Statistical methods in pricing

No article about artificial intelligence would be very complete or credible without a look into statistical methods. And since our work with SAMPO (acronym for Sniffie’s Automatic Market Price
Optimizer) is based on these fundamentals of statistical methods and models, they of course deserved their own chapter in this article.

How to model sales volume and profit (the faster way)

Modelling sales volume and profit can be done in different ways depending on your needs. You are
able to model them in a quick and simple way or a more robust and accurate way. If you have limited
resources, a fast and simple method is better than nothing. This can be useful in e.g. B2B businesses
where you can just estimate your sales people’s efficiency and costs and aggregate those for all of
your sales representatives. The following data points are a simple example of what you need to be
able to model sales volume and profit:

1. How many leads generated per representative per time unit
2. How many phone calls per representative per time unit
3. How many meetings per representative per time unit
4. How many closes per representative per time unit
5. Estimate costs for lead generation (e.g. marketing budget over time)

These data points allow you to calculate the amount of revenue generated per representative and
aggregate all costs per representative to know how much profit they will bring over time.
A similar super fast and easy method can be used for online businesses by taking into account the
following data points:

1. Estimate cost of marketing budget over time, e.g. 5000\$
2. Estimate average cost per click, e.g. 1\$ => 5000 clicks
3. Estimate conversion rate, e.g. 5% => 250 sales
4. Estimate average basket, e.g. 50\$ => 12500\$
5. This gives you a monthly profit forecast of 7500\$

After you have put together an estimate like the one above, remember to double check it from topdown; estimate your market share and see how realistic it is. If you have a limited capacity of e.g. 200
sales per month, you can’t have more than 200 sales even if your model shows 250 sales per month.
A capacity usage of >90% is usually quite high for any business (unless you are heavily invested
in automation and service scaling, on a level to a company like Amazon. If you achieve such high
capacity usage on a regular basis, we congratulate you for good efforts!). In certain cases you can
also take seasonality into account e.g. Christmas decorations are mostly sold before Christmas, not
during summer time.

Fast and simple methods like the ones described above are all you need to have in place to get

How to model sales volume and profit (the more robust way)

When you have more time and resources available, you can then start to consider using a more robust and accurate modelling system. For this kind of modelling to be possible, you’ll need to have systems in place to gather and record data points like:

1. Data of units sold per SKU (stock keeping unit) per price point over a time unit (either aggregated
to a time unit or then data of each individual sale for each SKU)
2. Potential interest per SKU over a period of time (e.g. visitor tracking using Google Analytics or
such)
3. Supplier data per SKU
4. Geographical location of sales per SKU
5. Data on past promotions per SKU
6. Data on customer reviews per SKU
7. Market data from competitors per SKU
8. Data of the product

Another important factor that you need to take into consideration and something you really need to know about your business is your cost structure. These typically involve purchase price as well as a attributed part of all fixed operating costs to each sale. On top of these, you also need to figure out your gross margin target.

For plotting sales volume vs. price and profit vs. price graphs and sales volume / profit vs. time it is
very beneficial to use a graphing software like Excel. Remember that the values are not constant:

they change over time as the market evolves and internal and external factors change. You may for example need to have multiple price points per SKU in the data set in order for some of the graphs to provide useful information. This could be the case where you have a standard price and a standard discounted price that you switch between.
Once you have gathered enough data, you are able to calculate some simple statistical key figures that give you some useful insight like:

• Average sales per SKU in a time unit, e.g. 1 unit sold every 10 days in average => 3 units sold per month in
average
• Sales variance per SKU, e.g. if 1 unit sold every 10 days means that you have 9 days in every 10 days when
there are no sales

With these data points and key figures you can then model your sales and profit using:

1. Normal distribution (often not the optimal choice for pricing models since it can be negative)
2. Gamma distribution (often quite good for pricing models)
3. Poisson distribution (often quite good for pricing models)

Why using average (mean) is not the best way to model sales volumes and profit

Average, also known as mean, is a simple statistical figure taken from a list of numbers to represent
them. Depending on the usage it can be calculated in different ways. Mostly average is used to describe the statistical populations that fit a normal distribution (Bell curve), e.g. height of population.

