Taller Woman Dance Partner

In the book Think Stats, there's an exercise to work out the percentage of dance couples where the woman is taller, when paired up at random. Mean heights (cm) & their variances are given as 178 & 59.4 for men, & 163 & 52.8 for women.

The two height distributions for men & women can be assumed Normal. The solution is to work out the total area under the two curves where the condition height_woman > height_men holds. This will need to be done for every single height point (h), i.e.  under the  entire spread of the two curves (-∞, ∞). In other words, the integral of the height curve for men from (-∞, h) having height < h,  multiplied by the integral of the height curve for women from (h, ∞) having height > h.

There are empirical solutions where N data points are drawn from two Normal distributions with appropriate mean & sd (variance) for men & women. These are paired up at random (averaged over k-runs) to compute the number of pairs with taller women.

The area computation can also be approximated by limiting to the ±3 sd range (includes 99.7%) of height values on either side of the two curves (140cm to 185cm). Then by sliding along the height values (h) starting from h=185 down in steps of size (s) s=0.5 or so, compute z at each point:

z = (h - m)/ sd, where m & sd are the corresponding mean & standard deviation of the two curves.

Refer to the one-sided standard normal value to compute percentage women to the right of h (>z) & percentage of men to left of h (<z). The product of the two is the corresponding value at point h. A summation of the same over all h results in the final percentage value. The equivalent solution using NORMDIST yields a likelihood of ~7.5%, slightly below expected (due to the coarse step size of 0.5). 

C1 plots the percent. of women to the right & percent. of men to the left at different height values. C2 is the likelihood of seeing a couple with a taller woman within each step window of size 0.5. Interestingly, the peak in C2 is between heights 172-172.5, about 1.24 sd from women's mean (163) & 0.78 sd from men's mean (178). The spike at the end of the curve C2 at point 185.5 is for the likelihood of all heights > 185.5, i.e. the window (185.5,∞).

Playing around with different height & variance values yields other final results. For instance at the moment the two means are separated by 2*sd. If we reduced this to 1*sd the mean height of women (or men) to about 170.5 cm, the final likelihood jumps to about 23%. This is understandable since the population now has far more taller women. The height variance for men is more than women, setting them to identical values 52.8 (fewer shorter men) results in lowering of the percentage to about 6.9%, vs. setting them to 59.4 (more taller women) increases the percentage to 8.1%.

Sample data points from Confidence_Interval_Heights.ods worksheet:

Point (p)	Women				Men		Likelihood (l)
	z_f Using p	r_f = % to Right	r_f_delta = % in between		z_m Using p	l_m= % to Left	l = l_m * r_f_delta
185.5	3.0964606	0.0009792	0.0009792		0.9731237	0.8347541	0.0008174
185	3.0276504	0.0012323	0.0002531		0.9082488	0.8181266	0.0002071
184.5	2.9588401	0.0015440	0.0003117		0.8433739	0.8004903	0.0002495
184	2.8900299	0.0019260	0.0003820		0.7784989	0.7818625	0.0002987
183.5	2.8212196	0.0023921	0.0004660		0.7136240	0.7622702	0.0003553
183	2.7524094	0.0029579	0.0005659		0.6487491	0.7417497	0.0004197
182.5	2.6835992	0.0036417	0.0006838		0.5838742	0.7203475	0.0004926
182	2.6147889	0.0044641	0.0008224		0.5189993	0.6981194	0.0005741

...

Poincaré and the Baker

There is a story about the famous French mathematician Poincaré, where he kept track of the weight of the loaves of bread that he bought for a year from one single baker. At the end of the year he complained to the police that the baker has been cheating people selling loaves with a mean weight of 950g instead of 1000g. The baker was issued a warning & let off.

Poincaré continued weighing the loaves that he bought from the baker through the next year at the end which he again complained. This time even though the mean weight of the loaves was 1000g, he pointed out that the baker was still continuing to bake loaves with lower weights, but handing out the heavier ones to Poincaré. The story though made up makes for an interesting exercise for a stats intro class.

The crux of the solution, as shown by others, is to model the baked loaves as a Normal distribution.

At the end of year 1

The expected  mean m=1000g, since the baker had been cheating the mean shows up as 950g. The standard deviation (sd) is assumed to remain unaffected. To know if the drop in mean from the expected value m1=1000 to the actual m2=950 for identical sd, a hypothesis test can be performed to ascertain if the sampled breads are drawn from the same Normal distribution having mean value m1=1000, or from a separate one having mean m2=950. The null hypothesis is H0(m1 = m2) vs. the alternate hypothesis H1(m1 != m2).

|z| = |(m1-m2)/(sd/sqrt(n))| where n is the no of samples.

