# If a wholesaler sells to 500 stores and one store shows a 50% uptick in sales, how can the wholesaler determine if this uptick is significant or if it is expected for a few stores to randomly see an uptick of 50%?

##### 1 Answer

There is no single simple answer. It depends on additional parameters that are not given.

See explanation below.

#### Explanation:

Important parameters that are not given in this problem are distribution of goods among stores and the number of customers buying in these stores.

Let's try to address a problem generally, and then we will make certain reasonable assumptions.

The distribution of goods among stores is related to probability of customers to buy goods in each specific store.

Assume that the probability of a single item to be bought at store

Assume further that the total number of items purchased is

Consider now a store

This is a Bernoulli random variable.

Its mathematical expectation is

its variance is

its standard deviation is

The wholesaler has certain number

For instance, if we are talking about bottles of soda, it must be thousands per store.

Consider now

Here random variable

Obviously, the sum of the above random variable is a random variable equal to the number of items bought at

Let's analyse the distribution of probabilities of

First of all, according to the Central Limit Theorem, this distribution should be very close to Normal.

Since it's a sum of independent identically distributed random variables, its expectation is a sum of expectations of its components and its variance is a sum of variances:

It's time to make some additional assumption. To simplify the problem, let's assume that all stores are approximately equal in the number of customers who buy there. Therefore, the probability of a single item to be bought in store

That makes all

Let's say, we want to determine the probability of purchases in store

In this case

According to the "rule of 2

So, under the condition of equal probabilities of purchase in different stores

The second part of this problem is related to probability of **ANY** store purchase not to exceed 50% of its average. With certain degree of precision it can be calculated as the **product of corresponding probabilities** in **EACH** store.

To achieve 95% certainty that number of purchases in any store would not exceed 95%, we need the probability of each store to be

To achieve this probability for each store we need the number of purchases to be very high. "Rule of 3

Thus, with

which is about 43% of the average, so it's sufficient to have 100,000 items to distribute to make sure that none of the store would have more than 50% extra purchases with certainty of 95%.

If, evenly distributing 100,000 items among 500 relatively equivalent (in average number of purchases) stores, **at least one** store exceeded its sale by more than 50%, something abnormal and unexpected happened.

Please refer to Unizor for details on probabilities and statistics.