Source: https://www.xkcd.com/2118/

## All posts in category **Statistics**

# How To Annoy a Statistician

*Posted by John Quintanilla on March 6, 2020*

https://meangreenmath.com/2020/03/06/how-to-annoy-a-statistician/

# My Favorite One-Liners: Part 116

This awful pun is just in time for Valentine’s Day.

Source: https://www.facebook.com/NeuroNewsResearch/photos/a.479172065434890/2989385557746849/?type=3&theater

*Posted by John Quintanilla on February 14, 2020*

https://meangreenmath.com/2020/02/14/my-favorite-one-liners-part-116/

# The End of “Statistical Significance”

I’ve linked to a number of articles about the misuse of *p*-values. Recently, I read a nice article in the October/November 2019 issue of MAA Focus summarizing a conversation between the Executive Directors of the Mathematical Association of America and the American Statistical Association about the ASA’s call to eliminate the use of *p*-values. Per copyright, I can’t copy the entire article here, but let me quote the lead paragraph:

In March 2016, the American Statistical Association took the extraordinary step of issuing a Statement on

p-Values and Statistical Significance. This spring, the association went even further, publishing a massive special issue of its journal The American Statistician entitled Statistical Inference in the 21st Century: A World Beyondp<0.05. The lead editorial in that special issue called for the end of the use of the concept of statistical significance.

It’s going to be a while before entrenched statistics textbooks catch up with this new standard of professional practice.

Here’s an NPR article on the issue: https://www.npr.org/sections/health-shots/2019/03/20/705191851/statisticians-call-to-arms-reject-significance-and-embrace-uncertainty

Other articles cited in the MAA Focus article:

- http://sciencealert.com/statisticians-argue-it-s-time-to-ditch-significance-in-science-and-replace-it-with-uncertainty
- http://psychologytoday.com/us/blog/the-athletes-way/201903/rethinking-p-values-is-statistical-significance-useless
- http://theness.com/neurologicablog/index.php/get-rid-of-statistical-significance
- http://sciencenews.org/article/statisticians-standard-measure-significance-p-values
- http://vox.com/latest-news/2019/3/22/18275913/statistical-significance-p-values-explained

*Posted by John Quintanilla on February 3, 2020*

https://meangreenmath.com/2020/02/03/the-end-of-statistical-significance/

# Engaging students: Defining a function of one variable

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for *engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Phuong Trinh. Her topic, from Algebra: defining a function of one variable.

How have different cultures throughout time used this topic in their society?

The understanding of functions is crucial in the study of both math and science. Not only that, some functions, especially function with one variable, are often used by everyone in their daily life. For example, a person wants to buy some cookies and a cake. The person will need to figure how much it will cost them to buy a cake and however many cookies they want. If the cost of the cake is $12, and the price for each cookie is $1.50, the person can set up a function of one variable to find the total cost for any number of cookies, expressed as c. The function can be written as f(c) = 1.50c + 12. With this function, the person can substitute any number of cookies and find out how much they would spend for the cookies and cake. Aside from the situation given by this example, function with one variable can also be used in various different scenarios.

What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

Function with one variable can be used in many real life situations. Word problems can be derived from every day scenarios that the students can relate to.

Problem 1: John is transferring his homework files into his flash drive. This is the formula for the size of the files on John’s drive S (measured in megabytes) as a function of time t (measured in seconds): S (t) = 3t + 25

How many megabytes are there in the drive after 10 seconds?

This problem allows the students to get familiar with the function notation as well as letting the students work with a different variable other than x.

Problem 2: (Found at https://www.vitutor.com/calculus/functions/linear_problems.html )

“A car rental charge is $100 per day plus $0.30 per mile travelled. Determine the equation of the line that represents the daily cost by the number of miles travelled and graph it. If a total of 300 miles was travelled in one day, how much is the rental company going to receive as a payment?”

Besides giving the students practice with finding a solution from a function, this problem let the students practice setting up the equation. This also shows the students’ understanding of the subject.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

There are multiple resources that can be used to help the students understand what a function is as well as how they should approach a problem with function. One of the resources can be found at coolmath.com. The layout of the website makes it easy to locate the topic of “Functions” under the “Algebra” tab. By comparing a function with a box, Coolmath defines a function in a way that can be easily understood by students, while also showing how a function can be thought of as visually. The site also provides the explanation for function notation with visuals and examples that are easy to understand. On Coolmath, the students will also have the chance to practice with randomly generated questions. They can also check their answers afterward. On other hands, the site also provides definitions and explanations to other ideas such as domain and range, vertical line tests, etc. Overall, coolmath.com is great to learn for students in and out of the classroom, as well as before and after the lesson.

http://www.coolmath.com/algebra/15-functions

References:

“Linear Function Word Problems.” Inicio, www.vitutor.com/calculus/functions/linear_problems.html.

“Welcome to Coolmath.” Cool Math – Free Online Cool Math Lessons, Cool Math Games & Apps, Fun Math Activities, Pre-Algebra, Algebra, Precalculus, www.coolmath.com/algebra/15-functions.

