Hi everyone and welcome! In this video, we’ll talk about the real-world application of one
of the most widely used theorems in data science: The Central Limit Theorem. For more super
practical videos like this one, make sure to subscribe to our channel right now.
The Central Limit Theorem is the core of ‘hypothesis testing’- an approach in statistics that
lets you use data to evaluate your ideas. In fact, this theorem can be applied to a
variety of real-life problems. Let’s illustrate with an example.
Say, you own a business connected to the fish market area, more specifically - a trout farm.
You own hundreds of fish reservoirs where you keep and breed trout in order to sell
them to the main fish stores which supply the biggest cities in the country. The farm
operates in the following way: you buy and breed fish, which you later sell. Your clients
range from single fish vendors, to supermarket chains.
Quite straightforward, right? But to transform the above procedure into
a cycle, you need to use your capacity of reservoirs appropriately. Here’s how it
works. First, you have to label the reservoir depending on the approximate size of fish
in it. The labels are three - newly hatched, middle size, and first-class. As the fish
grow, you move them from newly hatched to middle size and from middle size to first-class
and once they’re fully grown, you sell them. Meanwhile, you’ve stocked the pool with
newly hatched trout and the process goes on. Now, what’s crucial to know is that first-class
fish are the largest among all the fish-groups. Why is that so important? Well, as a business
owner, your goal is to maximize profit, and, to achieve that, you must sell fish when they
reach the largest possible size, as customers pay by the pound. What’s more, there’s
a regulation set by the government that allows you to keep 1,000 fish maximum in the first-class
reservoirs. All things considered, selling the first-class
fish as large as possible would be your best strategy to increase profit. Therefore, you
need to maximize the length of each fish in every single tank.
Easy to say, but how can you do that? How long is it going to take? Is the effort worth
the time you will lose while in competition with other fish farms?
Let’s think about it for a second. One option is to try to measure each fish separately...
But there are 1,000 fish in each tank, and more than 20 tanks stocked with first-class
fish. So, manually measuring each fish doesn’t sound like a good idea. It will simply take
too long. You and your employees will be stuck measuring fish from dawn till dusk which is
highly inefficient. What’s more, if you can quickly find out the average size of your
fish, you can project how long it will take for each tank to grow the necessary size.
This will allow you to plan key resources such as staff and fish food supplies.
Finally, having an edge in the business depends on your ability to stay competitive and agile.
Knowing what kind of sales volumes you can produce daily helps you to be prepared when
a customer calls to purchase a certain number of tanks due a particular date.
And this is where you can truly benefit from some maths knowledge to optimize the process.
More precisely, you can use the Central Limit Theorem - it will help you tremendously with
time-saving and, what’s more, you will also maximize your profit at the same time.
So, what is the Central Limit Theorem and how does it work?
The Central Limit Theorem is a theorem in probability theory, whose first version was
proposed by the French mathematician Abraham de Moivre in 1733. Moivre published an article
where he used a normal distribution to approximate the distribution of the number of heads resulting
from many tosses of a fair coin. The finding was nearly forgotten until the French mathematician
Pierre-Simon Laplace expanded it in his monumental work in the 19th century. Over the years,
numerous versions of it have been discovered and proven by other mathematicians.
In its base form, the Central Limit Theorem states that if we have a population and we
take sufficiently large random samples from it, then the sample means will be approximately
normally distributed. We call the average of each sample a sample mean.
In case you’re wondering what a sample mean is, let’s go back to our fish example and
see. If you simply take groups of fish from each
first-class reservoir and record the average size in each group, that would be the so-called
sample means. In fact, that’s exactly how the Central Limit Theorem can be applied to
improve our fish measuring dilemma. Let’s see it in practice!
It’s a rule of thumb that the minimum sample size to apply the CLT on is 30. That’s why
we start with a sample size of 30. That means, each group of fish you select from your 1,000
first-class fish reservoir will consist of 30 fish. Then, you’ll increase the sample
size to 50,100 etc. And each time you record the sample mean. The idea is to gradually
increase the sample size because the bigger the sample the better the theorem applies.
However, you must not try with too many values for sample sizes because you want to be able
to act as quickly as possible for each first-class tank. So, there are two possible outcomes:
The size of fish in the respective pool is the maximized one, 50 cm, which means you
can sell the whole tank; Or you must keep feeding them and take measurements
at a future date; Then you do this for each first-class reservoir.
After recording and plotting the sample means, you can see the plot fits under the bell-shaped
curve, illustrating the Normal distribution. Therefore, going forward, you are in a position
to perform a statistical analysis using the properties of this distribution.
From the normal distribution graph, you see that the middle is denoted by µ-the mean
of sample means, which divides the area into two equal and symmetric halves. Moreover,
the area under the curve, which is this area here is equal to 1.
The first key observation is that approximately two thirds of the collected means are one
standard deviation away from the mean of sample means and approximately all of the data lies
within two standard deviations away from it. But how does all this affect your fish measuring
business? Well, for example if you have a sample mean
of 48 and standard deviation of 2 for a tank, then the theorem says that approximately two
thirds of your observed sample means are in the range between 46 and 50. Moreover, almost
all of the sample means are between 44 and 52. This tells you that you must feed the
fish in the tank a little bit more and on the next measurements. This can have a massive
effect on your planning, as it helps you track the rate of growth in length of fish.
Also, the sample means are normally distributed random variables, which yields that we can
standardize them. More precisely, standardization in this case stands for transforming each
of our variables’ mean to 0 and variance to 1. This way, we can easily find information
in statistical tables about the area under the curve of the standard normal distribution.
Knowing all this, now we can answer some very interesting questions:
What is the probability of seeing the average of the 5th sample of 30 fish in the range
between 45 and 50? Or what is the probability to obtain that the mean of 10th sample is
bigger than 48cm? Extracting those probabilities from the table
above, will give you a more intuitive picture of what’s happening in your tanks.
Alright! So, this is how the Central Limit Theorem
can be applied in a real-world scenario. The power of this theorem lies in the fact that
it makes it possible to analyze data even with incomplete information about it and allows
large datasets to be well approximated in a highly accurate manner.
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