Back when I used to work at the Gokhale Institute, I would get a recurring request every year without fail. What request, you ask? To get AB InBev to come on campus. To the guys at AB InBev – if you’re reading this, please do consider going to GIPE for placements. The students are thirsty to, er, learn.
But what might their work at AB InBev look like?
I don’t know for sure, but it probably will not involve working with barley and hops. It should, though, if you ask me, because today, building statistical models about other aspects of selling beer today might rake in the moolah. But there will be a pleasing historical symmetry about using stats to actually brew beer.
You see, you can’t – just can’t – make beer without barley and hops. And to make beer, these two things should have a number of desirable characteristics. Barley should have optimum moisture content, for example. It should have high germination quality. It needs to have an optimum level of proteins. And so on. Hops, on the other hand, should have beefed up on their alpha acids. They should be brimming with aroma and flavor compounds. There’s a world waiting to be discovered if you want to be a home-brewer, and feel free to call me over for extended testing once you have a batch ready. I’ll work for free!
But in a beer brewing company, it’s a different story. There, given the scale of production, one has to check for these characteristics. And many, many years ago – a little more than a century ago, in fact – there was a guy who was working at a beer manufacturing enterprise. And this particular gentleman wanted to test these characteristics of barley and hops.
So what would this gentleman do? He would walk along the shop-floor of the firm he worked in, and take some samples from the barley and hops that was going to be used in the production of beer. Beer aficionados who happen to be reading this blog might be interested to know the name of the firm bhaisaab worked at. Guinness – maybe you’ve heard of it?
So, anyway, off he’d go and test his samples. And if the results of the testing were encouraging, bhaisaab would give the go-ahead, and many a pint of Guinness would be produced. Truly noble and critical work, as you may well agree.
But Gosset – for that was his name, this hero of our tale – had a problem. You see, he could never be sure if the tests he was running were giving trustworthy results. And why not? Well, because he had to make accurate statements about the entire batch of barley (and hops). But in order to make an accurate statement about the entire batch, he would have liked to take larger samples.
Imagine you’re at Lasalgaon, for example, and you’ve been tasked with making sure that an entire consignment of onions is good enough to be sold. How many sacks should you open? The more sacks you open, the surer you are. But on the other hand, the more sacks you open, the lesser the amount left to be sold (assume that once a sack is open, it can’t be sold. No, I know that’s not how it works. Play along for the moment.)
A student of statistics faced with a real world problem (Source)
So how many sacks should you open? Well, unless your boss happens to be a lover of statistics and statistical theory for its own sake, the answer is as few as possible.
The problem is that you’re trying to then reach a conclusion about a large population by studying a small sample. And you don’t need high falutin’ statistical theory to realize that this will not work out well.
Your sample might be the Best Thing Ever, but does that mean that you should conclude that the entire population of barley and hops is also The Best Thing Ever? Or, on the other hand, imagine that you have a strictly so-so sample. Does that mean that the entire batch should be thought of as so-so? How to tell for sure?
Worse, the statistical tools available to Gosset back then weren’t good enough to help him with this problem. The tools back then would give you a fairly precise estimate for the population, sure – but only if you took a large enough sample in the first place. And every time Gosset went to obtain a large enough sample, he met an increasingly irate superior who told him to make do with what he already had.
Or that is how I like to imagine it, at any rate.
So what to do?
Well, what our friend Gosset did is that he came up with a whole new way to solve the problem. I need, he reasoned, reasonably accurate estimates for the population. Plus, khadoos manager says no large samples.
Ergo, he said, new method to solve this problem. Let’s come up with a whole new distribution to solve the problem of talking usefully about population estimates by studying small samples. Let’s have this distribution be a little flatter around the centre, and a little fatter around the tails. That way, I can account for the greater uncertainty given the smaller sample.
And if my manager wants to be a little less khadoos, and he’s ok with me taking a larger sample, well, I’ll make my distribution a little taller around the center, and a little thinner around the tails. A large enough sample, and hell, I don’t even need my new method.
And that, my friends, is how the t-distribution came to be.
You need to know who Gosset was, and why he did what he did, for us to work towards understanding how to resolve Precision and Oomph. But it’s going to be a grand ol’ detour, and we must meet a gentleman, a lady, and many cups of tea before we proceed.
If you were to watch a cooking show, you would likely be impressed with the amount of jargon that is thrown about. Participants in the show and the hosts will talk about their mise-en-place, they’ll talk about julienned vegetables, they’ll talk about a creme anglaise and a hajjar other things.
But at the end of the day, it’s take ingredients, chop ’em up, cook ’em, and eat ’em. That’s cooking demystified. Don’t get me wrong, I’m not for a moment suggesting that cooking high-falutin’ dishes isn’t complicated. Nor am I suggesting that you can become a world class chef by simplifying all of what is going on.
But I am suggesting that you can understand what is going on by simplifying the language. Sure, you can’t become a world class cook, and sure you can’t acquire the skills overnight. But you can – and if you ask me, you should – understand the what, the why and the how of the processes involved. Up to you then to master ’em, adopt ’em after adapting ’em, or discard ’em. Maybe your paneer makhani won’t be quite as good as the professional’s, but at least you’ll know why not, and why it made sense to give up on some of the more fancy shmancy steps.
