A probability distribution is a function that describes the likelihood of obtaining the possible values that a random variable can assume. In other words, the values of the variable vary based on the underlying probability distribution.

Suppose you draw a random sample and measure the heights of the subjects. As you measure heights, you can create a distribution of heights. This type of distribution is useful when you need to know which outcomes are most likely, the spread of potential values, and the likelihood of different results.

In this blog post, you’ll learn about probability distributions for both discrete and continuous variables. I’ll show you how they work and examples of how to use them.

## General Properties of Probability Distributions

Probability distributions indicate the likelihood of an event or outcome. Statisticians use the following notation to describe probabilities:

p(x) = the likelihood that random variable takes a specific value of x.

The sum of all probabilities for all possible values must equal 1. Furthermore, the probability for a particular value or range of values must be between 0 and 1.

Probability distributions describe the dispersion of the values of a random variable. Consequently, the kind of variable determines the type of probability distribution. For a single random variable, statisticians divide distributions into the following two types:

- Discrete probability distributions for discrete variables
- Probability density functions for continuous variables

You can use equations and tables of variable values and probabilities to represent a probability distribution. However, I prefer graphing them using probability distribution plots. As you’ll see in the examples that follow, the differences between discrete and continuous probability distributions are immediately apparent. You’ll see why I love these graphs!

**Related post**: Data Types and How to Use Them

## Discrete Probability Distributions

Discrete probability functions are also known as probability mass functions and can assume a discrete number of values. For example, coin tosses and counts of events are discrete functions. These are discrete distributions because there are no in-between values. For example, you can have only heads or tails in a coin toss. Similarly, if you’re counting the number of books that a library checks out per hour, you can count 21 or 22 books, but nothing in between.

For discrete probability distribution functions, each possible value has a non-zero likelihood. Furthermore, the probabilities for all possible values must sum to one. Because the total probability is 1, one of the values must occur for each opportunity.

For example, the likelihood of rolling a specific number on a die is 1/6. The total probability for all six values equals one. When you roll a die, you inevitably obtain one of the possible values.

If the discrete distribution has a finite number of values, you can display all the values with their corresponding probabilities in a table. For example, according to a study, the likelihood for the number of cars in a California household is the following:

## Types of Discrete Distribution

There are a variety of discrete probability distributions that you can use to model different types of data. The correct discrete distribution depends on the properties of your data. For example, use the:

- Binomial distribution to model binary data, such as coin tosses.
- Poisson distribution to model count data, such as the count of library book checkouts per hour.
- Uniform distribution to model multiple events with the same probability, such as rolling a die.

To learn more in depth about several probability distributions that you can use with binary data, read my post Maximize the Value of Your Binary Data.

To learn how to determine whether a specific discrete distribution is appropriate for your data, read my post Goodness-of-Fit Tests for Discrete Distributions.

## Example of How to Use Discrete Probability Distributions

All of the examples I include in this post will show you why I love to graph probability distributions. The case below comes from my blog post that presents a statistical analysis of flu shot effectiveness. I use the binomial distribution to answer the question—how many times can I expect to catch the flu over 20 years with and without annual vaccinations?

This example uses binary data because the two possible outcomes are either being infected by the flu or not being infected by the flu. Based on various studies, the long-term probability of a flu infection is 0.07 annually for the unvaccinated and 0.019 for the vaccinated. The graph plugs these probabilities into the binomial distribution to display the pattern of outcomes for both scenarios over twenty years. Each bar indicates the likelihood of catching the flu the specified number of times. Additionally, I’ve shaded the bars red to represent the cumulative probability of at least two flu infections in 20 years. The left panel displays the expected outcomes with no vaccinations while the right panel shows the outcomes with annual vaccinations.

A significant difference jumps out at you—which demonstrates the power of probability distribution plots! The largest bar on the graph is the one in the right panel that represents zero cases of the flu in 20 years when you get flu shots. When you vaccinate annually, you have a 68% chance of not catching the flu within 20 years! Conversely, if you don’t vaccinate, you have only a 23% of escaping the flu entirely.

