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%?
There is no single simple answer. It depends on additional parameters that are not given.
See explanation below.
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.
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
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.
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