Diversity as Business KPI – Alpha and Beta Diversity

cluster-analysis
Similarity analysis using beta diversity

The Concept of diversity is well introduced in Ecology, Economy and Information theory. The underlying mathematical theory relates to statistics (probabilities), multivariate analysis, cluster analysis etc. Diversity can be partitioned into two independent components: alpha and beta diversity. In the following the concept of alpha and beta diversity is explained,  using a simple example of selling drinks in different sales areas. It helps to understand beta diversity as a measure of variation (similarity and overlap) between different samples of data distributions, and gives some practical applications in the field of business analysis.

Introduction

To understand the basic concept of diversity, you might watch my video here; it explains how diversity can be characterized using diversity indices – like the Simpson index – taking into account richness and evenness.

In general the concept of diversity can be formulated using the power mean. The Simpson index is based on the arithmetic mean, in the general concept of diversity it corresponds to a “true” diversity of order two.

Shannon Entropy

In the following we will use the Shannon diversity index H – in other applications also named Shannon entropy – which is based on the geometric mean, and the “true” diversity of order one. It uses the logarithm, and we will write it here with the natural logarithm

H = – ∑ pi ln pi.

For an equal distribution – all types in the data set are equally common – the Shannon entropy has the value of the natural logarithm of Richness H = ln(R), the more unequal the proportional abundances, the smaller the Shannon entropy. For only one type in the data set, Shannon entropy equals zero. Therefore high Shannon entropy stands for high, low Shannon entropy for low diversity.

Let us go back to our example of selling different drinks in a restaurant.

With seven types of drinks – each selling with 1/7 or 14% – the Shannon entropy equals ln (7) = 1.95

Selling only one type of drink, the Shannon entropy takes a value of zero, the natural logarithm of 1.

Now let us assume we manage a couple of restaurants in different locations, and we get a monthly summary report of total sales of the different type of drinks.

Comparison of samples

Does it mean we are selling all drinks evenly in all locations?

There are actually two possibilities.

1. The first one: yes, at each location we sell evenly all types of drinks.

High diversity – a Shannon entropy of 1.95 – in Boston, NY, Denver, Austin, etc., resulting in a high diversity of sales for the total sales area.

2. What is the second possibility?

In Boston we are selling coffee only: low diversity with Shannon entropy of zero. Similar in NY; here we are selling tea only, low diversity with Shannon entropy of zero, but selling a different type of drink: tea instead of coffee! Similar in Denver with milk, Austin with coke, and so on.

Looking at our total sales – it looks the same as in the first case – the total diversity is high, as overall we are selling all drinks equally.

Partitioning Diversity – Introducing Alpha- and Beta-Diversity

Diversity in the individual location is called alpha diversity. Our total sales report – the consolidation of all sales location gives us the gamma diversity, and the difference – gamma minus alpha diversity reflects the beta diversity.

Now I can also explain the reason, why we selected the Shannon entropy instead of the Simpson index: only for the Shannon entropy as a measure of diversity, the partitioning of the overall (gamma) diversity into two independent alpha and beta components follows the simple relation: Hα + Hβ = Hγ

Beta Diversity – How to interpret?

As we have seen in our simple example:

In case one we find a high alpha diversity in each location, resulting in the same high consolidated gamma diversity taking all locations together. So the difference between alpha and gamma, i.e. the beta diversity, is zero – we have the same sales distribution and a total overlap in all locations.

In case two we find a low alpha diversity in each location, but a high consolidated gamma diversity taking all locations together: In this case the difference between alpha and gamma diversity, i.e. the beta diversity, is high – we have totally different sales distributions among the locations, selling only one, but a different type of drinks in each location – we got totally different distributions without overlap.

Beta diversity is a measure for similarity and overlap between samples of distributions. Partitioning diversity in alpha and beta diversity allows us to gain insight in the variation of distributions – relative abundances – across samples.

Diversity Calculation in Excel

Alpha, beta and gamma diversity can be calculated in a spreadsheet program. You might find more information about an Excel template for free download here.

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New Podcast Episode – Diversity Index

My video “Diversity Index as Business Performance Indicator” is now added as episode #14 in my BPMSG podcast feed.

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Diversity Index as Busines Performance Indicator – The Concept of Diversity

A business relying on a single type of product or services can be highly risky. Any change in the market place – a change of customer behavior, a new competitor – might have a high impact on the business results. Diversification is an opportunity to spread and reduce the potential risks.

The mathematical concept of diversity has many applications, ranging from ecology over demography to information science. In the context of business performance diversity indices can be used as key performance indicators (KPIs) to analyze markets, define targets for diversification, and track the success of derived business actions.

Download Presentation Slides (pdf)

See also

Video: Diversity as Business KPI – Alpha and Beta Diversity

Diversity Calculation in Excel – Diversity Indices and True Diversity

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  • diversity and inclusion performance indicator policy
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Strategic and Operational Business Performance

dashboardDecisions can have a short- or long-term impact; they can be strategic or operational. Measuring strategic or operational performance might require a different set of key performance indicators (KPIs). You need both: Operational KPIs help to measure the short term performance of an organization, strategic KPIs help to measure the implementation of a long term strategy.

This new podcast episode #13 explains the differences, using the picture of driving a car.

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Combined Performance Index – Example in Excel

Here you find a simple example, how to realize the calculation of a combined performance index in an Excel sheet.

Download the excel sheet:  Comb_Index.xlsx

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Combined Performance Index

BPM-combIn order to limit the number of Key performance Indicators (KPIs), and maintain clarity and conciseness of a dashboard, sometimes it might be useful, to combine a set of performance indicators into one single key performance indicator. This latest episode of my podcast will show, step-by-step, how you can build up a combined performance index. using

  • a target value,
  • a base value and
  • a weighting

for each individual indicator. It also gives an example how to implement this concept in an Excel spreadsheet.

You might subscribe to the podcast with this latest video episode here

 

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Order Income per Sales Person

Any business needs sales activities to get orders from potential customers. Often you have dedicated sales people in an organization, promoting and selling your products or services to customers. But how much order income should each sales person achieve? What is the minimum required, what is the maximum possible? We introduce a new key performance indicator “Order Income per Headcount Sales”

You might subscribe to the podcast with this latest video episode here

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Value Add and Local Content

Operating a sales company, the gross margin percentage is often given by the reseller discount you are getting from your suppliers. If you want to increase profitability you could provide “value added services” with your products, for example maintenance contracts, on-site installations etc. In this presentation we will introduce the key ratios “value add percentage” VAP and “profitability value add” PRVA.

You might subscribe to the podcast with this latest video episode here

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The Dashboard – Driving Business to Success

DashboardA dashboard is the panel of instruments and controls facing the driver of a vehicle. Driving a companies business, a dashboard should give you a condensed summary, which allows you to recognize critical areas and take actions to keep the business moving in the right direction – provided you know your destination.

Any dashboard without a clearly defined strategy for the organization will be useless. Only if you know the destination, you can derive the right set of key performance indicators and arrange them in such a way that they give you the current position of the organization, highlight deviations and allow you to make necessary corrections.

You might subscribe to the podcast with this latest video episode here.

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Operating Expenses

The operating expenses of a company have a high impact, not only on the company’s result, but also on other parameters, like profit per costs and minimum order income to be achieved. Even the same operating result and profitability of two companies could be of a different quality, because of different relations in fixed and variable expenses. In this presentation two performance indicators will be developed to monitor these important differences.

You might subscribe to the podcast with this latest video episode here

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