In the left panel, the distribution spreads out much further than in the right panel. Without vaccinations, you have a 41% chance of getting the flu at least twice in 20 years compared to 5% with annual vaccinations. Some unlucky unvaccinated folks will get the flu four or five times in that time span!

## Continuous Probability Distributions

Continuous probability functions are also known as probability density functions. You know that you have a continuous distribution if the variable can assume an infinite number of values between any two values. Continuous variables are often measurements on a scale, such as height, weight, and temperature.

Unlike discrete probability distributions where each particular value has a non-zero likelihood, specific values in continuous distributions have a zero probability. For example, the likelihood of measuring a temperature that is exactly 32 degrees is zero.

Why? Consider that the temperature can be an infinite number of other temperatures that are infinitesimally higher or lower than 32. Statisticians say that an individual value has an infinitesimally small probability that is equivalent to zero.

## How to Find Probabilities for Continuous Data

Probabilities for continuous distributions are measured over ranges of values rather than single points. A probability indicates the likelihood that a value will fall within an interval. This property is straightforward to demonstrate using a probability distribution plot—which we’ll get to soon!

On a probability plot, the entire area under the distribution curve equals 1. This fact is equivalent to how the sum of all probabilities must equal one for discrete distributions. The proportion of the area under the curve that falls within a range of values along the X-axis represents the likelihood that a value will fall within that range. Finally, you can’t have an area under the curve with only a single value, which explains why the probability equals zero for an individual value.

## Characteristics of Continuous Probability Distributions

Just as there are different types of discrete distributions for different kinds of discrete data, there are different distributions for continuous data. Each probability distribution has parameters that define its shape. Most distributions have between 1-3 parameters. Specifying these parameters establishes the shape of the distribution and all of its probabilities entirely. These parameters represent essential properties of the distribution, such as the central tendency and the variability.

**Related posts**: Understanding Measures of Central Tendency and Understanding Measures of Variability

The most well-known continuous distribution is the normal distribution, which is also known as the Gaussian distribution or the “bell curve.” This symmetric distribution fits a wide variety of phenomena, such as human height and IQ scores. It has two parameters—the mean and the standard deviation. The Weibull distribution and the lognormal distribution are other common continuous distributions. Both of these distributions can fit skewed data.

Distribution parameters are values that apply to entire populations. Unfortunately, population parameters are generally unknown because it’s usually impossible to measure an entire population. However, you can use random samples to calculate estimates of these parameters.

To learn how to determine which distribution provides the best fit to your sample data, read my post about How to Identify the Distribution of Your Data.

## Example of Using the Normal Probability Distribution

Let’s start off with the normal distribution to show how to use continuous probability distributions.

The distribution of IQ scores is defined as a normal distribution with a mean of 100 and a standard deviation of 15. We’ll create the probability plot of this distribution. Additionally, let’s determine the likelihood that an IQ score will be between 120-140.

Examine the properties of the probability plot above. We can see that it is a symmetric distribution where values occur most frequently around 100, which is the mean. The probabilities drops-off as you move away from the mean in both directions. The shaded area for the range of IQ scores between 120-140 contains 8.738% of the total area under the curve. Therefore, the likelihood that an IQ score falls within this range is 0.08738.

**Related Post**: Using the Normal Distribution

## Example of Using the Lognormal Probability Distribution

As I mentioned, I really like probability distribution plots because they make distribution properties crystal clear. In the example above, we used the normal distribution. Because that distribution is so well-known, you might have guessed the general appearance of the chart. Now, let’s look at a less intuitive example.

Suppose you are told that the body fat percentages for teenage girls follow a lognormal distribution with a location of 3.32317 and a scale of 0.24188. Furthermore, you’re asked to determine the probability that body fat percentage values will fall between 20-24%. Huh? It’s probably not clear what the shape of this distribution is, which values are most common, and how often values fall within that range!