With an assumption of Poincaré's frequency of buying at 1 loaf/ day (n=365), & sd=50,

z=19.1, much larger than 1% (2.576) level of significance, so the null hypothesis can be rejected. Poincaré was right in calling out the baker's cheating!

From the original story we are only certain about the values of m1=1000 & m2=950. Playing around with the other values, for instance changing n=52 (1 loaf/ week), or sd to 100g (novice baker, more variance), could yield other z values & different decisions about the null hypothesis.     

At the end of year 2

What Poincaré observes is a different sort of curve with a mean=1000g, but that is far narrower than Normal. A few quick checks make him realize that he's looking at an Extreme Value Distribution (EVD) or the Gumbell distribution. Using data from his set of bread samples, he could easily work out the parameters of the EVD (mean, median, shape β, location μ, etc.). Evident from the EVD curve was the baker's modus-operandi of handing out specially earmarked heavier loaves (heaviest ones would yield narrowest curve) to Poincaré.

Shanon Entropy and Information Gain

Shanon's Information Gain/ Entropy theory gets applied a lot in areas such as data encoding, compression and networking. Entropy, as defined by Shanon, is a measure of the unpredictability of a given message. The higher the entropy the more unpredictable the content of the message is to a receiver.

Correspondingly, a high Entropy message is also high on Information Content. On receiving a high Entropy/ high Information Content laden message, the receiver has a high Information Gain.

On the other hand, when the receiver already knows the contents (or of a certain bias) of the message, the Information Content of the message is low. On receiving such a message the receiver has less Information Gain. Effectively once the uncertainty about the content of the message has reduced, the Entropy of the message has also dropped and the Information Gain from receiving such a message has gone down. The reasoning this far is quite intuitive.

The Entropy (& unpredictability) is the highest for a fair coin (example 1.a) and decreases for a biased coin (examples 1.b & 1.c). Due to the bias the receiver is able to predict the outcome (favouring the known bias) in the later case resulting in a lower Entropy.

The observation from the (2-outcomes) coin toss case generalizes to the N-outcomes case, and the Entropy is found to be highest when all N-outcomes are equally likely (fair).

Need Support to Lift with Confidence

Brace up terminologies coming your way...

Support: A measure of the prevalence of an event x in a given set of N data points. Support is effectively a first level indicator of something occurring frequent enough (say greater than 10% of the times) to be of interest.

In the case of two correlated events x & y,

Confidence: A measure of predictability of two events occurring together. Once confidence is above a certain threshold (say 70%), it means the two events show up together often enough to be used for rules/ decision making, etc.

Lift: A measure of the power of association between two events. For an event y that has occurred, how much more likely is event y to occur once it is known that event x has occurred

False Negative, False Positive and the Paradox

First a bit about the terms False Positive & False Negative. There terms are associated with the nature of error in the results churned out by a system trying to answer an unknown problem, based on a (limited) set of given/ input data points. After analysing the data, the system is expected to come up with a Yes (it is Positive) or a No (it is Negative) type answer. There is invariably some error in the answer due to noisy data, wrong assumptions, calculation mistakes, unanticipated cases, mechanical errors, surges, etc.

A False Positive is when the system says the answer is Positive, but the answer is actually wrong. An example would be a sensitive car's burglar alarm system that starts to beep due to heavy lightning & thunder on a rainy day. The alarm at this stage is indicating a positive hit (i.e. a burglary), which is not really happening.

On the other hand, a False Negative is when the system answers in a Negative, where the answer should have been a Positive. False negatives happen often with first level medical tests and scans which are unable to detect the cause of pain or discomfort. The test report of "Nothing Abnormal Detected" at this stage is often a False Negative, as revealed by more detailed tests performed later.

The False Positive Paradox is an interesting phenomenon where the likelihood of a False Positive shoots up significantly (& sometimes beyond the actual positive) when the actual rate of occurrence of a condition within a given sample group is very low. The results are thanks to basic likelihood calculations as shown below.

Let's say in a group of size 1,000,000 (1 Mn.), 10% are doctors. Let's say there's a system wherein you feed in a person's Unique ID (UID) and it tells you if the person is a doctor or not. The system has a 0.01% chance of incorrectly reporting a person who is not a doctor to be a doctor (a False Positive).

Now, let's work out our confidence levels of the results given out by the system.


On the other hand if just 0.01% of people in the group are actually doctors (while the rest of the info. remains same) the confidence level works out to be quite different.


This clearly shows that the likelihood of the answer being a False Positive has shot up from much under 1% to as much as 50%, when the occurrence of a condition (number of doctors) within a given population dropped from 10%  (i.e. 100,000) to a low value of 0.1% (i.e. 1,000).