Arithmetic mean (AM) is calculated as the sum of all occurrences divided by the number of occurrences in any data set. There are also other types of means such as: geometric mean (GM) and harmonic mean (HM), which have the mathematical properties of AM ≥ GM ≥ HM in any data set.
From now on, in this article, we are going to discuss the use of arithmetic mean (AM) and just call it
mean or average. If the population data set follows the bell curve, AM has the property of being equal
to mode (the most common data point in a data set) and median (the 50 percentile in the data set).

AM should never be considered as the only statistical figure in any decision making due to skewness.
In a data set where a large majority of values are small, a sufficient number of large figures can skew
the data set to the right. If you at this point assume that you are dealing with an unskewed normal
distribution your decision making will be compromised. The same scenario is also true in an opposite
situation where the majority of the data points are large and a significant number of small figures will
then skew the data to the left.

AM also has a hard time dealing with sales estimates where sales are infrequent, which can happen
quite often. Here are a few scenarios that illustrate the problem of using AM in sales forecasting:

Scenario A)

Scenario B)

Scenario C)

1 sale occurring every 10 days means that there are 3 sales in 30 days (1 month), 30 sales over 300 days, 27 sales days in a month are “empty” sales. Mean is now 1/10 sales per day.

3 sales occurring on one day in a 30 day period (1 month) means that there are 29 sales days with empty sales in one month (30 sales over 300 days). Mean is still 1/10 sales per day.

30 sales occurring once in a 300 day period means that there are 299 days with empty sales in that period. Mean is still 1/10 sales per day.

In all three scenarios the calculated mean is nowhere near reality and if this would be the method you
would use it would lead to you either not having stock when you need to, or having too much. Now
that we’ve looked at the challenges of using average in forecasting sales volumes and profit, it is time
to turn to something more positive and see what the optimal distribution model would look like.

What is a good distribution model for forecasting sales volumes and profit?

Since normal distribution poses some challenges it is good to find a model that takes into account
time. This following text should provide you with enough information whether you can use normal
distribution in your sales modelling or not.

• Normal distribution can sometimes be useful for SKUs that have a high volume or high frequency of sales per time unit
• Normal distribution is less useful for SKUs that have a low volume or a a low frequency of salesper time unit

Inelastic products do not react on price changes.
Example is on critical medicine like insulin or Covid-19 vaccine.

Elastic products do react on price changes. Example of 1€ coin to be sold less than 1€.

If you lower the price of a product, you are likely to sell more units. This in turn will create an S-shaped
curve when the price is lowered and then again raised. Normal distribution has the problem that it
can be negative in such cases if we start the slope at zero. If we would want to create something that
looks like a Bell curve, we would need to start at a low price point and that would have to lead to low
or no sales, then increase the price to mid level and see the highest sales and then again raise the
price and see lower sales. This is of course very unlikely to happen which again proves that the Bell
curve is rarely a very realistic outcome.

With infrequent sales over a period of time, it is often more useful to use a probability distribution to
model sales, profits and pricing. Here are a few examples of good statistical models for forecasting
sales volumes and price, especially if your sales are infrequent during certain times:

Poisson distribution

Gamma distribution

Negative binomial distribution

To model the # of events in the future, for discrete usage (e.g. sales occurrences). For example, if you buy infinite amounts of lottery tickets, the distribution of winning tickets is Poisson distributed.

To predict the wait time until future events occurs (for any
number of future occurrence, not only the first occurance).

When buying two lottery tickets, the probability of winning is
modelled with binomial distribution. When the sample size
increases, it will start to look very much like Poisson. (combi
nation of Poisson and Gamma)

Like Lumen learning says, Poisson is “a discrete probability distribution the probability of a given
number of events occurring in a fixed interval of time and/or space if these events occur with a known
average rate and independently of the time since the last event”. This applies very well to people
buying stuff, and thus purchase behaviour can be modelled with Poisson.

Hoarding of certain products results in overdispersed data, which can be modelled with negative
binomial distribution. When the sample size increases, the distribution will start to look very much like
Poisson.

Thus Poisson or gamma or binomial distribution or a combination of some or all of them is a good
choice for modelling. Just remember to take into account seasonality and promotions and all other

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“We don’t have enough historical transaction data.” “We don’t have big enough sales volumes.” “We don’t want to be the cheapest.” “We can’t change prices