Most statistical software allow you to plot probability distributions and answer all of these questions at once.

The graph displays both the shape of the distribution and how our range of interest fits within it. We can see that it is a right-skewed distribution and the most common values fall near 26%. Furthermore, our range of interest falls below the curve’s peak and contains 18.64% of the occurrences.

As you can see, these graphs are an effective way to report complex distribution information to a lay audience.

This distribution provides the best fit for data that I collected for a study. Learn how I identified the distribution of these data.

## Hypothesis Testing Uses Special Probability Distributions

Statistical hypothesis testing uses particular types of probability distributions to determine whether the results are statistically significant. Specifically, they use sampling distributions and the distributions of test statistics.

### Sampling distributions

A vital concept in inferential statistics is that the particular random sample that you draw for a study is just one of a large number of possible samples that you could have pulled from your population of interest. Understanding this broader context of all possible samples and how your study’s sample fits within it provides valuable information.

Suppose we draw a substantial number of random samples of the same size from the same population and calculate the sample mean for each sample. During this process, we’d observe a broad spectrum of sample means, and we can graph their distribution.

This type of distribution is called a sampling distribution. Sampling distributions allow you to determine the likelihood of obtaining different sample values, which makes them crucial for performing hypothesis tests.

The graph below displays the sampling distribution for energy costs. It shows which sample means are more and less likely to occur when the population mean is 260. It also displays the specific sample mean that a study obtains (330.6). The graph indicates that our observed sample mean isn’t the most likely value, but it’s not wholly implausible either. Hypothesis tests use this type of information to determine whether the results are statistically significant.

To learn more about sampling distributions, read my post about How Hypothesis Tests Work.

### Distributions for test statistics

Each type of hypothesis test uses a test statistic. For example, t-tests use t-values, ANOVA uses F-values, and Chi-square tests use chi-square values. Hypothesis tests use the probability distributions of these test statistics to calculate p-values. That’s right, p-values come from these distributions!

For instance, a t-test takes all of the sample data and boils it down to a single t-value, and then the t-distribution calculates the p-value. The probability distribution plot below represents a two-tailed t-test that produces a t-value of 2. The plot of the t-distribution indicates that each of the two shaded regions that corresponds to t-values of +2 and -2 (that’s the two-tailed aspect of the test) has a likelihood of 0.02963—for a total of 0.05926. That’s the p-value for this test!

To learn more about how this works for different hypothesis tests, read my posts about:

- How t-Tests Work
- How the F-test Works in One-Way ANOVA
- Degrees of Freedom (There’s a section about probability distributions.)

I hope you can see how crucial probability distributions are in statistics and why I think graphing them is a powerful way to convey results!

If you’re learning about statistics and like the approach I use in my blog, check out my Introduction to Statistics eBook!

Matt Augeri says

Hi Jim. I’m learning a lot from your site! Here’s a question for you. I work with the homeless population in New Hampshire. Many of these folks have 50 or 60 visits to the ED per year. We’re trying to help divert them to more appropriate services. I know that John Doe has had 50 visits and that the longest time between visits is 20 days, If John hasn’t had an ED visit in the last 15 days, what is the probability he’ll show up in the ED in the next 5? Our goal is to reach out to John before he goes back to the ED. Thanks for considering my question!

diego says

Hey Jim, I’m new to statistics and I’m really trying to understand deeply the concepts. Can you recommend some material that explains how these distributions were created?

The intuition and thoughts behind, the proofs, etc?

Jim Frost says

Hi Diego! Scroll down these comments and you’ll see a recent one where I answer this same question and provide a reference. There’s an interesting history! Click here to go to the comment I mention.

shepard ngwaru says

hie Jim can you assist me on this one

By investing in a particular stock for one year, an investor hopes to make a profit of $1000, $4000, $5000, or $10000 with probabilities 0.4, 0.3, 0.18, or 0.12 respectively. Calculate the mean and standard deviation of expected profits.