Can we deconstruct the process of statistical inference?
Let’s find out.
Let’s assume, for the sake of discussion, that you and your team of intrepid stats students have been hired to find out the weight of the average Bangalorean. Maybe some higher-up somewhere has heard about how 11% of India is diabetic, and they’ve decided to find out how much the average Bangalorean weighs. You might wonder if that is the best fact-finding mission to be sent on, but as Alfred pointed out all those years ago, ours not to reason why.
Does it make sense to go around with a weighing scale, measuring the weight of every single person we can find in Bangalore?
Nope, it doesn’t. Apart from the obvious fact that it would take forever, you very quickly realize that it will take literally forever. Confused? What I mean is, even if you imagine a Bangalore where the population didn’t change at all from the time you started measuring the weight to the time you finished – even in such a Bangalore, measuring everybody’s weight would take forever.
But it won’t remain static, will it – Bangalore’s population? It likely will change. Some people will pass away, and some babies will be born. Some people will leave Bangalore, and some others will shift into Bangalore.
Not only will it take forever, it will take literally forever.
And so you decide that you will not bother trying to measure everybody’s weight. You will, instead, measure the weight of only a few people. Who are these few people? Where in Bangalore do they stay, and why only these and none other? How do we choose them? Do we pick only men from South Bangalore? Or women from East Bangalore? Only rich people near MG Road? Or only basketball players near National Games Village? Only people who speak Tamil near Whitefield? Only people who have been stuck for the last thirteen months at Silk Board? The ability to answer these questions is acquired when we learn how to do sampling.
What is sampling? Here’s our good friend, ChatGPT:
“Sampling refers to the process of selecting a subset of individuals from a larger population to gather data. In the case of a survey of the sort you’re talking about, you would need to define the target population, which could be the residents of Bangalore city. From this population, you would need to employ a sampling method (e.g., random sampling, stratified sampling) to select a representative sample of individuals whose weights will be measured.”
One sample or many samples? That is, should you choose the most appropriate sampling method and collect only one humongous sample across all of Bangalore city, or many different (and somewhat smaller) samples? This is repeated sampling. Monsieur ChatGPT again:
“You might collect data from multiple samples, each consisting of a different group of individuals. These multiple samples allow you to capture the heterogeneity within the population and account for potential variations across different groups or locations within Bangalore city. By collecting data from multiple samples, you aim to obtain a more comprehensive understanding of the weight situation.”
So all right – from the comfort of your air-conditioned office, you direct your minions to sally forth into Namma Bengaluru, and collect the samples you wish to analyze. And verily do they sally forth, and quickly do they return with well organized sheets of Excel. Each sheet containing data pertaining to a different, well-chosen sample, naturally.
What do you do with all these pretty little sheets of samples? Well, you reach a conclusion. About what? About the average weight of folks in Bangalore. How? By studying the weights of the people mentioned in those sheets.
So, what you’re really doing is you’re reaching a conclusion about the population by studying the samples. This is statistical inference. Our friend again:
“Statistical inference involves drawing conclusions about the population based on the information collected from the sample. Statistical inference helps you make generalizations and draw meaningful conclusions about the larger population using sample data.”
Remember those pretty little Excel sheets that your minions bought back as gifts for you? If you so desire, you can have Excel give you the average weight for each of those samples. Turns out you do so desire, and now you have many different averages in a whole new sheet. Each average is the average of one of those samples, and let’s say you have thirty such samples. Therefore, of course, thirty such averages.
What to do with these thirty averages, you idly wonder, as you lazily swing back and forth in your comfortable chair. And you decide that you will try and see if these averages can’t be made to look like a distribution. Let’s say the first sample has an average weight of sixty five kilograms. And the second one sixty-three kilograms. And the seventh one is sixty-seven kilograms. And the twenty-first is seventy kilograms. Can we arrange these averages in little groups, lightest to the left and heaviest to the right, and draw little sticks to indicate how frequently a particular value occurs?
That’s a sampling distribution.
“A sampling distribution is the distribution of a statistic, such as the mean, calculated from multiple samples. In the context of estimating the average weight of citizens in Bangalore, you would collect weight data from multiple samples of individuals residing in the city. By calculating the mean weight in each sample, you can examine the distribution of these sample means across the multiple samples. The sampling distribution provides insights into the variability of the estimate and helps you assess the precision of your findings regarding the average weight in the population.”
What’s that mean – “the sampling distribution provides insights into the variability of the estimate and helps you assess the precision of your findings regarding the average weight in the population”?
Well, think of it this way. Let’s say you have to report back what the average weight is. That is, what is the average weight, in your opinion, of all of Bangalore. Should you pick the first sample and report it’s value? Or the eighth sample? Or the twenty-third?
