Introduction

When building a data based model, it is important to know what kind of distribution the data follows. There are different types of distributions: discrete and continuous. Each type has its own set of characteristics and implications for analysis, making the ability to visualize and interpret these distributions fundamental for anyone building data-driven models. This discussion aims to delve into the methods and tools used for visualizing discrete and continuous distributions, highlighting their differences, and demonstrating how these visualizations can influence decision-making in data science.

In the following tutorial, I will provide visualizations to all distribution and provide tips on what to do when you get that distribution. Whenever you are working with data, it is important to visualize its distribution when doing exploratory data analysis. That will guide you which models to select next.

Discrete Distributions

Discrete distribution cannot take a value between 0 and 1. They are whole numbers only. An example of a discrete value is the number of persons in a room. There is no such thing as half a person each person is counted as 1.

Bernoulli Distribution

This distribution is used to model binary outcomes (success or failure). When analyzing Bernoulli data, focus on proportions and differences in proportions between groups.

library(ggplot2)

plot_distribution <- function(distribution, title, ...) {
  data <- data.frame(x = distribution(...))
  p <- ggplot(data, aes(x=x)) + 
    geom_histogram(stat="count", binwidth = 1, fill="steelblue", color="black") +
    ggtitle(title) +
    theme_minimal()
  print(p)
}

plot_distribution(rbinom, "Bernoulli Distribution", size = 1, prob = 0.5, n = 1000)
## Warning in geom_histogram(stat = "count", binwidth = 1, fill = "steelblue", :
## Ignoring unknown parameters: `binwidth`, `bins`, and `pad`

Binomial Distribution

Ideal for data representing the number of successes in a fixed number of independent Bernoulli trials. Use this distribution to estimate probability of success or test hypotheses about proportion differences.

plot_distribution(rbinom, "Binomial Distribution", size = 10, prob = 0.5, n = 1000)
## Warning in geom_histogram(stat = "count", binwidth = 1, fill = "steelblue", :
## Ignoring unknown parameters: `binwidth`, `bins`, and `pad`

Geometric Distribution

This models the number of trials until the first success. It is useful in survival analysis and studies where the time until an event is of interest.

plot_distribution(rgeom, "Geometric Distribution", prob = 0.5, n = 1000)
## Warning in geom_histogram(stat = "count", binwidth = 1, fill = "steelblue", :
## Ignoring unknown parameters: `binwidth`, `bins`, and `pad`

Negative Binomial Distribution

Similar to the geometric, but counts the number of trials until a specified number of successes occurs. It’s especially useful in over-dispersed count data where the variance exceeds the mean.

plot_distribution(rnbinom, "Negative Binomial Distribution", size = 10, prob = 0.5, n = 1000)
## Warning in geom_histogram(stat = "count", binwidth = 1, fill = "steelblue", :
## Ignoring unknown parameters: `binwidth`, `bins`, and `pad`

Hypergeometric Distribution

Use this when sampling without replacement. It’s crucial for understanding population subsets, like quality control or sampling species from a finite population.

plot_distribution(rhyper, "Hypergeometric Distribution", m = 20, n = 20, k = 10, nn = 1000)
## Warning in geom_histogram(stat = "count", binwidth = 1, fill = "steelblue", :
## Ignoring unknown parameters: `binwidth`, `bins`, and `pad`

Poisson Distribution

Appropriate for data representing the number of events in a fixed interval of time or space. This distribution is key in fields like inventory management or queueing theory where events occur independently.

plot_distribution(rpois, "Poisson Distribution", lambda = 3, n = 1000)
## Warning in geom_histogram(stat = "count", binwidth = 1, fill = "steelblue", :
## Ignoring unknown parameters: `binwidth`, `bins`, and `pad`

Continuous Distributions

Continouous distribution can take up any value infinite in range from 0 to 1. An example is temperature. Temperature can be 32, 32.5, 32.534, etc.

Uniform Distribution

Uniformly distributed data has equal probability across its range. When dealing with such data, consider using models that assume uniform effects across a range, such as in simulations or Monte Carlo methods.

plot_density <- function(distribution, title, n, ...) {
  data <- data.frame(x = distribution(n, ...))
  p <- ggplot(data, aes(x=x)) + 
    geom_density(fill="steelblue", alpha=0.5) +
    ggtitle(title) +
    theme_minimal()
  print(p)
}


plot_histogram <- function(distribution, title, n, ...) {
  data <- data.frame(x = distribution(n, ...))
  p <- ggplot(data, aes(x=x)) + 
    geom_histogram(bins = 30, fill="steelblue", color="black") +
    ggtitle(title) +
    theme_minimal()
  print(p)
}

plot_histogram(runif, "Uniform Distribution", 1000, min = 0, max = 1)

Exponential Distribution

This is typically used to model the time between events in a Poisson point process. It is crucial in reliability analysis and survival analysis. Focus on rates and time-to-event data.

plot_density(rexp, "Exponential Distribution", 1000, rate = 1)

Gamma Distribution

Useful in modeling waiting times for multiple Poisson events. It’s applicable in areas like queuing theory and financial services for risk assessment. Analyze shape and scale to understand the behavior of the process.

plot_density(rgamma, "Gamma Distribution", 1000, shape = 2, rate = 1)

Chi-Square Distribution

Often used in hypothesis testing or confidence interval estimation for variance from normally distributed samples. It is also used to assess goodness-of-fit for observed data against theoretical models.

plot_density(rchisq, "Chi-Square Distribution", 1000, df = 2)

Normal Distribution

The most common distribution in statistical analysis, used to model errors, natural variations in measurements, and more. Techniques like z-tests or t-tests are relevant when data follows this distribution.

plot_density(rnorm, "Normal Distribution", 1000, mean = 0, sd = 1)

Beta Distribution

Good for modeling variables that are limited to intervals like 0 to 1, useful in Bayesian statistics, and for proportions and percentages. Analyze the parameters to understand the skewness and central tendency of your data.

plot_density(rbeta, "Beta Distribution", 1000, shape1 = 2, shape2 = 5)

Conclusion

In conclusion, this tutorial has provided a exploration of both discrete and continuous distributions. From the discrete simplicity of a Bernoulli distribution to the continuous subtleties of a Beta distribution, understanding these fundamental concepts is essential for anyone engaged in data science. As we have seen, each distribution has specific applications and considerations that can significantly influence our approach to data analysis and decision-making.