Jim Frost says

Hi Shepard,

It sounds like you just multiply each profit by its probability to get the expected profits. Then take the average and standard deviation of those values.

Neda says

Hello Jim,

Thank you for the comprehensive explanation. I have a question about the determination of the distribution type. I have 50 data per minute. The data are continuous. I expect that the data should be closed to each other. It means the distribution of data around the mean should be higher than far from it. But for different minutes, the distributions are not symmetrical. Is there any way to determine the type of distribution?

thank you in advance for the answer.

regards,

Neda

Jim Frost says

Hi Neda,

Typically, these distributions assume you are working with independent, random observations. It sounds like your data are not independent and that you can’t use these distributions. Perhaps you’d need to create some type of time series model that explains your observations over time?

Mitch says

Hey Jim, I’ve always been curious about how the probablity distributions were constructed originally/historically, such as the normal cuve, t distribution, or X2 distribuion. Were they originally constructed theoretically with mathamatics or instead with actual (sampled) data. Any insights or a good reference for this?

Posts are nice- wish your books came out in hard copy (guess I’m old-fashion) Mitch

Jim Frost says

Hi Mitch,

I’m glad you enjoy my posts. I hope to have hard copy books down the road. Unfortunately, I need to redo the output so they’ll look good in print. Perhaps a project for 2021. Currently, I’m cramming to finish my third ebook, which is about hypothesis testing.

There is a long and interesting history behind the normal distribution. We associate it with Carl Gauss (hence, it’s also known as the Gaussian distribution), but it’s ideas go further back.

Interestingly, the normal distribution was introduced by the French mathematician Abraham de Moivre in a 1738 article book “The doctrine of chance” but at that point it was in relation to how the binomial distribution increased to a smooth curve how as the number of events of increased. Additionally he believed large errors are rarer than small errors. And that errors are evenly distributed around the arithmetic mean at the peak. This describes the familiar bell shaped curve. Moivre needed actuarial methods for calculating life insurance.

Others work on this distribution. And Carl Gauss wrote about it in 1809 and used it to analyze errors in astronomical data. From there it’s use expanded to other areas in applied probability.

Gosset, a brewer and a statistician, created the t-distribution in the early 1900s because he needed a probability distribution that worked with the small samples that he worked with at the brewery whereas the normal distribution works with larger samples.

By and large, it sounds like these distributions were developed with practical applications in mind. Often starting with one application but then expanding to others. Unfortunately, I don’t know much about the historical development of the chi-squared distribution.

I ran across a good article about the normal distribution. A Brief Historical Overview Of the Gaussian Curve: From Abraham De Moivre to Johann Carl Friedrich Gauss, International Journal of Engineering Science Invention (IJESI)

ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726, http://www.ijesi.org, Volume 7 Issue 6 Ver V, June 2018, PP 28-34.

Thanga says

Thanks.it is very good explanation on probability distribution. Can you please say about rectangular and triangular distribution related with uncertainties.

Michael Thomas says

How do I find x using a binomial graph

Ash says

Hi Jim. Its an excellent blog. Love the simplicity with which you explain. However, recently I was trying to re-read your blogs and found that plots & graphs are not loading propoerly.

Jim Frost says

Hi Ash,

Hmmm. Someone else mentioned a graph that didn’t load. However, they seem to load for most people. I will look into it. I’m not sure what is happening. Do they load when you click Refresh?

dbadrysys says

Hi Mr. Jim,

I appreciate your post. It make everything is clear for beginner like me. I hope that you have a post related to the Cumulative Distribution Function topic.

Thanks.

Qiang Heng says

Let me find and go through that post. Thanks

QIANG HENG says

Jim

love your example about “flu shot“, would you please share your raw data?

Jim Frost says

Hi Heng,

The flu shot graph is based on the binomial distribution. It’s not directly based on data itself. I simply entered the two probabilities of 0.07 and 0.019 and the binomial distribution calculates the probability for each number of occurrences.

