Mathe Für Dummies

[Xena Donovan]

StuartDonnelly, PhD, earned his doctorate in mathe-matics from Oxford University at the age of 25. Since then, he has established successful tutoring services in both Hong Kong and the United States and is considered by leading educators to be one of the most experienced and qualified private tutors in the country.

In regression analysis, a dummy variable (also known as indicator variable or just dummy) is one that takes a binary value (0 or 1) to indicate the absence or presence of some categorical effect that may be expected to shift the outcome.[1] For example, if we were studying the relationship between biological sex and income, we could use a dummy variable to represent the sex of each individual in the study. The variable could take on a value of 1 for males and 0 for females (or vice versa). In machine learning this is known as one-hot encoding.

Dummy variables are commonly used in regression analysis to represent categorical variables that have more than two levels, such as education level or occupation. In this case, multiple dummy variables would be created to represent each level of the variable, and only one dummy variable would take on a value of 1 for each observation. Dummy variables are useful because they allow us to include categorical variables in our analysis, which would otherwise be difficult to include due to their non-numeric nature. They can also help us to control for confounding factors and improve the validity of our results.

As with any addition of variables to a model, the addition of dummy variables will increase the within-sample model fit (coefficient of determination), but at a cost of fewer degrees of freedom and loss of generality of the model (out of sample model fit). Too many dummy variables result in a model that does not provide any general conclusions.

Dummy variables are useful in various cases. For example, in econometric time series analysis, dummy variables may be used to indicate the occurrence of wars, or major strikes. It could thus be thought of as a Boolean, i.e., a truth value represented as the numerical value 0 or 1 (as is sometimes done in computer programming).

Dummy variables may be extended to more complex cases. For example, seasonal effects may be captured by creating dummy variables for each of the seasons: D1=1 if the observation is for summer, and equals zero otherwise; D2=1 if and only if autumn, otherwise equals zero; D3=1 if and only if winter, otherwise equals zero; and D4=1 if and only if spring, otherwise equals zero. In the panel data fixed effects estimator dummies are created for each of the units in cross-sectional data (e.g. firms or countries) or periods in a pooled time-series. However in such regressions either the constant term has to be removed, or one of the dummies removed making this the base category against which the others are assessed, for the following reason:

If dummy variables for all categories were included, their sum would equal 1 for all observations, which is identical to and hence perfectly correlated with the vector-of-ones variable whose coefficient is the constant term; if the vector-of-ones variable were also present, this would result in perfect multicollinearity,[2] so that the matrix inversion in the estimation algorithm would be impossible. This is referred to as the dummy variable trap.

Wonderful article--fascinating. One question: On the importance of next point chart--when the team on the Left is at 25 and the team on the Top is at 23, the chart shows 25%. Shouldn't it be 0% as the team on the Left would have already won the match?

Beautiful idea-I'm so happy that I could read this article! Reading this, I thought this could be used in many the other sports. I'm a high school student in Korea, and I want to use this to do a study about the mathematics of sports. Could you let me use this for reference?

Hi Mr.Shirriff, I was looking over your mathematics of volleyball pascal's triangle and I was confused as to what your (n) and (i) values meant in your formula following the triangle? Would you mind explaining the significance of them? Thanks!

Hello. I came across this after I had essentially derived its equivalent using a cumulative binomial probability model(Pascal's Triangle link you mention). I decided to allow for a future point win rate assumption different than 50%, if the reader desired this to be variable. The fact that serving win rates and receiving win rates are typically differing by nearly 25% throws off the model in terms of game flow (average rally lengths are less than a cumulative binomial model would suggest, closer to 1.6 than 2.0, I believe),but regarding the end state probability to win the game it remains very useful. Below is a link to a newsletter I am working on and it has links to Googlesheets which help inform the ideas within. I am apparently a little late to the discussion, but still found your post interesting!

-6IOZnK3Xz-rWHOpw/view?usp=sharing

The $64,000 question is this...how does knowing this help us earn a point? The concept is interesting of course. The big assumption necessary to do any of this work is that the teams are evenly matched, which they never are, but that's why it's an assumption. What can we say or do in the match to make this useful?

Hello Sir, really like your work! I would like to know how did you come up with both the numerator and the denominators for these two equations:

T(a+b-1, b-1)/ 2^a+b-1

(a+b-2

a-1 ) / 2^ a+b-2

Also, for the first equation, when we calculate the numerator, i understood we can use nCr, but i dont understand why we have to add the numbers in the row in pascal's triangle, then divide that number by the denominator and X100 to get the percentage.

I hope you find time to answer my question Sir.

Thank you in advance!!

I am coming very late to the party but let me say that your basic assumption is wrong.

"If we assume each team has a 50-50 chance of scoring each point and the score is tied, each team obviously has a 50% chance of winning the game. (With side-out scoring, it makes a difference which team is serving, but for rally point scoring we avoid that complication.)"

It is actually the other way round. With the rally scoring in place it makes a huge difference which team is serving because in volleyball the serving team is at a disadvantage - many more points are scored by the receiving team. So if Team A is serving the most likely outcome is Team B taking the 25-24 lead, with the side-out scoring the most likely outcome is the score staying the same, Team B just gaining the serve.

-is-a-disadvantage-in-some-olympic-sports/

Konrad Ciborowski

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