Interactive and Relevant Applications of Statistics

Hawkes statistics courses include games and simulations that help students apply key concepts to the world outside of the classroom. Check these out below! If you’re an instructor who would like to explore these games and simulations yourself, sign up for free student access today.

1. Games of Chance


Relevant Application:
This lesson helps students apply the concept of the expected value of a random variable to winning or losing games. Students develop a rational approach to analyzing decisions that involve risk. After all, many business decisions—such as purchasing new equipment, hiring additional employees, and expanding into new markets—involve some kind of risk, and students need to assess these situations as best as they can.

Learn Key Concepts:

  • Basic probability distribution
  • Binomial distribution function
  • Hypergeometric distribution function

2. Direct Mail


Relevant Application:
Even in today’s digital world, direct mail marketing remains one of the most viable and proven strategies to connect with customers.

Active Learning Approach:
By assuming the role of a direct mail marketing manager, students start off with $20,000. They are then tasked with developing a strategy by finding mailing lists that will produce sufficient sales, using confidence intervals to determine which lists to use to reach their $40,000 goal.

They win when they correctly formulate which questions they need to solve, collect the data, and analyze the data to evaluate potential risk and profitability for each mailing list.

Learn Key Concepts:
The game provides an environment in which students apply statistical concepts while making business decisions. They also learn the following:

  • Confidence intervals
  • Experimentation
  • Statistical analysis
  • Inference

3. Estimating Population Proportions


Relevant Application:
Students might not realize at first how many decisions involve measurements of a population attribute. For example, television stations base advertising charges on
ratings that reflect the percentage of viewers who watch a particular show. Political analysts are concerned with the fraction of voters who prefer a certain candidate. No
matter the field, estimating population proportions gives us greater insight into the data given to us.

Active Learning Approach:
In the game, students see a box filled with red and blue balls, and are asked to estimate the proportion of red balls in the population. They can draw sample sizes of 20, 50, or
100 to help them estimate the population proportion.

Learn Key Concepts:

  • Determine the minimum sample size for a particular confidence level.
  • Construct a confidence interval for a population proportion.

4. Central Limit Theorem with Proportions

Relevant Application:
In many decisions, the variable of interest is a proportion. A university may want to know the fraction of first-year students with low grades in order to provide more support and resources for them. Manufacturers may be concerned with the fraction of parts that are defective.

Active Learning Approach:
Students see a box of red and blue balls, then draw three samples to calculate the sample proportions for each sample taken. Students draw samples again after being informed that samples of first 20 balls and then 40 balls were drawn 200 times to determine the proportion of the number of red balls to the total number of balls chosen. Students then view the data, including tables and histograms, to understand that the sampling distribution of the sampling proportion is approximately normal.

Learn Key Concepts:
Determine p-hat using the Central Limit Theorem for population proportions.


1. Name That Distribution


Relevant Application:
This concept builder strengthens analytical skills in distribution recognition and data analysis. By detecting symmetric or skewed data, students will begin to understand how to apply this knowledge in the real world.

Active Learning Approach:
Students are asked to identify the type of distribution from a given histogram, frequency/relative frequency distribution, statistics table, or set of sample data. They
can increase the number of intervals on the histogram or frequency distribution, view different sample displays, or choose to view a hint before submitting their answer.

Learn Key Concepts:

  • Analyze the histogram, frequency, statistics, and sample data of a distribution.
  • Identify different distribution types: uniform, normal, exponential, chi-square, Poisson, and mystery.

2. Central Limit Theorem

The simulation can run automatically and in bursts. This image shows histograms for n=5, n=15, and n=30. It includes the histogram of the parent function.

Relevant Application:
This simulation shows students how to use samples to make useful predictions about a population. Since many population sizes are too large to have their data collected and analyzed, we turn to the Central Limit Theorem for help.

The visual nature of this simulation lets students truly comprehend how the sample means from any population are normally distributed, regardless of the original
population’s distribution.

Active Learning Approach:
Students select a parent distribution and set the sample sizes and the burst rate. They choose the desired distribution type: exponential, chi-square, normal, Poisson, or bi-modal. Students can decide to run the simulation a set number of times or automatically, which will keep the simulation running.

Learn Key Concepts:

  • Sample population
  • Mean
  • Variance
  • Standard deviation
  • Distribution type

3. Type II Error

You can select the plus or minus buttons for alpha, true mean, and sample size to change the graph. The shaded part increases or decreases depending on the number, and the bell curve moves forward or backward when you change the true mean.

Relevant Application:
Understanding hypothesis testing and type II error is essential to fields like evidence-based medicine, quality engineering, and reliability engineering, among others.