However, I did obtain those two probabilities from a number of flu shot studies. I average the results from a number of studies to obtain an average probability of getting the flu when you’re vaccinated (0.019) and unvaccinated (0.7). You can read about that process in my post about the effectiveness of flu shots. In that post, I look at the studies, their results, etc. In fact, that particular graph first appeared in that other post, along with several other similar graphs. I describe them a bit more fully in the flu shot post. I think that post will answer your questions and I show you the numbers and reference the studies.

The Wizard Of Blog says

Dear Dr. Frost:

Every time I need an intuitive description of a rather abstract statistical concept, I come to your blog. I have been visiting your blog for quite a while now. I have even shared the link to your blog among my colleagues who I think need to strengthen their grasp of the fundamrntal of statistics. That is how much I trust your blog.

Love the way you explain.

Thanks very much and best wishes.

Jim Frost says

Hello Wizard!

Thanks so much for your kind words and your trust! They absolutely make my day!

Bil says

Hi, i m interesting with the determination of histogram with frequencies and in the same plot, the fitting with normal law. I want to do this with my experimental data. I m not good in statistic. Thanks

Moses Owoicho Audu says

Sir, please which software can one use to plot the PDF?

Jim Frost says

Hi Moses,

I use Minitab in my post. However, I’m sure applications can do that.

Sergio Nguyen (@Sergio35103374) says

Before reading your blog, I hardly understand the concept of probability distribution. But, after reading your blog, I understand the concept deeply due to your excellent explaining. Anyway, thanks so much, Jim.

Jim Frost says

Aw, thanks Sergio! That means a lot to me! I’m happy that this post was so helpful!

Bill says

I am a bit confused about your flu example. Can you please explain how you came up with the individual probabilities. For example: There is a 23% of not getting t he flu if you don’t get a vaccine.

Jim Frost says

Hi Bill,

The first step was calculating an average annual infection rate for the unvaccinated (7.0%) and vaccinated (1.9%). These values come from a number of published studies. For more information, see my post about the effectiveness of flu vaccinations.

After that, the next step is to use these probabilities in probability distributions that are designed for binary data. The properties of these distribution allow you to find the probabilities for different outcomes. There are formulas you can look up in textbooks if you’re really interested. But, my focus here is to teach when to use each distribution and then use statistical software to calculate the answers for you. For more information, read my post about distributions for binary data.

For this specific example, you need to calculate the probability of non-infection annually (1 – 0.07 = 0.93). Then you raise it to the power of 20 for twenty years.

0.93^20 = 0.23

In the graphs, the value of 23% comes from the bar that represents zero infections in the left-hand plot for unvaccinated people. When you have a 7% chance of infection annually, your chances of zero infections over 20 years is 23%.

I hope this helps!

Frank says

Well espose, because of ur publication i bougth ur ebook, hope u can make a book about this publication, and make a publication about Gaussian Process

Jim Frost says

Thank you, Frank! I really appreciate you buying my ebook and I hope you find it to be helpful. I have a blog post about the Normal (Gaussian) Distribution that you might find helpful. In the near future, I will be writing an introductory book to statistics that talks about things such as the normal distribution.

Uendel Rocha says

Oi, Jim! Seu blog é claro, simples e agradável de ler. Parece que estamos conversando contigo, ouvindo suas explicações. Sua maneira de ensinar torna a compreensão de estatística mais fácil do que estou acostumado. Gostei muito do seu artigo. O livro que você publicou segue essa mesma linha? Seria interessante uma pequena amostra, não? Com faço para conseguir um código promocional?

Muito obrigado.

Jim Frost says

Oi UendelMuito obrigado! Sim, eu uso o mesmo estilo de escrita no meu livro que uso em meus posts. Se você gosta do meu estilo de escrita no meu blog, você vai adorar o livro. Atualmente, não tenho uma amostra disponível, mas você pode considerar as postagens do blog como uma boa representação.

