## Video Tutorial – Rolling 2 Dice: An Intuitive Explanation of The Central Limit Theorem

According to the central limit theorem, if

• $n$ random variables, $X_1, ..., X_n$, are independent and identically distributed,
• $n$ is sufficiently large,

then the distribution of their sample mean, $\bar{X_n}$, is approximately normal, and this approximation is better as $n$ increases.

One of the most remarkable aspects of the central limit theorem (CLT) is its validity for any parent distribution of $X_1, ..., X_n$.  In my new Youtube channel, you will find a video tutorial that provides an intuitive explanation of why this is true by considering a thought experiment of rolling 2 dice.  This video focuses on the intuition rather than the mathematics of the CLT.  In a later video, I will discuss the technical details of the CLT and how it applies to this example.

You can also watch the video below the fold!

## Mathematical and Applied Statistics Lesson of the Day – The Central Limit Theorem Applies to the Sample Mean

Having taught and tutored introductory statistics numerous times, I often hear students misinterpret the Central Limit Theorem by saying that, as the sample size gets bigger, the distribution of the data approaches a normal distribution.  This is not true.  If your data come from a non-normal distribution, their distribution stays the same regardless of the sample size.

Remember: The Central Limit Theorem says that, if $X_1, X_2, ..., X_n$ is an independent and identically distributed sample of random variables, then the distribution of their sample mean is approximately normal, and this approximation gets better as the sample size gets bigger.

## Exploratory Data Analysis: Combining Histograms and Density Plots to Examine the Distribution of the Ozone Pollution Data from New York in R

#### Introduction

This is a follow-up post to my recent introduction of histograms.  Previously, I presented the conceptual foundations of histograms and used a histogram to approximate the distribution of the “Ozone” data from the built-in data set “airquality” in R.  Today, I will examine this distribution in more detail by overlaying the histogram with parametric and non-parametric kernel density plots.  I will finally answer the question that I have asked (and hinted to answer) several times: Are the “Ozone” data normally distributed, or is another distribution more suitable?

Read the rest of this post to learn how to combine histograms with density curves like this above plot!

This is another post in my continuing series on exploratory data analysis (EDA).  Previous posts in this series on EDA include