and number of observation of it.

There are too many observations, this why:

> 
> As the sample size grows the null hypothesis will be rejected and the
>  p value goes to zero for any small but nonzero deviation from the null
>  hypothesis. With counts, i.e. total number of observations, of more
>  than 50,000 the proper hypothesis test will most likely reject even
>  small differences that are statistically significant but irrelevant in
>  applications.
> 
> 
> 

Anyway I have applied a Chi-squared test to this K-2 contingency, and as a result get a `p_value = 6.3723954051318158e-126` . This is not so bad considering that applying this test to completely irrelevant data sets - the p value will be zero.

Using F-test as I know I will get the same result.

Another approach that comes to mind is applying a binomial test. I did it this way:

```
 stats.binom_test(1500, store.answered.sum(), 0.0233, alternative = 'two-sided')  
# 1500 observed amount, store.answered.sum() - sum over all observations, 0.0233 - expected frequency of that label 

```

P\_value = 0.00023472778370252812. The result is better, because we don't want to reject null hypothesis. However here is another point, that we have to keep in mind:

> 
> One of the underlying assumptions is that all observations are drawn
>  independently from the same distribution. This will not hold if there
>  is correlation within a store or heterogeneity in the
>  probabilities/distribution. In those cases the variance assumption of
>  the multinomial/binomial/Poisson model would not hold and we get
>  either under or over dispersion
> 
> 
> 

Generally I can make such an assumption, but I'm not sure.

So my question is, how can we check that data distribution within these data sets are the same? My final goal is to check that the second smaller data set is not shifted (English term may be different) to the bigger original one.