Mathematically Modelling Twitter Impressions (TI)

Twitter is an increasingly useful social medium by which businesses can gain a significant amount of exposure. But just how much exposure are we talking about? How can this be quantified? What are the best measures that organisations can take to actually increase exposure through Twitter? A number of metrics do exist already, such as average number of retweets per tweet, but these only provide snapshots or insights into the popularity of a business on Twitter. They do not give a quantifiable estimation of how large the audience really is, nor give any indication on how exposure can be increased.

Hence, the focus of this post is to try and model the audience size in Twitter (we will call this Twitter Impressions or TI) using simple mathematics. We will then play around with the variables and offer insights into which variables would be better for businesses to focus on in terms of getting the best ROI through Twitter marketing.

So, lets begin with the simplest mathematical model of all, where TI represents total number of impressions (or total audience size per post) and f represents number of followers the user has:

TI = f

Assuming this simple model, the number of impressions per tweet is equal to the number of followers a business has. For example, a coffee shop with 100 followers writes a tweet, it will appear one hundred times. Once for each follower. However this simple model ignores perhaps the most important marketing tool in Twitter, the retweet, or ability of the follower to repost the original tweet. For an example of the power of the retweet, imagine the coffee shop in question posts the original tweet again (100 impressions) but each follower retweets the post to each of their fifty followers. Thus the total number of impressions now would be 100 + (50*100) or 5100 impressions. Clever, huh? Now lets try and model this mathematically, now introducing two new variables, average number of retweets per post (r) and average retweeter follower width (w). The equation now becomes:

TI = f + rw

OK this is starting to look better, and finding out the number of followers one has is easy, but how on earth do we find out our average number of retweets per post? Let alone our average retweet follower width? Agggghhhh!

Let’s take this apart, and begin by looking at the average number of retweets per post (r). The metric is simple enough, if I were to post a tweet right now, how many retweets do I reckon I would get? What is the average? This is easy enough to work out, and really depends on how much time you have. You can go through every single tweet you have ever posted (this may be huge) and sum all of the retweets that you have had. You can then divide this by the number of tweets you have ever posted and you will arrive at the lucrative metric. However counting up your entire history on Twitter is extremely time consuming and may not be all the representative anyway. How useful are my figures from three years ago when I only had 20 followers? Now I have over 10,000 followers! Another way of calculating this metric is to choose a sample group that is large enough and representative enough to give a good estimate of your average number of retweets per post. For example, you could choose your last fifty posts, or last hundred tweets to gauge this metric.

Now the next thing we need to work out is the average retweet follower width (w). This is also fairly straight forward, and can be done on your entire twitter history (population) or a cross section (sample) as above. Pay attention here! What you are looking for is the average number of followers that each of your retweeters are retweeting to. A mouthful huh? So for example, say you have a total of 20 retweets in your sample group. Through Twitter, you can look at who has retweeted your post, and sum all of their followers up. Say you arrive at a total of 1000 followers. Divide this by your original 20 retweeters and you arrive at an average retweet follower width of 50.

Now the maths is out of the way, we can try calculating our own TI at ONMedia, and offer some suggestions on how we can improve our own marketing through Twitter. These are methods you can use yourself of course! Don’t worry, we don’t own any intellectual property here! We like to share our ideas!

Today, our number of Twitter followers is 6000, so:

f = 6000

Our total number of retweets is 209 and we have posted 23 tweets, hence:

r = 9 (1.s.f)

Now we move on to calculating w.  We could calculate this our self  as discussed above, but in the interests of time (and also to later highlight a limitation in the model) I will choose to use the average number of followers from a survey in Telegraph instead.  This average is 209 followers (The Telegraph, October 2012).  So we assume that the average in our own group of retweeters is commensurate with the worldwide average (hmm a big assumption as we will find out later).

Hence ONMedia’s TI = 6000 + (9 * 209)
TI = 6000 + 1881
TI = 7881

Soooo, our estimated Twitter audience is 7881 people per tweet, cool eh? Now, if you have followed this so far, you are probably asking which metric provides a better ROI to focus upon? Do I focus on increasing my own number of followers (f)? Or perhaps I should try and post more retweetable / viral material and try to increase my number of retweets (r)? Or maybe the best thing to do is try and get my material retweeted by more influential people and increase my (w)? Let’s try adjusting one variable at a time and try to answer this question.

Before we do this, it’s important to move away from the numbers, and try to rationalise the fact that it’s way easier to gain fifty random followers (increasing f) on Twitter than it is to gain fifty followers with large amounts of followers themselves (increasing w and f). Imagine how difficult it can be to gain influential followers, such as ones with over a million followers? Just ask any of the fans of popular musicians at the moment! Also, it can be extremely time consuming to write what you would consider to be retweetable, or viral material (increasing r).

Based solely on our own experience at ONMedia, the amount of time it takes to get 200 standard followers is roughly equivalent to the time investment taken to increase your average number of retweets by 1 point, and approximates to the time investment required to increase your average follower width by 10 points. So increasing our follower count (f) to 7000 is equivalent to working on increasing our average number of retweets to 14 (r) or increasing our average retweeter follower width to 309 (w). Putting these numbers into our equation:

TI = 8881 (when f is increased to 7000)
TI = 8926 (when r is increased to 14)
TI = 8781 (when w is increased to 309)

Based upon our simple model, adjusting r appears to give the better improvement in TI, and is therefore a better metric for a business to focus on.  Furthermore, by focusing upon gaining retweets, a business is exposing their ideas to a wider audience, increasing their sphere of exposure.  However, as we discussed earlier, this is merely an attempt at modelling an organisation’s Twitter audience, to seek to understand how a better ROI can be obtained.  As we also mentioned, there are a number of limitations to this model, which we will skim over now.

Firstly, clearly you cannot increase w without increasing f right? In order to increase r you need to add more influential followers! Secondly, in order to increase r you not only need to write more influential articles, you also have to show them to the right people! It’s no good writing a novel on Twitter, but only showing it to your one follower. It’s axiomatic that f, r and w are all inter-related.

Also, as we demonstrated, it is never as simple as coming up with textbook simple ratios to explain the amount of workload involved in adjusting f, r and w. We based ours on business experience. Not really an exact science!

Thirdly, remember we chose to use the industry average for w? Consider this again, the industry average. This is skewed heavily by influencers such as Barack Obama and David Beckham. It may be interesting to consider that 80% of people on Twitter have less than 50 followers, yet the average is 209 (The Telegraph, October 2012). Hmmmm. Of course, you could always undertake the time consuming process to calculate w yourself right?

To wrap up, this mathematical modelling has come to light due to the lack of useable metrics within Twitter itself by which a business can ascertain the best use of it’s resources and conversion rates. Our simple model has suggested that it is better to focus upon gaining retweets, rather than gaining standard followers or retweets from individuals with a large network of followers.  There is plenty of room for refinement and improvement, and comments are welcome! For example, could we construct a parametric equation to roughly link increases in r, w and f? Could we use a logarithmic scale to consider f?

Thanks for reading!


The Telegraph, 2012 (available at

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