Active Learning Approach:
The variance, hypotheses, and critical values are given. Students can increase or decrease the level of significance (α), true mean (μ), and sample size to see how these
changes affect the other factors involved.

Learn Key Concepts:

  • Examine the interrelationship between α, sample size, and β (the probability of making a type II error).
  • Develop an understanding of the concept of type II errors and the calculation of beta.
  • Explore the relationship between α and β.


Are you an instructor who would like to explore these lessons further?

Sign up for FREE student access today!

Instructor advice on motivating students

Having trouble motivating your students to stay active and engaged in class? We understand that some days, it can be a struggle. Current and former instructors here at Hawkes Learning have provided advice on how to keep students motivated. Check it out below, then let us know what advice YOU have!


  • Consider announcing a 3-point bonus question before your first test, and make it a scavenger hunt. Ask for three things (one point each): 1. What is written on your office door? (This encourages students to find your office.) 2. What is one name of a tutor in the tutoring lab? (This encourages them to find the tutoring lab.) 3. What are the hours for the tutoring lab? (This knowledge helps them if they need to schedule an appointment.)
  • Take attendance. Even if attendance isn’t part of the grade, it shows students that you’re aware whether or not they come to class and participate.
  • Get students to speak. A few will always take the lead and constantly ask questions, while some will never open their mouths. Directly ask those students a question. Hearing their voice and knowing it’s being heard has a positive effect and can lead them to speak up without being prompted later on.


  • Post discussions and message boards. Since you can’t talk face-to-face, the next best thing is to utilize these communications threads.
  • Remind students that they never stop learning because technology changes so often. Use the online environment to your advantage by showing students new communications tools and apps that they can adapt to and learn from.
  • Hold virtual office hours for students who have questions or need a little extra help.


  • Have a large class? Consider the “shared birthday” problem. A class of 30 students has over a 70% chance of having at least one shared birthday among them. A class of 40 students has almost 90%. If you happen to have one or more shared birthdays in the class, they never forget it and it gets them interested from the start.
  • Collect noninvasive data from your class to use throughout the semester. Asking at the beginning of the term for information like students’ majors, favorite sport, and number of siblings gives you data to incorporate in your lessons that will keep students interested.
  • Math courses have historically had a stigma for math anxiety for some students. Be reassuring and encouraging to your students, and provide opportunities for success that will help supply confidence and a positive momentum through the course.


  • Give students options! Anytime students can decide on an element of their learning, they get more invested in the outcome. Let them choose a project partner, reading selection, or project option.
  • Allow students to revise and resubmit assignments based on your feedback to improve their grades and strengthen their learning.
  • Put students in the role of instructor. Assign them a reading passage that they are responsible for teaching to part or all of the class. Teaching is the best way to learn a new concept!

Have more tips? We’d love to hear them! Comment below with your tried and true tips on keeping students motivated and engaged.

3 Ways to Strengthen Your Students’ Critical Thinking Skills

Psychologist Benjamin Bloom published his now widely spread document in education, “Taxonomy of Educational Objectives,” in 1956. In it, he and his team specify three domains of learning: affective, psychomotor, and cognitive. While the affective domain refers to the emotions, motivations, and attitudes of students, the psychomotor domain focuses on their motor skills.

The cognitive domain—arguably the most influential in a student’s success—covers six categories (according to the revised version of Bloom’s Taxonomy by Anderson & Krathwohl, et al (2001)):

  • Remembering
  • Understanding
  • Applying
  • Analyzing
  • Evaluating
  • Creating

These categories start with memorizing and defining what’s learned in class, build toward drawing connections among different ideas and applying them outside of class, then lead to creating your own work by using what you’ve learned (Armstrong). Building upon these processes develops students’ critical thinking and reasoning skills, which are more important today than ever before. (Pssst! Check out key definitions and verbs to describe each category here from Vanderbilt University’s Center for Teaching.)

So, how can you help students strengthen their critical thinking and reasoning? Below are three ways to incorporate these skills into any curriculum.

1. Allow time within class to brainstorm after asking an open-ended question.

Students need time on their own to think about how to solve a problem, as well as time to talk out their strategies with other students. Problem solving is a key component to critical thinking, and brainstorming gives students the opportunity to explore different perspectives and possible solutions in a low-pressure environment. According to Lee Crockett Watanabe from Global Digital Citizen Foundation, asking a question that can’t simply be answered with a yes or no encourages students to seek out the necessary knowledge on their own (“12 Strong Strategies for Effectively Teaching Critical Thinking Skills.”). Students must use the skills associated with the cognitive domain, such as recalling what they already know about the problem, analyzing different strategies to solve it, and evaluating the quality of each solution.

