This is a short note to talk about Harmonic Averages. Most people are familiar with Weighted Averages, as they are a valuable tool for aggregation. For instance, with the data below, the average profitability (~1735) can be easily calculated using weighted averages.

Avg_Profit = (1000*1200 + 200*300 + 500*2500 + 10*600 + 100*300) / (1000 + 200 + 500 + 10 + 100)

or

Avg_Profit = SUMPRODUCT(UnitsSold,ProfitPerUnit)/SUM(UnitsSold)

I’m using Excel notation, and assuming it is clear from the context that UnitsSold is a range that covers the second column, for all models, etc.

A less known way of averaging are Harmonic Averages. It is relevant when the data to aggregate is actually a ratio whose denominator is proportional to the weighting factor. A typical case is miles per gallon (MPG) for a bunch of vehicles. Gas consumption is directly proportional to the number of units.

Let’s add some MPG data to the table above.

Using Weighted Averages for an inverse ratio like MPG is plain wrong (24.3 MPG is NOT the average fuel economy)

The right thing is to use Harmonic Average:

Harm_Avg_MPG = (1000 + 200 + 500 + 10 + 100) / (1000/22.5 + 200/15.0 + 500/32.0 + 10/12.0 + 100/24.0)

As Excel doesn’t have a similar function to SUMPRODUCT for adding 1000/22.5, 200/15.0, etc. I will not use Excel notation, but plain math notation:

UPDATED formula

If you have to deal with Harmonic Averages, you may find interesting this note on how to do PivotTable Multidimensional Analysis with Harmonic Averages. There’s a similar one for Weighted Averages as well.

Let me know what you think.

I have received a number of comments regarding the Simplified Excel Model for market adoption published a few months ago. Reader Vince asked how to extend the math behind it to comprehend effects like cross-segment interactions.

There is no simple answer, and this post is an attempt to point readers to ways to think about what they want to model, as well as giving helpful resources for further study

In my opinion, one of the best approaches to understand market adoption is through system dynamics. One of the advantages of the methodology is that it allows you to conceptually link business effects and relationships to the equations. I touched on this issue on on a previous entry, and here I will try to explain further.

The logistic equation (shown below) is a commonly used way to model market adoption.

From a System Dynamics perspective, the logistic model can be explained looking at the following model (click for full size): The boxes, called “stocks” in SD terminology, represent an accumulated quantity over time. One way to think of stocks is a bathtub. The amount of water in the tub is the accumulation over time of how much water you added through the faucets, less how much water you let out through the drain.

On the model, there are two stocks: how many potential adopters are out there (left side) and how many adopters are (right side). The pipe that connects the boxes is called a “flow”, and it shows a valve, whose value represents how fast potential adopters turn into actual adopters (thus we call it Adoption Rate). Again, in the bath tub analogy, we can think of the value of the flow as how open or closed the faucet is.

Adoption rate depends on how big the population is (the larger the population, the larger the adoption rate), how much the adopters interact with potential adopters (creating the “word of mouth” benefits), etc.

As stocks are accumulations of whatever flows in minus what flows out, from a mathematical perspective, the value of a stock is calculated integrating over time the values of the net flow. On the logistic model, the arrow that links the stock and the adoption rate flow means that the flow changes proportionally to the stock – i.e. if I have more potential adopters, there are more possibilities for contagion, when a user talks favorably to a potential user about the product. The net result is an exponential behavior, which, after some mathematical reduction, is represented by the formula above.

If I want to explain a business audience some market adoption dynamic, it possible to do it talking in terms of stocks and flows (once the audience is comfortable with these terms). It’s almost a guaranteed failure if I try to explain it by using a mathematical formula with exponentials and integrals

The Bass model addresses one limitation of the simple logistic model, regarding how the system “gets started”: with no adopters, there is no chance for interactions, so there is no inflow to the adopters stock. It does it through the use of an external force, like advertising.

