New Release for Kindle: Introduction to Stars – Spectra, Formation, Evolution, Collapse

I’m happy to announce my new e-book release “Introduction to Stars – Spectra, Formation, Evolution, Collapse” (126 pages, $ 2.99). It contains the basics of how stars are born, what mechanisms power them, how they evolve and why they often die a spectacular death, leaving only a remnant of highly exotic matter. The book also delves into the methods used by astronomers to gather information from the light reaching us through the depth of space. No prior knowledge is required to follow the text and no mathematics beyond the very basics of algebra is used.

If you are interested in learning more, click the cover to get to the Amazon product page:


Here’s the table of contents:

Gathering Information
Spectrum and Temperature
Gaps in the Spectrum
Doppler Shift

The Life of a Star
Stellar Factories
From Protostar to Star
Main Sequence Stars
Giant Space Onions

The Death of a Star
Slicing the Space Onion
Electron Degeneracy
Extreme Matter
Black Holes

Sources and Further Reading

Enjoy the reading experience!

CTR (Click Through Rate) – Explanation, Results and Tips

A very important metric for banner advertiesment is the CTR (click through rate). It is simply the number of clicks the ad generated divided by the number of total impressions. You can also think of it as the product of the probability of a user noticing the ad and the probability of the user being interested in the ad.

CTR = clicks / impressions = p(notice) · p(interested)

The current average CTR is around 0.09 % or 9 clicks per 10,000 impressions and has been declining for the past several years. What are the reasons for this? For one, the common banner locations are familiar to web users and are thus easy to ignore. There’s also the increased popularity of ad-blocking software.

The attitude of internet users is generally negative towards banner ads. This is caused by advertisers using more and more intrusive formats. These include annoying pop-ups and their even more irritating sisters, the floating ads. Adopting them is not favorable for advertisers. They harm a brand and produce very low CTRs. So hopefully, we will see an end to such nonsense soon.

As for animated ads, their success depends on the type of website and target group. For high-involvement websites that users visit to find specific information (news, weather, education), animated banners perform worse than static banners. In case of low-involvement websites that are put in place for random surfing (entertainment, lists, mini games) the situation is reversed. The target group also plays an important role. For B2C (business-to-consumer) ads animation generally works well, while for B2B (business-to-business) animation was shown to lower the CTR.

The language used in ads has also been extensively studied. One interesting result is that often it is preferable to use English language even if the ad is displayed in a country in which English is not the first language. A more obvious result is that catchy words and calls to action (“read more”) increase the CTR.

As for the banner size, there is inconclusive data. Some analysis report that the CTR grows with banner size, while others conclude that banner sizes around 250×250 or 300×250 generate the highest CTRs. There is a clearer picture regarding shape: in terms of CTR, square shapes work better than thin rectangles having the same size. No significant difference was found between vertical and horizontal rectangles.

Here’s another hint: my own theoretical calculations show that higher CTRs can be achieved by advertising on pages that have a low visitor loyalty. The explanation for this counter-intuitive outcome as well as a more sophisticated formula for the CTR can be found here. It is, in a nutshell, a result of the multiplication rule of statistics. The calculation also shows that on sites with a low visitor loyalty the CTR will stay constant, while on websites with a high visitor loyalty it will decrease over time.

Sources and further reading:

  • Study on banner advertisement type and shape effect on click-through-rate and conversion

  • The impact of banner ad styles on interaction and click-through-rates

  • Impact of animation and language on banner click-through-rates

Mathematics of Banner Ads: Visitor Loyalty and CTR

First of all: why should a website’s visitor loyalty have any effect at all on the CTR we can expect to achieve with a banner ad? What does the one have to do with the other? To understand the connection, let’s take a look at an overly simplistic example. Suppose we place a banner ad on a website and get in total 3 impressions (granted, not a realistic number, but I’m only trying to make a point here). From previous campaigns we know that a visitor clicks on our ad with a probability of 0.1 = 10 % (which is also quite unrealistic).

The expected number of clicks from these 3 impressions is …

… 0.1 + 0.1 + 0.1 = 0.3 when all impressions come from different visitors.

… 1 – 0.9^3 = 0.27 when all impressions come from only one visitor.

(the symbol ^ stands for “to the power of”)

This demonstrates that we can expect more clicks if the website’s visitor loyalty is low, which might seem counter-intuitive at first. But the great thing about mathematics is that it cuts through bullshit better than the sharpest knife ever could. Math doesn’t lie. Let’s develop a model to show that a higher vistor loyalty translates into a lower CTR.

Suppose we got a number of I impressions on the banner ad in total. We’ll denote the percentage of visitors that contributed …

… only one impression by f(1)
… two impressions by f(2)
… three impressions by f(3)

And so on. Note that this distribution f(n) must satisfy the condition ∑[n] n·f(n) = I for it all to check out. The symbol ∑[n] stands for the sum over all n.

We’ll assume that the probability of a visitor clicking on the ad is q. The probability that this visitor clicks on the ad at least once during n visits is just: p(n) = 1 – (1 – q)^n (to understand why you have the know about the multiplication rule of statistics – if you’re not familiar with it, my ebook “Statistical Snacks” is a good place to start).

Let’s count the expected number of clicks for the I impressions. Visitors …

… contributing only one impression give rise to c(1) = p(1) + p(1) + … [f(1)·I addends in total] = p(1)·f(1)·I clicks

… contributing two impressions give rise to c(2) = p(2) + p(2) + … [f(2)·I/2 addends in total] = p(2)·f(2)·I/2 clicks

… contributing three impressions give rise to c(3) = p(3) + p(3) + … [f(3)·I/3 addends in total] = p(3)·f(3)·I/3 clicks

And so on. So the total number of clicks we can expect is: c = ∑[n] p(n)·f(n)/n·I. Since the CTR is just clicks divided by impressions, we finally get this beautiful formula:

CTR = ∑[n] p(n)·f(n)/n

The expression p(n)/n decreases as n increases. So a higher visitor loyalty (which mathematically means that f(n) has a relatively high value for n greater than one) translates into a lower CTR. One final conclusion: the formula can also tell us a bit about how the CTR develops during a campaign. If a website has no loyal visitors, the CTR will remain at a constant level, while for websites with a lot of loyal visitors, the CTR will decrease over time.