The Central Limit Theorem, Simplified!

Although the Central Limit Theorem may seem daunting at first, it’s really quite simple to comprehend once we understand the concept of the normal distribution first.

What Is the Normal Distribution?

The normal distribution, also known as the Gaussian curve or the bell curve, is a symmetrical distribution that is centered around its mean. In addition, the normal distribution will have a relatively higher amount of values around its center and a relatively lower amount of value at its tails. For a distribution to be normal, its mean, median, and mode have to be equal. In addition, approximately 68% of the distribution has to be within one standard deviation from the mean and around 95% of the distribution has to be within two standard deviations from the mean.

As long as the normal distribution respects the above rules, it can come in many different sizes:

What Does This Have to Do with the Central Limit Theorem?

Great question! The Central Limit Theorem states that as independent random variables are increasingly taken, the distribution of these variables will look increasingly similar to a normal distribution. This even holds true if the population itself is not evenly distributed! That’s right, the population itself could be skewed, but as an increasing number of samples are taken from the population, these samples will be normally distributed. However, there is a small caveat to this: if the population itself is not normally distributed, your n, or number of samples taken, has to be greater than 30 for it to be normally distributed.

Why Do We Care about the Central Limit Theorem, Anyway?

One of the biggest reasons why we care about the Central Limit Theorem is because it facilitates the gathering of political polls in an incredible way! You might often ask yourself: how do political polls work? How can a pollster take a poll of around 1,000 registered voters and say that it’s representative of the country as a whole? Without the concept of the Central Limit Theorem, a pollster would have to contact every single person in a country to accurately find out the opinions of its citizens. This would be an impossible task due to financial and time restraints. The Central Limit Theorem has many different uses, but political polling is one of the most important.