The idea of sampling is fundamental in data science and even though it is taught in every data science book or course out there, there is still a lot to be learned about it. The reason is that sampling is a very deep topic and just like every other data-related topic out there, conventional Statistics fails to do it justice. The reason is simple: good quality samples come about by obtaining an unbiased representation of a population and this is rarely the case from strictly random samples. Also, the fact that Statistics doesn’t offer any metric whatsoever regarding bias in a sample, doesn’t help the whole situation.
The Index of Bias (IB) of a Sample
There are two distinct aspects of a sample, its bias and its diversity. Here we’ll explore the former, as it is expressed in the two fundamental aspects of a distribution, its central point and its spread. For these aspects we’ll use two robust and fairly stable metrics, the median and the inter-quartile range, respectively. The deviation of a sample in terms of these metrics, with each deviation normalized based on the maximum deviation possible for the given data, yields two metrics, one of the central point and one for the spread. Each metric takes values between 0 and 1, inclusive. The average of these metrics is defined as the index of bias of a sample and takes values in the [0, 1] interval too. Note that the index of bias is always in relation to the original dataset we take the sample from.
Although the above definition applies to one-dimensional data only, it can be generalized to n-dimensional data too. For example, we can define the index of bias of a dataset comprising of d dimensions (features) as the arithmetic mean of the index of bias of each one of its features.
IB Scores for Various Samples
Strictly random samples tend to have a fairly high IB score, considering that we expect them to be unbiased. That’s not to say that they are always very biased, but they are definitely in need of improvement. Naturally, if the data we are sampling is multi-dimensional, the chances of a bias are higher, resulting to an overall biased sample.
Samples that are engineered with IB in mind are more likely to be unbiased in that sense. Naturally, this takes a bit of effort. Still, given enough random samples, it is possible to get a good enough sample that is unbiased based on this metric. In the attached file I include the IB scores of various samples, for both a random sampling process (first column) and a more meticulous one that aims to obtain a less biased sample (second column). Note that the latter did not use the IB metric in its algorithm, though a variant of it that makes use of that metric is also available (not free though). Also, you don’t need to be an expert in statistical tests to see that the second sampling method is consistently better than the first one. Finally, I did other tests on different data, and in every case, the results were very similar.
Hopefully, this small experiment goes on to show how sampling is not a trivial problem as it is made out to be by those who follow some old-fashioned paradigm for data analytics. Despite its simplicity, sampling has a lot of facets that need to be explored and understood, before it can be leveraged as a data science technique. After all, what good is an advanced model if the data it is trained on is biased? I believe we owe it to ourselves as data scientists to pay attention to every part of the data science process, including the less interesting parts, such as sampling.
Is It Possible to Have a Set of Numbers in a Variable Where the Majority of Them Is Higher than the Mean?
!ommon sense would dictate that this is not possible. After all, there are numerous articles out there (particularly on the social media) using that as a sign of a fallacy in an argument. Things like “most people claim that they have better than average communication skills, which is obviously absurd!” are not uncommon. However, a data scientist is generally cautious when it comes to claims that are presented without proof, as she is naturally curious and eager to find out for herself if that’s indeed the case. So, let’s examine this possibility, free from prejudice and the views of the know-it-alls that seem to “know” the answer to this question, without ever using a programming language to at least verify their claim.
The question is clear-cut and well-defined. However, our common sense tells us that the answer is obvious. If we look into it more deeply though and are truly honest with ourselves, we’ll find out that this depends on the distribution of the data. A variable may or may not follow the normal distribution that we are accustomed to. If it doesn’t it’s quite likely that it is possible for the majority of the data points in a variable to be larger than the average value of that variable. After all, the average value (or arithmetic mean as it is more commonly known to people who have delved into this matter more), is just a measure of central tendency, certainly not the only measure for figuring out the center of a distribution. In the normal distribution, this metric coincides in value with that of median, which is always in the center of a variable, if you order its values in ascending or descending order. However, the claim that mean = median (in value) holds true only in cases of a symmetric distribution (like the normal distribution we are so accustomed to assuming it characterizes the data at hand). If the distribution is skewed, something quite common, it is possible to have a mean that is smaller than the median, in which case, the majority of the data points will be towards the right of it, or in layman’s terms, higher in value than the average.
Don’t take my word for it though! Attached is a script in Julia that generates an array that is quite likely to have the majority of its elements higher in value than its overall mean. Feel free to play around with it and find out by yourselves what the answer to this question is. After all, we are paid to answer questions using scientific processes, instead of taking someone else’s answers for granted, no matter who that person is.
Lately everyone likes to talk big picture when it comes to data science and artificial intelligence. I’m guilty of this too, since this kind of talk lends itself for blogging. However, it is easy to get carried away and forget that data science is a very detailed process that requires meticulous work. After all, no matter how much automation takes place, mistakes are always possible and oftentimes unavoidable. Even if programming bugs are easier to identify and even prevent, to some extent, some problem may still arise and it is the data scientist’s obligation to handle them effectively.
