‘twas a glorious day in NYC today, where by mid-afternoon, the overnight snow had settled beautifully on rooftops - it looked like a post card of the Swiss countryside

I went to Central Park today and I thought I’d just use Central Park as our backdrop

Imagine you go to Central Park today, and you stand next to 59th and 7th, right at the front entrance of the park

You find five people as they come into the park, and ask them about how much they love pizza. Specifically, you ask them to rate about how much they love a slice of vintage NYC pizza - on a scale of 1 to 10, with ten being that they absolutely love a New York slice

The five people you found at random did give you an answer, and they rated it a 3, 4, 9, 9 and 10. The mean (average) of these five samples is (3+4+9+9+10)/5 = 7 (this is the mean of the five samples you took ~ it’s called the sample mean)

Great, on an average, the five people you asked at random, love pizza at a 7 out of 10

Now imagine, you do the same thing tomorrow. This time, the five random people you ask, respond with, 3, 3, 1, 9, and 9. The average of these five samples is (3+3+1+9+9)/5 = 5

Now, imagine you repeat this experiment for 90 days. For each day, you’d ask five random people who come to Central Park, and you estimate the sample mean each day

You’ll end up with 90 sample means right ? plot them, and it would look like a bell curve

**That - is one of the most powerful theorems in statistics. It’s called Central Limit Theorem (CLT)**

Some days, you might get people who are allergic to gluten and they cannot have a normal NYC slice of cheese or pepperoni. So, they all would be forced to say 0 or 1 and your sample mean for that day would be to the far left of the bell curve

Then on other days, you’d have people who absolutely adore a slice, and they’d all say a 9 or a 10. The sample mean from that day would be to the far right of the bell curve

But most days, you’d have people who love a New York slice enough to say a 7 (7.1, 7.5 or 7.7) and that would be where your middle part of the bell curve is. We’d call this the median (mid point), and it’s the highest point in the bell curve

It is high and in the middle because this is where most people would end up - in response to this question

CLT has important implications for sampling, polls, and machine learning algorithms. We don’t always know the underlying shape or distribution of things

A good fraction of things in our world approximately look like a bell curve. A lot of other things in our world follow the shape of other probability distributions, but here is the beauty of CLT,

even if you know nothin’ about how much, people going to Central Park, love pizza,

if you repeat this experiment for 90 days and plot the sample mean from each day, you will end up with a bell curve. That bell curve will tell you the entire range (i.e. dispersion) of - how many people love pizza and by how much

Here is more elegance to this, if you repeat this experiment for 90 days, you might get a fat bell curve (left end of the picture below). Where as if you repeat this experiment for the next 900 days, you’ll get a very narrow and skinny bell curve (right end of the picture below)

The more times you repeat this experiment, the narrower and taller your bell curve gets, because you’d have more and more people rate how much they love pizza - and your sampling results will get you closer and closer to the real truth of, how much, people in NYC love a slice of pepperoni

As your sample size tends to infinity, a stochastic problem becomes a deterministic one, or in other words, as you sample more and more people, you’ll get closer and closer to the real truth

There are about nine million people in NYC, and you cannot ask all of them to rate how much they love a slice. Say, hypothetically, on an average, the nine million people love pizza at about a 8 out of 10

The more times you repeat this experiment, the closer and closer your median (the highest and mid point of the bell curve) will get to 8

isn't there sheer beauty in statistics ?

Source : A homage to one of NYC’s best institutions - Joe’s Pizza

so taking 5 people 900 times ..,,, or 900 *5 people at once.. both move towards CLT right? actually it took time for me get this feel..? am i right? . wonder of statistics..