# How to find a consistency index for a binary variable?

I am working on a project where there are 2 variables to monitor, say sales (actual) and projected sales (target).

The sales can be volatile in terms of target achievement.

### 1️⃣ Case

Like there can be sales which achieves the target for say first two months, but then it doesn't and again it does... this way my sales variable will be called "inconsistent" for the target achievement.

### 2️⃣ Case

There can be sales which out of 12 months in a year it does achieve the target for 6 months.

If alternatively: Means one month it does achieve the target and next month it doesn't = Less consistent

If follows some consistancy: Like it achieves the target for say 3 months and then after a gap again 3 months, or say continuously 6 months then it will be called more consistent.

## 🤔 What do I want?

I am willing to find some consistency index that I can give to my sales variable (0-100) where 100 is most consistent and 0 is the least.

The variable would become binary following 0 1 1 0 1 1 1 0... sequence of target achievement.

## ➗ My current solution

I find the consistency index currently by:
Step - 1: Finding the achievement ratio (AR)
Step - 2: Finding the consistancy ratio (CR) (streak)
Step - 3: The Consistancy Index (CI)

Example: If the data shows like [1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1] for last 12 months, we can see that:

1. Total achievements are 6 so AR = 6/12 = 50%
2. Longest streak 3 so CR = 3/12 = 40%
3. So, the consistency index = (50 + 40) / 2 = 45%

Your thinking is not bad, as a certain degree of experience with Key Performance Indicators allows me to conclude.

As a clarification, in your $$CR$$, you count the longest streak of target-achievements, not the longest streak in general (which is reasonable).

But I would suggest a different way to compute your final Consistency Index, since in order to have a meaningful average, one needs to sum similar in nature/meaning/content things -and your $$AR$$ and $$CR$$ certainly describe different aspects of the situation.

What you could do is to "penalize" the achievement ratio by using multiplicatively the consistency ratio, in order to obtain your target-consistency Index:

$$CI = AR \times CR$$

So the first level of target achievement, $$AR$$, is adjusted downwards by the degree of consistency, $$CR$$.

This approach is a bit aggressive, since it penalizes more heavily discrepancies between $$AR$$ and $$CR$$.

For example, for a period of length $$T$$, for $$AR = CR = 4$$ we get $$CI = 16/T^2$$, while for $$AR = 5$$, $$CR = 3$$ we get $$CI = 15/T^2$$, while your arithmetic average would here be the same. But there is no meaningful sense in which "one target achievement less is exactly offset by an achievement streak longer by one". And if further you had $$AR = 6$$, $$CR = 2$$ you would get $$CI = 12/T^2$$, even lower.

So in the multiplicative approach the person who achieved the target 5 times gets a lower CI than the person who achieved the target only 4 times, because the second had a longer streak. And the person that achieved the target 6 times with a best streak of 2, is scored even lower. In other words the multiplicative approach does put a lot of weight in the consistency aspect, justifying the name of the final Index.

It sends the signal "it's not enough to achieve targets; you have to do it in a consistently" -perhaps because increased consistency means less uncertainty.

#### Make sure you're not making metrics for the sake of metrics

When analysing situations like this, it is important to genuinely consider what you actually care about and what you don't care about rather than just constructing metrics willy-nilly. For example, your proposed "consistency ratio" effectively measures streaks of months hitting the sales target. Does your firm actually care about this or not? If you have two salesmen who each meet their sales targets for six months out of twelve in the year, do you actually give a shit that one of them met their target for six months in a row and then failed to meet it for another six months in a row (yielding a consistency ratio of 50%) whereas the other met their target every second month (yielding a consistency ratio of 12.5%). If your firm actually prefers the former then by all means use this metric, but do they? Does the former situation actually mean that sales performance was better, or are you just constructing metrics for the sake of constructing metrics?

Here I would also ask a broader question: Since the firm (presumably) always wants higher sales, does a failure to hit the sales target reflect a failure of salesmanship or a failure in the formulation of the target? Presumably a result that is far above the target is good, yet it constitutes less "accuracy" in meeting the target. So again, what does the firm actually care about?

The reason I note these issues is that it is that it is a standard principle of business management that you have to be careful with the metrics you develop, because whatever you develop as a metric to optimise is going to incentivise how people act. Development of spurious KPIs creates incentives to do suboptimal things to meet those KPIs. Statistics can offer various suggestions on how to measure the "consistency" of a sequence of values (binary or otherwise) but ultimately you will need to have a good idea of what it is you want to know about that you are seeking to reduce to measurement. You can only do this by deciding why you want to formulate a metric and what you want to use it for --- the function of the metric will determine the appropriate way to formulate it.