Why not, you decide in a fit of inspiration, take the average of all these averages? Whatay idea, sirjee! Because even if you assume that one sample might be a little bit off from the population mean, our father what goes? Maybe the second sample will be a little bit off, but on the other side! That is, the first sample is a little lighter than the actual value. But the second sample might well be a little heavier than the actual value! The average of the two might result in the two errors canceling each other out! And if taking the average of two averages is a good idea, why, taking the average of the thirty averages is positively brilliant.
But hang on a second, you say. Just you wait, you say, for you’re on a roll now. If we can take – and follow me closely here – the average of these averages, well then. I mean to say, why not…
… why not calculate the standard deviation of these averages as well! Not only do we get the average value, but we also get the dispersion, on average, around the average value. Ooh, you genius, you.
This latest invention of yours, it has a name. It’s called the standard error:
“The standard error is a measure of the variability or uncertainty associated with an estimate. In the case of estimating the average weight of citizens in Bangalore, the standard error would quantify the uncertainty surrounding the estimated mean weight. It is typically calculated based on the observed variability in weight within the sample(s) and the information contained in the sampling distribution of the sample means. By considering the spread of the sample means around the population mean, the standard error provides an indication of how much the estimated average weight may deviate from the true population average. A smaller standard error suggests higher precision, indicating that the estimated average weight is expected to be closer to the true population average. Conversely, a larger standard error indicates greater uncertainty and variability, implying that the estimated average weight may have a wider range of potential values and may be less precise.”
Well, not quite. The standard error is actually the standard deviation of the sampling distribution divided by the square root of the number of samples. Ask, ask. Go ahead, ask. Here’s why:
“Imagine you have a larger sample size. With more observations, you have more information about the population, and the estimates tend to be more precise. Dividing the standard deviation by the sample size reflects this concept. It adjusts the measure of variability to match the precision associated with the sample size.”
Larger the sample size, lower the standard error. Also known as ” more data is always better”. Which is why, since time immemorial, every stats prof in the world has always responded to questions about how large your sample size should be with the following statement:
“As large as possible”.
They’re your friends, you see. They wish you to have smaller standard errors.
And so, the sampling distribution gives you the following gifts:
An estimate of what the average weight is for the city of Bangalore
This estimate is obtained by studying many samples, and taking the average of all of these samples. We can hope that errors in each samples are cancelled out by other errors in other samples
The calculation of the standard error of the sampling distribution tells how much the estimated weight varies around the population mean.
Not only do you have a very good guess about the value, but you also have a very good guess about the error that is implicit in your guess. Buy one get one free, you might smugly tell your superior.
And that, my friends, is the process of statistical inference.
But kahaani, as you might have guessed, abhi baaki hai. We’ll get back to this in a future post.
“The growing use of digital transactions—by consumers, investors, tax payers—as well as the rise of newer forms of data collection has the potential to revolutionise Indian public policy. It is unlikely that these newer forms of data will completely replace the more traditional numbers derived from surveys, national accounts and administrative data. They will more likely complement each other. Government agencies will increase their dependence on big data analytics in the coming years—though the risks to individual privacy should not be underestimated.” Niranjan Rajadhakshya on big data, data collection, and how we might reach conclusions on the basis of both over time. The papers he mentions towards the end are also worth reading in their own right!
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“According to the Handbook Of Statistics On Indian Economy 2016-17, since the 1991 reforms, the Union government’s revenue has increased 25 times and state government revenues have increased 28 times in nominal terms, and about 4 times in real terms. Economic growth and the consequent increase in revenue also increases the ability of the government to focus on inequality and deal with sector-specific distress.” The article is about why economic growth is important for poverty alleviation, an as the article itself says, this is both an obvious point, but also one that bears repetition. But the excerpt above was notable for me: obvious in retrospect, but worth thinking about. Government revenues have gone up significantly since liberalization.
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“In 2014, the Delhi High Court found both major parties guilty of violating foreign-exchange laws when they accepted a donation from London-based commodities giant Vedanta Resources Plc.(The suit, filed by a former top bureaucrat and the Association for Democratic Reforms, was against the political entities and Vedanta wasn’t a party. The company didn’t respond to request for comment. The BJP and Congress argued the donations weren’t foreign because the Vedanta units that channelled the money were registered under Indian law.)
The law passed last year changed the definition of a foreign company all the way back to 1976, effectively nullifying the court’s verdict because Vedanta’s overseas parent owned less than 50 percent of the Indian unit.” A somewhat depressing, if all too predictable read about campaign financing in India. There isn’t that much more to say do read the whole thing, though.
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“This jellyfish doesn’t mean to brag, but it’s both beautiful and immortal. If it gets sick, or even stressed, it just reverts into it’s younger self so it can get strong and mature again, bouncing between youth and adulthood forever.”
It’s impossible to choose one particular thing to highlight – please (for a change!) read every single comment. Nature is an impossibly weird thing, and we know far too little about it.
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“One of the great sources of leverage is other people. You can get leverage via directing folks to do things (a superpower whose impact I probably underappreciated when running my business solo). You can also get it by making them more effective at doing things.” A very long essay, perhaps rambling in part – but a great read nonetheless. About a whole variety of things, but mostly about productivity, I’d say – the self and the organization, both.
EconForEverybody 