Product Manager Math — 4 concepts you need to know
There are a million places where math can apply in a business context. By math I’m not talking about excel functions and number-crunching arithmetic but rather formulas, theorems and frameworks. Oftentimes these were created (and taught) in a purely conceptual setting in applied mathematics, statistics, CS or Engineering class. As a result even if you’ve learned some valuable concepts and useful tips and tricks back in the day, you never applied it, it is long-gone from your working memory, and not something you think about as you do your daily work.
This is certainly true for me, and probably you too. I’ve been spending a little time thinking about this, and I think there is particular value to unlock in the product management world, where analysis and decision making are a central component to the job. The increasing availability of raw data from myriad sources and computing power will undoubtedly make much of this automatic and invisible, but nevertheless I wanted to throw out a few concepts I think every PM should be familiar with.
First, as a general caveat, I use tools like these as a guide rather than something that gives me absolute answers. I need to be able to articulate the logic and insights in my own words independent of what any analytic conclusions pop out of some formula. If I can explain it and it makes sense, then good.
I won’t try to explain the actual variables and calculations here, you can find all that on the web when you need it. This is just a starting point to provide rough familiarity with the concepts before you dig in for more.
What Features Should We Build ? Use Discrete Choice Model
Product planning hits right at the heart of product management. There are lots of philosophies on and approaches to the topic, but here I am thinking about what analytic tools can we use to help answer the question.
The primary starting point is to think about making choices. In an engineering world of limited resources you have to make choices on what to build and what not. And from a business perspective it is about making choices around the package of features which yield the customer outcomes you want.
A well-known approach here is to use a discrete choice model. This involves giving a survey to a respondent with a set of mutually exclusive choices. These choices usually involve a combination of items, which can vary from choice to choice. Most often in my work, it has been a basket of features at some price and/or implementation time or technical topology.
For example, you may ask if a user would prefer the “standard edition” product with features a, b, c for $5/month or the “enterprise version” with a, b, c and d for $6/month or “pro version” with a, c, d for $25 one time fee, and so on. Usually you iterate through various sequences of questions with varying options to create a complete set of choices. If there are many combinations, each respondent may not be able to answer the full set, in which case you can distribute subsets of choices over many respondents.
With this data you can determine the probability of different types of users selecting each choice. And further, you can create a utility function of each variable you include inside your basket of choices — basically understanding what features are the most powerful driver of that choice.
What Customers Should We Target ? Use K-Means Clustering
Yet another topic that is near and dear to all PMs is customer segmentation. The starting point for segmentation is customer behaviors or attributes, often collected using surveys and compilation of individual customer-level financial and operational results.
The basis for good segmentation is cluster analysis. Conceptually this involves plotting various data and then formulaically identifying the set of variables that best distinguish groups of customers. Rarely is one variable the right factor for basing segmentation decisions, it is usually a combination of variables.
The first step is to look at the variables you have and see how similar customers are across these variables. Think of this as plotting them all on a multidimensional graph resulting in a point cloud, with each point representing a single customer. First you look at how close/far away customers are from each other for each variable. You do this by measuring the Euclidian distance between each of the points.
Using this distance data you then identify the center points on the graph for each cloud. A common approach is K-means Clustering. At the start you don’t necessarily know how many clouds (or clusters) there are, so you can just start with 1 or two.
The K-means is an iterative approach, where you start with the number of clusters and it will try to find the center points for that number of clusters. You then map customers to that center point based on their Euclidian distance. You then increment the number of clusters, identify center points, map the customers to the closest one and keep repeating until you see little or no reassignment of customers to new center points.
Who are our most/least valuable customers ? Use The Sigmoid Curve
I used to work with an ex-Goldman investment banker and I learned a lot about value-based management and economic levers from him. One of the tools he frequently pulled out was to rank order the population of companies based on whatever attribute he was thinking about and plot them on a graph. With a big enough population it inevitably would show a familiar S-curve shape. I don’t know why that struck me as so interesting, but it did and I’ve continued to break it out myself on occasion ever since.
In my experience, customer lifetime value is not typically a linear function. There are often long-tail cases — both good and bad. Sales and support costs, churn, product mix, special pricing deals even currency swings can pull outliers away from the pack — and if you rank order and plot them you just might find they distribute along the S curve — or more accurately a Sigmoid curve.
The Sigmoid curve is a special case of the logistic function which shows up in a lot of places, such as a learning curve. This is an example when you start small and slow and then accelerate rapidly until you reach some limit where only small incremental improvements are possible.
Looking at the Sigmoid curve you can derive the formula which describes the distribution, and additionally you can break out your calculus and look at the derivatives at any point along the curve. As you may remember this describes the slope of the graph at the point. Those points with very flat slope are those at the ends of the S curve and they identify the customers you will want to pull out and start asking lots of ‘why’ questions.
What will our Sales be next quarter ? Use Monte Carlo Simulation
I bet that if you asked what your sales will be next quarter, the answer will be whatever was is in the most recent forecast. I also bet that number will be wrong (unless you have some very crafty salespeople).
Clearly extrapolating out into the future means there is some level of uncertainty. A solid way of getting a handle on that is to run Monte Carlo simulations to determine probabilities of different sales results. The way I think about Monte Carlo simulations is that it replaces the key variables that determine sales results (close rate, funnel velocity, sales price and so on) with a random number generator which confirms to a normal distribution with realistic min/max values that you subjectively define.
Run this a bunch of time and you can see how your sales come out, and then you can put some probability estimates on each range of result. So instead of a single number sales forecast you can create a set of probability weighted expected results.
There’s a lot more clever applications of math and statics concepts you can pull out to help apply data and rigor to product work, but these four are my favorites.