Below is a Systems Dynamics interpretation of the Bass model. As you can see, the only difference is that now the Adoption Rate is the addition of two elements, adoption rate from advertising and adoption rate from word of mouth. The latter is exactly the same as the AR in the logistic model.

Returning to Reader Vince’s specific question on how to extend the logistic or Bass models to comprehend effects like cross-segment interactions, I would frame it like this:

• Identify the most important cross-segment interactions – How much “cross-shopping” exists between the segments? (using data like second choice selection); are there characteristics of the upper segment that consumers will translate into the lower segment favorably/unfavorably? consumers replace their vehicles within segment or they try to go up segment? etc.
• Incorporate the key cross-segment interactions on the model – They will most likely affect the Adoption Rate. It also may be necessary to model another stock or stocks (Upper Segment Adopters and Lower Segment Adopters, for instance)
• Check sensitivity of cross-segment assumptions – Understand how different the results are when the cross-segment assumptions are considered versus when they are not. What are the assumptions that most impact the results? A tornado diagram, as discussed in a previous entry, may provide a good way to show the sensitivity to the assumptions

As more dynamic effects are considered for inclusion in a model, it is better to move from a tool like Excel to something like Vensim, or iThink. Chapter 9 of John Sterman’s excellent book “Business Dynamics” talks about both the logistic and Bass models as shown here, and expands on ideas on how to extend them.

Here are some other very good references on the topic

• Forrester, J. W. 1980. Information Sources for Modeling the National
  Economy. Journal of the American Statistical Association 75 (371):
  555-574.
  Argues that modeling the dynamics of firms, industries, or the economy requires use of multiple data sources, not just numerical data and statistical techniques. Stresses the role of the mental and descriptive data base; emphasizes the need for first-hand field study of decision making.
• Legasto, A. A., Jr., J. W. Forrester & J. M. Lyneis, eds. 1980. System Dynamics. TIMS Studies in the Management Sciences. Vol. 14. Amsterdam:
  North-Holland.
  Collection of papers focused on methodology. Includes Forrester and Senge on Tests for Building Confidence in System Dynamics Models and Gardiner & Ford’s discussion on Which Policy Run is Best, and Who Says So?
• Randers, J., ed. 1980. Elements of the System Dynamics Method.
  Cambridge MA: Productivity Press. Includes Mass on Stock and Flow Variables and the Dynamics of Supply and Demand; Mass & Senge on Alternative Tests for Selecting Model Variables; and Randers’ very useful Guidelines for Model Conceptualization.
• Richardson, G. P., and A. L. Pugh, III. 1981. Introduction to System Dynamics Modeling with DYNAMO. Cambridge MA: Productivity Press.
  Introductory text with excellent treatment of conceptualization,
  stocks and flows, formulation, and analysis. A good way to learn the
  DYNAMO simulation language as well.
• Morecroft, J. D. W. 1982. A Critical Review of Diagramming Tools for
  Conceptualizing Feedback System Models. Dynamica 8 (part 1): 20-29.
• Critiques causal-loop diagrams and proposes subsystem and policy
  structure diagrams as superior tools for representing the structure of
  decisions in feedback models.
• Roberts, N., D. F. Andersen, R. M. Deal, M. S. Grant, & W. A. Shaffer.
  1983. Introduction to Computer Simulation: A System Dynamics Modeling
  Approach. Reading MA: Addison-Wesley.
• Easy-to-understand introductory text, complete with exercises.
• Homer, J. B. 1983. Partial-Model Testing As A Validation Tool for
  System Dynamics. In International System Dynamics Conference: 920-932
• How model validity can be improved through partial model testing when
  data for the full model are lacking.
• Sterman, J. D. 1984. Appropriate Summary Statistics for Evaluating the
  Historical Fit of System Dynamics Models. Dynamica 10 (2): 51-66.
• Describes the use of rigorous statistical tools for establishing model
  validity. Shows how Theil statistics can be used to assess
  goodness-of-fit in dynamic models.
• Forrester, J. W. 1985. ‘The’ Model Versus a Modeling ‘Process’. System
  Dynamics Review 1 (1): 133-134.
• The value of a model lies not in its predictive ability alone but
  primarily in the learning generated during the modeling process.
• Richardson, G. P. 1986. Problems with Causal-Loop Diagrams. System
  Dynamics Review 2 (2 ): 158-170.
• Causal-loop diagrams cannot show stock-and-flow structure explicitly
  and can obscure important dynamics. Offers guidelines for proper use
  and interpretation of CLDs.
• Forrester, J. W. 1987. Fourteen ‘Obvious Truths’. System Dynamics
  Review 3 (2): 156-159.
• The core of the system dynamics paradigm, as seen by the founder of the field.
• Forrester, J. W. 1987. Nonlinearity in High-Order Models of Social
  Systems. European Journal of Operational Research 30 (2): 104-109.
• Nonlinearity is pervasive, unavoidable, and essential to the
  functioning of natural and human systems. Modeling methods must
  embrace nonlinearity to yield realistic and useful models. Linear and
  nearly-linear methods are likely to obscure understanding or lead to
  erroneous conclusions.
• Barlas, Y. 1989. Multiple Tests for Validation of System Dynamics Type
  of Simulation Models. European Journal of Operational Research 42 (1):
  59-87.
• Discusses a variety of tests to validate SD models, including
  structural and statistical tests.
• Barlas, Y., & S. Carpenter. 1990. Philosophical Roots of Model
  Validation: Two Paradigms. System Dynamics Review 6 (2): 148-166.
• Contrasts the system dynamics approach to validity with the
  traditional, logical empiricist view of science. Finds that the
  relativist philosophy is consistent with SD and discusses the
  practical implications for modelers and their critics.
• Wolstenholme, E. F. 1990. System Enquiry – A System Dynamics Approach.
  Chichester: John Wiley.
• Describes a research methodology for building a system dynamics
  analysis. Emphasizes causal-loop diagramming, mapping of mental
  models, and other tools for qualitative system dynamics.
• Mass, N. 1991. Diagnosing Surprise Model Behavior: A Tool For Evolving
  Behavioral And Policy Insights (written in 1981). System Dynamics
  Review 7 (1): 68-86.

