The beta distribution is a continuous probability distribution that models random variables with values falling inside a finite interval. Use it to model subject areas with both an upper and lower bound for possible values. Analysts commonly use it to model the time to complete a task, the distribution of order statistics, and the prior distribution for binomial proportions in Bayesian analysis. [Read more…] about Beta Distribution: Uses, Parameters & Examples

# Probability

## Geometric Distribution: Uses, Calculator & Formula

## What is a Geometric Distribution?

The geometric distribution is a discrete probability distribution that calculates the probability of the first success occurring during a specific trial. In other words, during a series of attempts, what is the probability of success first occurring during each attempt? Use this distribution when you need to understand how many attempts are necessary to produce the first successful outcome. [Read more…] about Geometric Distribution: Uses, Calculator & Formula

## Permutation vs Combination: Differences & Examples

In mathematics and statistics, permutations vs combinations are two different ways to take a set of items or options and create subsets. For example, if you have ten people, how many subsets of three can you make? While permutation and combination seem like synonyms in everyday language, they have distinct definitions mathematically.

**Permutations**: The order of outcomes matters.**Combinations**: The order does not matter.

Let’s understand this difference between permutation vs combination in greater detail. And then you’ll learn how to calculate the total number of each. [Read more…] about Permutation vs Combination: Differences & Examples

## Binomial Distribution: Uses, Calculator & Formula

## What is a Binomial Distribution?

The binomial distribution is a discrete probability distribution that calculates the probability an event will occur a specific number of times in a set number of opportunities. Use the binomial distribution when your outcome is binary. Binary outcomes have only two possible values that are mutually exclusive. [Read more…] about Binomial Distribution: Uses, Calculator & Formula

## Odds Ratio: Formula, Calculating & Interpreting

## What is an Odds Ratio?

An odds ratio (OR) calculates the relationship between a variable and the likelihood of an event occurring. A common interpretation for odds ratios is identifying risk factors by assessing the relationship between exposure to a risk factor and a medical outcome. For example, is there an association between exposure to a chemical and a disease? [Read more…] about Odds Ratio: Formula, Calculating & Interpreting

## Lognormal Distribution: Uses, Parameters & Examples

## What is the Lognormal Distribution?

The lognormal distribution is a continuous probability distribution that models right-skewed data. The shape of the lognormal distribution is comparable to the Weibull and loglogistic distributions. [Read more…] about Lognormal Distribution: Uses, Parameters & Examples

## A Statistical Thanksgiving: Global Income Distributions

In the United States, our Thanksgiving holiday is fast approaching. On this day, we give thanks for the good things in our lives.

For this post, I wanted to quantify how thankful we should be. Ideally, I’d quantify something truly meaningful, like happiness. Unfortunately, most countries are not like Bhutan, which measures the gross national happiness and incorporates it into their five-year development plans.

Instead, I’ll focus on something that is more concrete and regularly measured around the world—income. By examining income distributions, I’ll show that you have much to be thankful for, and so does most of the world! [Read more…] about A Statistical Thanksgiving: Global Income Distributions

## Uniform Distribution: Definition & Examples

## What is a Uniform Distribution?

The uniform distribution is a symmetric probability distribution where all outcomes have an equal likelihood of occurring. All values in the distribution have a constant probability, making them uniformly distributed. This distribution is also known as the rectangular distribution because of its shape in probability distribution plots, as I’ll show you below. [Read more…] about Uniform Distribution: Definition & Examples

## Conditional Probability: Definition, Formula & Examples

## What is Conditional Probability?

A conditional probability is the likelihood of an event occurring given that another event has already happened. Conditional probabilities allow you to evaluate how prior information affects probabilities. For example, what is the probability of A given B has occurred? When you incorporate existing facts into the calculations, it can change the likelihood of an outcome. [Read more…] about Conditional Probability: Definition, Formula & Examples

## Empirical Rule: Definition & Formula

## What is the Empirical Rule?

The empirical rule in statistics, also known as the 68 95 99 rule, states that for normal distributions, 68% of observed data points will lie inside one standard deviation of the mean, 95% will fall within two standard deviations, and 99.7% will occur within three standard deviations. [Read more…] about Empirical Rule: Definition & Formula

## Gamma Distribution: Uses, Parameters & Examples

## What is the Gamma Distribution?

The gamma distribution is a continuous probability distribution that models right-skewed data. Statisticians have used this distribution to model cancer rates, insurance claims, and rainfall. Additionally, the gamma distribution is similar to the exponential distribution, and you can use it to model the same types of phenomena: failure times, wait times, service times, etc. [Read more…] about Gamma Distribution: Uses, Parameters & Examples

## Exponential Distribution: Uses, Parameters & Examples

## What is the Exponential Distribution?

The exponential distribution is a right-skewed continuous probability distribution that models variables in which small values occur more frequently than higher values. Small values have relatively high probabilities, which consistently decline as data values increase. Statisticians use the exponential distribution to model the amount of change in people’s pockets, the length of phone calls, and sales totals for customers. In all these cases, small values are more likely than larger values. [Read more…] about Exponential Distribution: Uses, Parameters & Examples

## Weibull Distribution: Uses, Parameters & Examples

## What is a Weibull Distribution?

The Weibull distribution is a continuous probability distribution that can fit an extensive range of distribution shapes. Like the normal distribution, the Weibull distribution describes the probabilities associated with continuous data. However, unlike the normal distribution, it can also model skewed data. In fact, its extreme flexibility allows it to model both left- and right-skewed data. [Read more…] about Weibull Distribution: Uses, Parameters & Examples

## Poisson Distribution: Definition & Uses

## What is the Poisson Distribution?

The Poisson distribution is a discrete probability distribution that describes probabilities for counts of events that occur in a specified observation space. It is named after Siméon Denis Poisson.

In statistics, count data represent the number of events or characteristics over a given length of time, area, volume, etc. For example, you can count the number of cigarettes smoked per day, meteors seen per hour, the number of defects in a batch, and the occurrence of a particular crime by county. [Read more…] about Poisson Distribution: Definition & Uses

## Using Combinations to Calculate Probabilities

Combinations in probability theory and other areas of mathematics refer to a sequence of outcomes where the order does not matter. For example, when you’re ordering a pizza, it doesn’t matter whether you order it with ham, mushrooms, and olives or olives, mushrooms, and ham. You’re getting the same pizza! [Read more…] about Using Combinations to Calculate Probabilities

## Using Permutations to Calculate Probabilities

Permutations in probability theory and other branches of mathematics refer to sequences of outcomes where the order matters. For example, 9-6-8-4 is a permutation of a four-digit PIN because the order of numbers is crucial. When calculating probabilities, it’s frequently necessary to calculate the number of possible permutations to determine an event’s probability.

In this post, I explain permutations and show how to calculate the number of permutations both with repetition and without repetition. Finally, we’ll work through a step-by-step example problem that uses permutations to calculate a probability. [Read more…] about Using Permutations to Calculate Probabilities

## Multiplication Rule for Calculating Probabilities

The multiplication rule in probability allows you to calculate the probability of multiple events occurring together using known probabilities of those events individually. There are two forms of this rule, the specific and general multiplication rules.

In this post, learn about when and how to use both the specific and general multiplication rules. Additionally, I’ll use and explain the standard notation for probabilities throughout, helping you learn how to interpret it. We’ll work through several example problems so you can see them in action. There’s even a bonus problem at the end! [Read more…] about Multiplication Rule for Calculating Probabilities

## Using Contingency Tables to Calculate Probabilities

Contingency tables are a great way to classify outcomes and calculate different types of probabilities. These tables contain rows and columns that display bivariate frequencies of categorical data. Analysts also refer to contingency tables as crosstabulation (cross tabs), two-way tables, and frequency tables.

Statisticians use contingency tables for a variety of reasons. I love these tables because they both organize your data and allow you to answer a diverse set of questions. In this post, I focus on using them to calculate different types of probabilities. These probabilities include joint, marginal, and conditional probabilities. [Read more…] about Using Contingency Tables to Calculate Probabilities

## Probability Definition and Fundamentals

## What is Probability?

The definition of probability is the likelihood of an event happening. Probability theory analyzes the chances of events occurring. You can think of probabilities as being the following:

- The long-term proportion of times an event occurs during a random process.
- The propensity for a particular outcome to occur.

Common terms for describing probabilities include likelihood, chances, and odds. [Read more…] about Probability Definition and Fundamentals