I’ll give an example from a recent project of mine, a PoC in the text analytics field. The idea was to develop a bunch of features from various texts and then use them to build an unsupervised learning model. Everything in the design and the core functions was smooth, even from the first draft of the code. Yet, when running one of the scripts, the computer kept running out of memory. That’s a big issue, considering that the text corpus was not huge, plus the machine used to run the programs is a pretty robust system, with 16GB of RAM, while it’s also running Linux (so a solid 15GB of RAM are available to the programming language to utilize as needed). Yet, the script would cause the system to slow down until it would eventually freeze (no swap partition was set up when I was installing the OS, since I didn’t expect to ever run out of memory on this machine!) Of course, the problem could be resolved by adding a swap option to the OS, but that still would not be a satisfactory solution, at least not for someone who opts for writing efficient code. After all, when building a system, it is usually built to scale well and this prototype of mine didn’t look very scalable. So, I examined the code carefully and came up with various hacks to manage resources better. Also, I got rid of some unnecessary array that was eating up a lot of memory, and rerouted the information flow so that other arrays can be used to provide the same result. After a couple of attempts, the system was running smoothly and without using too much RAM.
It’s small details like these that make the difference between a data science system that is practical and one that is good only on the conceptual level (or one that requires a large cluster to run properly). Unfortunately, that’s something that is hard to learn through books, videos, or other educational material. Perhaps even conventional experience may not trigger this kind of lesson, though perhaps a good mentor might be very beneficial in such cases. The morale of the story for me is that we ought to continuously challenge ourselves in data science and never be content with our aptitude level. Just because something runs without errors identifiable by the language compiler, doesn’t mean that it’s production-ready. Even in the case of a simple PoC, like this one, we cannot afford to lose focus. Just like the data that is constantly evolving into more and more refined information, data scientists follow a similar process, as we grow into more refined manifestations of the craft.
Are the Countries of One Hemisphere Different to Those of the Other in Terms of Income Level? A Case Study Using Stats in Python
That’s a quite interesting question and coincidentally one that we can answer using the Gapminder data. First we’ll need to establish the hemisphere of each country, a fairly irksome but doable task. Using the data from www.mapsofworld.com and Wikipedia (for the countries where the site failed to deliver any information), we can classify each country to the North or the South hemisphere. For countries that are more or less equally divided between the two, we just insert a missing value (denoted as “NA” in our dataset). Note that a country that crosses the equator is still classified to one or the other hemisphere, if the majority of its area or if the majority of its main cities are on one side of it.
As for the income level of a country, although no such variable exists in the original dataset, we can employ the same transformation as we did in the previous case study and end up with a tertiary variable having the values high, medium, and low. Now, on to the hypotheses. Our null hypothesis would be “there is difference between the two hemispheres, regarding the proportions of countries of high, medium, and low income level”. In other words, the proportions should be more or less the same. The alternative hypothesis would be opposite of that, i.e. “there is a difference in the proportions of countries of high, medium, and low income level, between the two hemispheres.”
To test this we’ll employ the chi-square test since:
1. Both our variables are discreet / categorical
2. There are enough countries in all possible combinations of hemisphere and income level
For our tests we’ll employ the significance level of alpha = 0.05, which is quite common for this sort of analysis.
Once we remove the missing data from the variables (in data-frame sub1) by first replacing them with NaNs, we can create the contingency table ct1 and perform the chi square test on it. As there are 2 and 3 unique values in the variables respectively, the contingency table will be a 2 x 3 matrix:
income_level high low medium
North 57 44 45
South 6 15 16
It is clear that the North hemisphere has more countries, so the high numbers of each income level category may be misleading. This problem goes away if we take the proportions of these hemisphere-income_level combos instead:
income_level high low medium
North 0.904762 0.745763 0.737705
South 0.095238 0.254237 0.262295
Wow! 90% of all rich countries (countries with income level = high) are on the North part of the globe, with only barely 10% in the South. For the other types of countries the proportions are somewhat different and more similar to each other. So, if there is a discrepancy among the proportions, it is due to the rich countries being mainly in the Northern hemisphere. The chi-square test supports this intuition:
Chi-square value: 6.8244834439401423 (quite high)
P-value: 0.032967214150344128 (about 3%)
As p < alpha, we can safely reject the Null hypothesis and confidently say that the proportions are indeed skewed. Of course there is a 3% chance that we are wrong, but we can live with it!
Before we go deeper (ad hoc analysis) and examine if the pairwise comparisons are also significant, we need to apply the Bonferroni correction to the alpha value (not to be confused with the pepperoni correction, which applies mainly to pizzas!):
Adjusted alpha = alpha / (number of comparisons)
In our case the comparison we can make are high to low income countries, medium to low, and high to medium, three in total. Therefore, our new alpha is a = 0.017 (1.7%), making the chi-square tests a bit less eager to yield significant results.
By calculating the contingency tables for each one of these comparisons, we the following results respectively:
Comparison: High income level to Low income level
Chi-square value: 4.3468868966915135
P-value: 0.037076661090159897 (relatively low but not significant)
Comparison: Medium income level to Low income level
Chi-square value: 0.011613712922753157
P-value: 0.91418056977624251 (quite high and definitely not significant)
Comparison: High income level to Medium income level
Chi-square value: 4.8370929759550796
P-value: 0.027853811382628445 (quite low but not significant)
So, although we obtained some pretty good results originally (rejecting the null hypothesis), the ad hocs are not all that great, since none of the in-depth comparisons yield anything statistically significant (as they are all larger than the adjusted alpha). In other words, if we are given a couple of proportions for the income_level variable, we can not confidently predict the hemisphere. However, if we are given all three proportions, predicting the hemisphere is quite doable.
In one of the next posts we’ll examine a more foxy way of analyzing the same data, so feel free to revisit this blog. In the meantime, you can validate these results by examining the code and the data, made available in the attached files.
Zacharias Voulgaris, PhD
Passionate data scientist with a foxy flair when it comes to technology, technique, and tests.