2. Compare and contrast different ideas. 

Once students learn and understand different approaches to solving a problem, they can evaluate the qualities of each approach. Which one is easier? Which is the most thorough? Which makes the most sense to use in this context? Students need to judge the strengths and weaknesses of varying solutions in order to decide their next steps in solving the problem. Creating a pro/con chart can help, as well as a pro/pro chart, according to instructor Jason Watt. In a pro/pro chart, students see the positives of different perspectives by listing out only the good traits of each, bringing a fresh take to an old decision-making strategy. Watt explains that a pro/pro chart can help students try to find the positives in what they originally thought of as a weakness, allowing them to get creative with their thinking and less intimated to do so (Schwartz).

3. Get them thinking about thinking.

In the revised version of Bloom’s Taxonomy, metacognitive knowledge includes strategy, self-knowledge, and contextual and conditional knowledge (Armstrong). To increase their critical thinking skills, students need to think about how they think. If they pause to reflect upon how they’re studying and learning the class content, they may just improve their grades. Dr. Patricia Chen, a postdoctoral researcher at Stanford, conducted a study in which she asked a group of her students several prompts asking them to think about how they’re studying for an upcoming test and how they could improve their studying. She only reminded a second student group that the test was coming up. The first group outperformed the students who did not reflect on their studying by 1/3 of a letter grade on average (Anderson). Check out more information on the study.

When students analyze their own thinking techniques and visualize how they want to perform on assessments, they develop critical strategies to set goals and determine which resources work best for their unique learning processes. These skills can help students improve their grades, and they’ll transfer over when students are learning in other classes, navigating the workplace, and facing the challenges of daily life.


Have other ways to help improve students’ cognitive domains and critical thinking skills? Please share them in the comments below!


Anderson, Jenny. “A Stanford researcher’s 15-minute study hack lifts B+ students into the As.” Quartz, 9 May 2017, Accessed 5 June 2017.

Armstrong, Patricia. “Bloom’s Taxonomy.” Vanderbilt University Center for Teaching, Accessed 12 June 2017.

Schwartz, Katrina. “Three Tools for Teaching Critical Thinking and Problem Solving Skills.” KQED News, 6 Nov. 2016, Accessed 13 June 2017.

Watanabe Crockett, Lee. “12 Strong Strategies for Effectively Teaching Critical Thinking Skills.” Global Digital Citizen Foundation, 13 March 2017, Accessed 12 June 2017.

Apply mathematical concepts to other fields

Sometimes, getting students excited about math isn’t easy. Nearly every math instructor has heard “When will I use this in real life?” at least once during their teaching career. Many students don’t see right away that they use math just about every day, and you can lose their interest in the subject if you don’t connect your course objectives to their lives outside of class. Thankfully, math applies to more fields than most students realize. Here are just a few ways to connect mathematical concepts to other areas and to get students more motivated to learn.

1. Create art with math.

Not all students see how subjects in STEM connect with the liberal arts. Some people mistakenly think the fields are separate and never the two shall meet. One great way to get rid of this misconception is to show how art can be created by using math. Creative Bloq shows eight examples of beautiful fractal art with suggestions on programs to use in order to create your own fractal masterpieces, such as Mandelbulb 3D and FraxHD.

The co-author of our Single Variable Calculus with Early Transcendentals textbook, Dr. Paul Sisson, used to incorporate art into his math classes when he taught at Louisiana State University – Shreveport. He encouraged students to use software to track complex numbers’ behaviors and create images to which students could assign different colors. Learn more from Math in the Media here.

2. Show students how to be fiscally responsible.

Chances are you have some students who don’t know much about personal finance beyond having a checking and savings account. Teaching them about budgeting, loans, interest, and more will benefit them now and in all the years to come. Students can start with concepts such as calculating tip and figuring out how much money they save when they buy discounted items before moving on to long-term financial decisions, such as putting a down payment on a house and paying a mortgage.

This post from Annenberg Learner summarizes the basics of simple and compound interest that you can incorporate into your class.

3. Calculate sports statistics.

Have students who want to be professional athletes, coaches, sports announcers, agents or just die-hard fans of the game? They’ll benefit from learning how much math goes into any sport. Everything from calculating batting averages in baseball to knowing touchdowns per pass attempt in football to determining the probability of winning a point in tennis can connect the concepts learned in class to some students’ favorite extracurricular activities. Plus, fantasy sports are especially popular, so you may even consider having your class join a fantasy league and see who wins!

Fantasy Sports and Mathematics is a website that includes the latest scores and injuries lists for various sports and sample math problems to use in class. This NYT blog post lists out ways to use sports analytics to teach math and includes additional resources ranging from a video demonstrating what it’s like to return a serve in professional tennis to a graphic showing how often football teams go for the fourth down.