The simplified market adoption model I described on previous postings (1,2) is an Excel implementation of a kind of logistic function. The Bass model is one of the most popular models used in marketing, and management of technology to think about product introductions. (See Wikipedia article). From a mathematical perspective, when the parameter p is 0, the Bass model reduces to the logistic function.

What is most interesting, from a business perspective, is how you arrive to each of those functions by modeling real-world interactions. On both models, you can conceptualize the world as two different pools of people (or stocks, in the system dynamics terminology). One is the pool of potential adopters, and the other is the pool of adopters. The flow between these two pools is controlled by the adoption rate, a variable that models how probable is that a potential adopter becomes “infected” by a current adopter. On the logistic model, it depends solely on how much they interact, how big the total population is, and how “contagious” the product is. On the Bass model, an additional parameter accounts for external factors, the most common being advertising. The Bass model overcomes what is called the startup problem of the logistic model: how a initial base of zero adopters can spread “infection” of the product.

There are more refinements that can be done to the Bass model: accounting for changes in the total population over time, learning and experience curves, etc. For projects where the sensitivity of the model to these factors is high, I definitely recommend to spend more time calibrating your model, understanding which of the different available curves fits better any data you may have, and most critical of all, whether the chosen coefficients for any of the functions have strong impacts on the critical business issues you want to model — capacity planning, pricing, profitability, etc.

For many projects like business plans, revenue projections, etc. I’m willing to sacrifice the ability to fine tune parameters in a model like the BDM for the clarity provided by a model like the Excel logistic function I described. I can generate more tangible conversations with executives by discussing what they believe will be the takeover time, when they believe it will be the start of the fast growth, how much share they believe will be reached in steady state, etc.