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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.

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Types Of Federal Student Loans

Students who look for financial aid during studies either go for federal student loans or private student loans. Federal student loans are offered by the US government, which are available directly through banks, student loan lenders, schools or from the Federal Family Education Loan program (FFELP). Federal loans are offered with very low interest rates, longer repayment periods and various kinds of repayment options with simpler credit requirements than private loans. In case of federal subsidized student loans, the interest is paid by the government to the financial institution while the student is enrolled as well as during a grace period. A federal loan may not be enough to cover all the expenses of the student and in that case, the student might have to take an additional private student loan. Note that the student will not get the full loan amount applied for and should only take the actual amount into account while preparing the budget.

There are different kinds of federal student loans from different institutions. Hence, it is advisable to take the guidance of parents or other financial aiding sources to decide on the type of federal direct student loan that suits the student the best.

Perkins loan option:

This loan is available for undergraduates and graduates in need at a fixed lower interest rate of five percent for a repayment period of ten years. The loan limits for undergraduates are $5,500 per year and $27,500 per lifetime. For graduate students the limit is $8,000 per year and $60,000 per lifetime (including undergraduate loans). The funds are handled directly by the school, making it easier to get the amount as soon as the student enrolls.

Stafford loan option:

This is the most common federal student loan and anyone can apply for it. It offers fixed interest rates and is available in the subsidized and unsubsidized form. When making use of the subsidized federal student loan, the government pays the interest while the student is enrolled. In the case of unsubsidized federal student loan, the student has to the pay the interest but can wait with payments until he completes his or her graduation. The interest rate for unsubsidized loans is currently at 6.8 %. Students applying for a Stafford Loan must complete the FAFSA (free application for federal student aid). Stafford Loans are available directly from the United States Department of Education through the Federal Direct Student Loan Program (FDSLP). It is important to apply much earlier than the closing date for the application to avoid any last minute trouble.

PLUS loan option:

Also known as parent loan for undergraduate students. It is given to the parents of undergraduate students who are dependent and have enrolled at least half-time. This loan option requires the applicant to be free from any adverse credit experiences like bankruptcy, default etc on their credit record. These loans are offered at a fixed interest rate that is higher than the Stafford loan rate and repayment starts while the student is enrolled.

For more information on student loans check out my post about Student Loan Consolidation.

Inflation: How long does it take for prices to double?

A question that often comes up is how long it would take for prices to double if the rate of inflation remained constant. It also helps to turn an abstract percentage number into a value that is easier to grasp and interpret.

If we start at a certain value for the consumer price index CPI0 and apply a constant annual inflation factor f (which is just the annual inflation rate expressed in decimals plus one), the CPI would grow exponentially according to this formula:

CPIn = CPI0 · f n

where CPIn symbolizes the Consumer Price Index for year n. The prices have doubled when CPIn equals 2 · CPI0. So we get:

2 · CPI0 = CPI0 · f n

Or, after solving this equation for n:

n = ln(2) / ln(f)

with ln being the natural logarithm. Using this formula, we can calculate how many years it would take for prices to double given a constant inflation rate (and thus inflation factor). Let’s look at some examples.

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In 1918, the end of World War I and the beginning of the Spanish Flu, the inflation rate in the US rose to a frightening r = 0.204 = 20.4 %. The corresponding inflation factor is f = 1.204. How long would it take for prices to double if it remained constant?

Applying the formula, we get:

n = ln(2) / ln(1.204) = ca. 4 years

More typical values for the annual inflation rate are in the region several percent. Let’s see how long it takes for prices to double under normal circumstances. We will use r = 0.025 = 2.5 % for the constant inflation rate.

n = ln(2) / ln(1.025) = ca. 28 years

Which is approximately one generation.

One of the highest inflation rates ever measured occurred during the Hyperinflation in the Weimar Republic, a democratic ancestor of the Federal Republic of Germany. The monthly (!) inflation rate reached a fantastical value of r = 295 = 29500 %. To grasp this, it is certainly helpful to express it in form of the doubling time.

n = ln(2) / ln(296) = ca. 0.12 months = ca. 4 days

Note that since we used the monthly inflation rate as the input, we got the result in months as well. Even worse was the inflation at the beginning of the nineties in Yugoslavia, with a daily (!) inflation rate of r = 0.65 = 65 %, meaning prices doubled every 33 hours.

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This was an excerpt from “Business Math Basics – Practical and Simple”. I hope you enjoyed it. For more on inflation check out my post about the Time Value of Money.

World Population – Is Mankind’s Explosive Growth Ending?

According to the World Population Clock there are currently about 7.191 billion people alive. This year there have been 118 million births (or 264 per minute) and 49 million deaths (or 110 per minute), resulting in a net growth of 69 million people. Where will this end? Nobody can say for sure. But what we can be certain about is that the explosive growth has been slowing down for the past 40 years. I’ll let the graphs tell the story.

Here is how the world population has developed since the year 1700. The numbers come from the United Nations Department of Economic and Social Affairs. From looking at the graph, no slowdown is visible:

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However, another graph reveals that there’s more to the story. I had the computer calculate the percentage changes from one decade to the next. From 1960 to 1970 the world population grew by 22 %. This was the peak so far. After that, the growth rate continuously declined. The percentage change from 2000 to 2010 was “only” 12 %.

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Of course it’s too early to conclude that this is the end of mankind’s explosive growth. There have been longer periods of slowing growth before (see around 1750 and 1850). But the data does raise this question.

Talk to me again when it’s 2020 or 2030.

Just by the way: according to estimates, about 108 billion people have been born since the beginning of mankind (see here). This implies that about 101 billion people have died so far and that of all those born, 6.5 % percent are alive today.

Did somebody say dust in the wind?

The Probability of Becoming a Homicide Victim

 Each year in the US there are about 5 homicides per 100000 people, so the probability of falling victim to a homicide in a given year is 0.00005 or 1 in 20000. What are the chances of falling victim to a homicide over a lifespan of 70 years?

 Let’s approach this the other way around. The chance of not becoming a homicide victim during one year is p = 0.99995. Using the multiplication rule we can calculate the probability of this event occurring 70 times in a row:

 p = 0.99995 · … · 0.99995 = 0.9999570

 Thus the odds of not becoming a homicide victim over the course of 70 years are 0.9965. This of course also means that there’s a 1 – 0.9965 = 0.0035, or 1 in 285, chance of falling victim to a homicide during a life span. In other words: two victims in every jumbo jet full of people. How does this compare to other countries?

 In Germany, the homicide rate is about 0.8 per 100000 people. Doing the same calculation gives us a 1 in 1800 chance of becoming a murder victim, so statistically speaking there’s one victim per small city. At the other end of the scale is Honduras with 92 homicides per 100000 people, which translates into a saddening 1 in 16 chance of becoming a homicide victim over the course of a life and is basically one victim in every family.

 It can get even worse if you live in a particularly crime ridden part of a country. The homicide rate for the city San Pedro Sula in Honduras is about 160 per 100000 people. If this remained constant over time and you never left the city, you’d have a 1 in 9 chance of having your life cut short in a homicide.

Liked the excerpt? Get the book “Statistical Snacks” by Metin Bektas here: http://www.amazon.com/Statistical-Snacks-ebook/dp/B00DWJZ9Z2. For more excerpts check out Missile Accuracy (CEP), Immigrants and Crime and Monkeys on Typewriters.