Robert Pieczykolan says

hi Jim,

when talking about continuous distribition , yuo should also mention uniform. Used to model say duration of a waiting time when calling Home Revenue say from 5 min to 5 hours. It is quite commonly used distribution.

https://en.wikipedia.org/wiki/Uniform_distribution_(continuous)

Robert Pieczykolan

Freelance statistician/data analyst

P.S. You do explain probability really nicely.

David N'Dri Kan says

Thank you for sharing these informations. How can we stay tunned to your blog?

Jim Frost says

Hi David,

The easiest way is to subscribe by email. You’ll find that in the right column. You’ll receive an email every time I publish a new blog post. I don’t do anything else with those email addresses and never give them to anyone else.

Shan Murali says

Hi, Jim,

You are doing a divinely job…. helping other to learn and understood…. in simple and powerful means…I appreciate your help…. even though I have been teaching for the past 18 years….your work is exemplary… my prayers to your wellbeing….

Jim Frost says

Hi Shan,

Thank you so much! Your kind words and thoughts mean so much to me.

Best wishes to you and your loved ones.

Simona Va says

This is the best article written about probability distributions. It’s hard for beginners to understand these concepts and you wrote it so clearly. Thank you for sharing this!

Jim Frost says

Thank you so much, Simona! I work hard to make these concepts as easy as possible to understand. Your kind words mean a lot to me!

Josh says

Jim, I really like your approach to teaching stats. I’ve frustratingly studied it for years and have never felt satisfied in my understanding or application. Hoping your blog and upcoming book can help get me there.

On a separate note, what’s your take on simulation-based statistics (i.e., bootstrap)? Does it make more sense to learn resampling or the traditional analytical approach? Would appreciate your take.

Jim Frost says

Hi Josh,

Thanks so much. I really appreciate your kind words. I really strive to help people to understand. I don’t think it has to be so difficult to learn if educators would just use more intuitive explanations. So, it means a lot me that you’ve found my site to be helpful.

So, my background is in the more traditional approach, but I really like the concepts behind resampling (bootstrapping). It kind of gets to what I was saying above. I have a blog post about confidence intervals, but they’re really hard to explain how they work. However, I think bootstrap confidence intervals are much more intuitive. They’re definitely easier to explain what is happening. But, truth be told, I’ve never actually used them in an analysis but have stuck with the traditional CIs. But, I think they’re great tools and I can well imagine that teaching them would actually be more enlightening about the process behind inferential statistics where the sample you obtain is actually only one of an infinite number of possible samples that you could have obtained. Resampling really runs with that idea. I know there are some statisticians who think that resampling is the way of future! I’m thinking I need to write a blog post about this method!

Udbhav says

Great Post!

May I know how did you get the discrete probability plot? I tried in Matlab but got quite different plot.

My code –

x = 1:20

y = arrayfun(@(a) binopdf(a, 20, 0.019), x)

bar(x,y)

Jim Frost says

Hi Udbhav,

Unfortunately, I’m not that familiar with MATLAB, so I can’t help you there. I use Minitab for my graphs.

Dileep Kumar M says

Great post…thank you sir.

Ananth says

Good

Bimal Thapa says

Hi Jim,

Thank you for your post. Have you written any book?

Jim Frost says

Hi Bimal,

You’re very welcome! Also, I’m currently writing my first book. Stay tuned!

PRIYANSHU KUMAR says

I Love Statistics, I want To learn and Teach Statistics Like You, This Blog is Very Good, Understanding Probability Distribution By Graphs Make a Crystal Clear Image of given Data.

THANK YOU For such an Intersting Blog.

Jim Frost says

Hi Priyanshu, thanks so much for your kind words. You made my day! And, it’s great that you love statistics! 🙂

Sami econ says

Mr. Jim

I appreciate you for your such type of contribution. You deserve it.

Jim Frost says

Thanks, Sami! I appreciate that!