The progression from the top left to the bottom right plot demonstrates the process of moving from a discrete representation of data (using histograms with large bins) to a continuous representation (using a smooth curve). This transition helps in understanding the underlying distribution of the data more accurately, especially when the number of data points is large.

In summary, this image shows how increasing the number of bins in a histogram can lead to a more detailed and eventually continuous approximation of the data distribution. The final continuous representation (bottom right) is useful in statistical analysis and probability theory, where we often deal with continuous probability distributions to model real-world phenomena.

What are the benefits of standardizing a distribution?

Standardizing a distribution has several benefits. Firstly, it transforms datasets into a standard scale, making it easier to compare between different datasets. Secondly, it simplifies statistical analysis, particularly when using techniques that assume a standard normal distribution, Finally, standardizing features in machine learning can improve the convergence rate of optimization algorithms and prevent some features from dominating others, leading to improved model performance.

The purpose of standardizing a distribution is to transform the data such that it has a mean of 0 and a standard deviation of 1. Many statistical techniques assume data is normally distributed with a mean of 0 and a standard deviation of 1. This is often done to normalize data, making it easier to compare different datasets and apply various statistical methods. This standardized distribution is often called a Z-distribution.

Skewness:

Skewness measures the asymmetry of a probability distribution of a real-valued random variable about its mean. In other words, it indicates whether the data points in a dataset are skewed to the left (negative skewness) or to the right (positive skewness) of the mean.

$$ \text{Skewness} = \mathbb{E} \left[ \left( \frac{X - \mu}{\sigma} \right)^3 \right] $$

Direction of Skewness:

• Positive Skewness (Right Skewness): If the skewness is positive, the distribution has a longer tail on the right side. This means there are more extreme high values. The bulk of the values lie to the left of the mean, and the right tail is longer. This is often seen in distributions of incomes or house prices.
• Negative Skewness (Left Skewness): If the skewness is negative, the distribution has a longer tail on the left side. This means there are more extreme low values. The bulk of the values lie to the right of the mean, and the left tail is longer. This can occur in distributions like test scores, where most students score high but a few score very low.
• Zero Skewness (Symmetrical Distribution): If the skewness is zero, the distribution is perfectly symmetrical around the mean. This is characteristic of a normal distribution.

Kurtosis: