Statistics By Jim

Making statistics intuitive

Probability

Joint Probability: Definition, Formula & Examples

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What is Joint Probability?

Joint probability is the likelihood that two or more events will coincide. Knowing how to calculate them allows you to solve problems such as the following. What is the probability of:

  • Getting two heads in two coin tosses?
  • Consecutively drawing two aces from a deck of cards?
  • The next customer being a woman who buys a Mac computer?
  • A bike rental customer getting both a flat front tire and a flat rear tire?

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Random Variable: Discrete & Continuous

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What is a Random Variable?

A random variable is a variable where chance determines its value. They can take on either discrete or continuous values, and understanding the properties of each type is essential in many statistical applications. Random variables are a key concept in statistics and probability theory. [Read more…] about Random Variable: Discrete & Continuous

Probability Mass Function: Definition, Uses & Example

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What is a Probability Mass Function?

A probability mass function (PMF) is a mathematical function that calculates the probability a discrete random variable will be a specific value. PMFs also describe the probability distribution for the full range of values for a discrete variable. A discrete random variable can take on a finite or countably infinite number of possible values, such as the number of heads in a series of coin flips or the number of customers who visit a store on a given day. [Read more…] about Probability Mass Function: Definition, Uses & Example

Cumulative Distribution Function (CDF): Uses, Graphs & vs PDF

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What is a Cumulative Distribution Function?

A cumulative distribution function (CDF) describes the probabilities of a random variable having values less than or equal to x. It is a cumulative function because it sums the total likelihood up to that point. Its output always ranges between 0 and 1. [Read more…] about Cumulative Distribution Function (CDF): Uses, Graphs & vs PDF

Monte Carlo Simulation: Make Better Decisions

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What is Monte Carlo Simulation?

Monte Carlo simulation uses random sampling to produce simulated outcomes of a process or system. This method uses random sampling to generate simulated input data and enters them into a mathematical model that describes the system. The simulation produces a distribution of outcomes that analysts can use to derive probabilities. [Read more…] about Monte Carlo Simulation: Make Better Decisions

Negative Binomial Distribution: Uses, Calculator & Formula

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What is a Negative Binomial Distribution?

The negative binomial distribution describes the number of trials required to generate an event a particular number of times. When you provide an event probability and the number of successes (r), this distribution calculates the likelihood of observing the Rth success on the Nth attempt. Statisticians also refer to this discrete probability distribution as the Pascal distribution. [Read more…] about Negative Binomial Distribution: Uses, Calculator & Formula

Benford’s Law Explained with Examples

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What is Benford’s Law?

Benford’s law describes the relative frequency distribution for leading digits of numbers in datasets. Leading digits with smaller values occur more frequently than larger values. This law states that approximately 30% of numbers start with a 1 while less than 5% start with a 9. According to this law, leading 1s appear 6.5 times as often as leading 9s! Benford’s law is also known as the First Digit Law. [Read more…] about Benford’s Law Explained with Examples

Hazard Ratio: Interpretation & Definition

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What are Hazard Ratios?

A hazard ratio (HR) is the probability of an event in a treatment group relative to the control group probability over a unit of time. This ratio is an effect size measure for time-to-event data. Use hazard ratios to estimate the treatment effect in clinical trials when you want to assess time-to-event.

For example, HRs can determine whether a medical treatment reduces the duration of symptoms or prolongs survival in cancer patients. [Read more…] about Hazard Ratio: Interpretation & Definition

Relative Risk: Definition, Formula & Interpretation

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What is Relative Risk?

Relative risk is the ratio of the probability of an adverse outcome in an exposure group divided by its likelihood in an unexposed group. This statistic indicates whether exposure corresponds to increases, decreases, or no change in the probability of the adverse outcome. Use relative risk to measure the strength of the association between exposure and the outcome. Analysts also refer to this statistic as the risk ratio. [Read more…] about Relative Risk: Definition, Formula & Interpretation

Probability Density Function: Definition & Uses

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What is a Probability Density Function (PDF)?

A probability density function describes a probability distribution for a random, continuous variable. Use a probability density function to find the chances that the value of a random variable will occur within a range of values that you specify. More specifically, a PDF is a function where its integral for an interval provides the probability of a value occurring in that interval. For example, what are the chances that the next IQ score you measure will fall between 120 and 140? In statistics, PDF stands for probability density function. [Read more…] about Probability Density Function: Definition & Uses

T Distribution: Definition & Uses

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What is the T Distribution?

The t distribution is a continuous probability distribution that is symmetric and bell-shaped like the normal distribution but with a shorter peak and thicker tails. It was designed to factor in the greater uncertainty associated with small sample sizes.

The t distribution describes the variability of the distances between sample means and the population mean when the population standard deviation is unknown and the data approximately follow the normal distribution. This distribution has only one parameter, the degrees of freedom, based on (but not equal to) the sample size. [Read more…] about T Distribution: Definition & Uses

Beta Distribution: Uses, Parameters & Examples

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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

Geometric Distribution: Uses, Calculator & Formula

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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

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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

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What is a Binomial Distribution?

The binomial distribution is a discrete probability distribution that calculates the likelihood an event will occur a specific number of times in a set number of opportunities. Use this distribution when you have a binomial random variable. These variables count how often an event occurs within a fixed number of trials. They have only two possible outcomes that are mutually exclusive. [Read more…] about Binomial Distribution: Uses, Calculator & Formula

Odds Ratio: Formula, Calculating & Interpreting

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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

A Statistical Thanksgiving: Global Income Distributions

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

Painting of the first Thanksgiving by Brownscombe.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