Data Distribution and its types - Probability and Statistics

Introduction to Distribution

In statistics, a distribution describes how the values of a variable or dataset are spread or distributed. It provides insights into the central tendency, variability, and shape of the data. Understanding distributions is fundamental for statistical analysis and inference.

Types of Distributions

Discrete Probability Distributions

Discrete probability distributions apply to variables that have specific, countable outcomes.

Examples:

• Binomial Distribution: Counts the number of successes in a fixed number of trials, like flipping a coin. Parameters: Number of trials $n$, probability of success $p$. Example: Number of heads in 10 coin flips, with $p = 0.5$ means each flip has a 50% chance of landing heads.
• Poisson Distribution: Counts the number of events in a fixed time or space interval. Parameter: Average rate $\lambda$. Example: Number of emails received in an hour. If you typically get 10 emails per hour, the distribution predicts how many emails you might get in any specific hour.

Continuous Probability Distributions

Continuous probability distributions apply to variables that can take any value within a range.

Examples:

• Normal Distribution: Symmetrical, bell-shaped distribution defined by its mean $\mu$ and standard deviation $\sigma$. Parameters: Mean $\mu$, standard deviation $\sigma$. Example: Heights of adult men, where most men are around average height, and fewer are much shorter or taller.
• Exponential Distribution: Measures the time between events in a Poisson process. Parameter: Rate $\lambda$. Example: Time between buses arriving at a bus stop. If buses arrive every 15 minutes on average, the distribution describes actual wait times.

Key Characteristics of Distributions

Central Tendency - Measures where the center of a distribution lies.

• Mean: Average value of the data.
• Median: Middle value when data is sorted.
• Mode: Most frequent value.

Dispersion : Measures the spread of the data.

• Range: Difference between the maximum and minimum values.
• Variance: Average of the squared deviations from the mean.
• Standard Deviation: Square root of the variance.

Shape - Describes the form of the distribution.

• Skewness: Measure of asymmetry. Positive Skew: Tail on the right. Negative Skew: Tail on the left.
• Kurtosis: Measure of the "tailedness". Leptokurtic: Heavy tails. Platykurtic: Light tails.