4. Delve into the history of mathematics.

Students gain a deeper appreciation of the subject when they know who’s behind all those theories, formulas, and discoveries. Plus, they just might connect with the subject more when they know that people from similar demographics advanced the field.

A Buzzle article introduces readers to several achievements of African American mathematicians, ranging from those in the 18th century like Benjamin Banneker to the present day like Dr. William A. Massey.

This post highlights five influential female mathematicians throughout history, including Ada Lovelace and Emmy Noether. It gives a little background into these women’s lives, explains their accomplishments, and kicks the blatantly false stereotype that women aren’t good at math to the curb!

5. Have students write about how they think they’ll use math in their future careers.

Are your students still not feeling connected with the course content? Dedicate some class time to brainstorming how they’ll use math in the careers they’re planning to pursue. While at first some may assume they won’t use math at all in their chosen professions, they might surprise themselves once they think a little harder and dig deeper into a job’s tasks and expectations. They may want to interview someone in their field via email or phone to get an insider’s perspective into the kind of math skills needed to excel in the workplace.

On the blog Math for Grownups, author Laura Laing interviewed several professionals—including writers, academic advisors, and artists—asking them how they use math in their jobs. Her books Math for Grownups and Math for Writers delve into more detail on these topics and encourage folks who are hesitant about math or think they’re bad at it to rethink their perspective.

What are some lessons you’ve taught that encouraged students to apply math to other subjects and think outside the box? Let us know in the comments!

Encourage students to develop positive academic mindsets with this new question bank

Many students struggle with math. Most don’t realize that developing their soft skills as learners contributes to success with the subject. How can you make time to teach both curricular content along with critical study habits to your students in just one course?

The NEW study skills question bank is now available!
We’ve partnered with learning assessment expert Dr. Paul Nolting to incorporate exercises from Winning at Math directly into the Hawkes online courseware.

Build more independent learners by integrating instruction and assessment on proper study practices into your homework, assignments, and tests. The online questions are automatically graded in Hawkes so you have time to do what you do best: teach!

These questions promote students’ positive academic mindsets by encouraging better academic behaviors. Students will learn how to:

  • rework class notes on their own
  • control test anxiety
  • understand and improve their memory process
  • use positive self-talk
  • coordinate a Supplemental Instruction study group
  • get the most out of online text, tests, and homework
  • and more!

FREE Study Skills Assessment
When you adopt Winning at Mathyour students will also receive the Math Study Skills Evaluation—an ungraded, penalty-free assessment that asks students to reflect on their test-taking, study, and homework habits before providing feedback on how to improve these skills.

Get in touch with us today at 1-800-426-9538 to learn more!

More simulations, stat!

Remember that in the appendix of your courseware, great resources for student engagement help you bring the real world into your classroom! Check out two examples:

1. Direct Mail Game

In the game, students assume the role of a direct mail marketing manager for a company that markets inexpensive computer software. Their task is to develop a mailing strategy by finding mailing lists that will produce sufficient sales to be profitable.

The game provides an environment in which students apply statistical concepts while making business decisions.

While students can explore the game on their own, we recommend playing it in class.

Dr. Hawkes created this game when he was a statistics professor. He says that this lesson was always his students’ favorite each semester.

A row of lists 1-3 shows each list's price, number, and how many sampled. Below that, the mailing results are displayed, including the list number, cost per name, sample size, number of responses and their data, the mailing costs, the old balance, profit, and new balance. The cash on hand is displayed below, which is $19,968.50.

2. Name That Distribution

Name that Distribution is a concept builder that strengthens analytical skills in distribution recognition and data analysis.

Students view the histogram, frequency, statistics, and sample data of a distribution. They can increase the number of intervals and choose to view a hint if they’re unsure of the answer. As they play with the different options to analyze the data, students combine that information to make an educated guess about the distribution type.

Check out a hint:
A text box called Hints is shown, which more information on uniform distributions. It displays the theoretical variance, and explains how to identify this distribution type.

Below is an example of a distribution type:

A histogram is shown. Below it is the question "What type of distribution is described by the sample?" with the multiple choice options of uniform, normal, exponential, chi-square, Poisson, and mystery. An arrow points to the answer uniform, and another arrow points to a submit button.
Students new to this kind of data analysis will begin to understand how they can apply this knowledge to the many real-world scenarios that they can evaluate through detecting typical or skewed data.

The Direct Mail lesson is available in:

  • Discovering Statistics Appendix A.9
  • Discovering Business Statistics Appendix A.10
  • Beginning Statistics Appendix A.5

The Name that Distribution lesson is available in Appendix A.3 in all three statistics courses.