*Posted by John Quintanilla on June 7, 2019*

https://meangreenmath.com/2019/06/07/engaging-students-defining-a-function-of-one-variable-2/

# Engaging students: Box and whisker plots

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for *engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This submission comes from my former student Chris Brown. His topic: how to engage students when teaching box and whisker plots.

How could you as a teacher create an activity or project that involves your topic?

My all-time favorite TV show as a child was Pokémon. This show is still a staple amongst the young and even adult generation of today. The activity that I have created, was designed to take place after a formal lesson over how to create Box and Whisker plots. For this activity, students will be given a labeled bar graph of the Pokémon Type Distribution for generations 1 through 6 of Pokémon, which I have listed an online data source below. The students will be tasked with identifying the top 7 Pokémon types and creating a Box and Whiskers plots for each of those types. They will then go through and analyze the consistency of the creation of Pokémon for that specific type and then compare contrast this same box plot to any other box plot of their choice. The students will then make predications for the number of Pokémon for each of the top 7 Pokémon types, for generation 7 and base their reasoning in the box plots they created. Then the student will finally research the type distributions for the 7^{th} generation of Pokémon, and discuss how the actual number compares to their prediction.

This is the online source for the type distributions for generations 1 – 6:

https://plot.ly/~powersurge360/6.embed

How does this topic extend what your students should have learned in previous courses?

From my experience, Box Plots are first taught in the early middle school years, in 6^{th} or 7^{th} grade. When constructing box plots by hand, in its essence, box plots require knowledge of how to order sets of numbers from least to greatest; an understanding and ability to find the maximum, minimum, median, and mean of a data set; and lastly, critical thinking and analytic skills developed from general course content. Box plots allow students to combine each of these skills to effectively analyze data sets with ease and compare different data sets with precision and accuracy. If any or all of these skills are not quite up to par, students will have an opportunity to develop them through box plots as they spend time creating them. For all students no matter their level, they will still gain better insight on how to properly analyze data and grow as analytical thinkers as they take the represented data and turn it into meaningful interpretations.

How can technology be used to effectively engage students with this topic?

In a classroom, I personally believe that Desmos is a wonderful online tool that can aid students in the understanding of how box and whisker plots function, and also a great place to check their work. Desmos, which is linked below, gives students the ability to list as many data points as they need to, and concurrently creates a box plot as they do so. In this way, students are able to see how singular data points can skew the data in significant and insignificant amounts. What I also love about Desmos is that, the list of data points does not have to be in any kind of order, so students do not have to worry about that tedious step! Desmos also lists the 5-point summary in two different places, on the box plot itself, and also on a drop-down menu, which is super convenient. Lastly, I love how Desmos also displays the mean of the data set as well, students can calculate the skew of the data, and definitively determine how it is skewed. This is a super visual, and interactive tool that will allow the student to manipulate box plots so seamlessly they will not be focused on the tediousness of the setup and solely on the concept.

The link to the Desmos setup is here: https://www.desmos.com/calculator/h9icuu58wn

*Posted by John Quintanilla on April 19, 2019*

https://meangreenmath.com/2019/04/19/engaging-students-box-and-whisker-plots-2/

# Statistics for People in a Hurry

The following article was recommended to me by a former student: https://towardsdatascience.com/statistics-for-people-in-a-hurry-a9613c0ed0b. It’s synopsis is in the opening paragraph:

Ever wished someone would just tell you what the point of statistics is and what the jargon means in plain English? Let me try to grant that wish for you! I’ll zoom through all the biggest ideas in statistics in 8 minutes! Or just 1 minute, if you stick to the large font bits.

*Posted by John Quintanilla on January 4, 2019*

https://meangreenmath.com/2019/01/04/statistics-for-people-in-a-hurry/

# Engaging students: Approximating data by a straight line

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for *engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Caroline Wick. Her topic, from Algebra: approximating data to a straight line.

How can this topic be used in your students’ future courses in mathematics or science?

Though approximating data by a straight line is a subject that is brought up in Algebra 2, it is something that students will need to use in a number of subjects down the line. Probably the most obvious subject would be statistics. Finding an approximate trend line is extremely important for a statistician so that they can predict future, unobserved data. Another example that might not be as readily noticeable would be anthropology. Anthropology is the study of humans in various parts of life. In this case, according to Brian Hopkins, anthropology can be used by stores to figure out what types of products they should stock on their shelves during different types of the year. They do this by collecting the data, then approximating the trend lines to predict how the product will sell during the same season of the next year. For example, Orange Juice and tissues are known to be sold more often during the winter seasons, so stores know that they want to stock up on orange juice and tissue during the colder season each year.

A1: Applications

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Using the data given below:

(a) plot the points on a graph

(b) Then, using a ruler, do your best to approximate a trend line that fits the points

(c) Write an equation (y=mx+b) that best fits the trend line

(d) Approximate the next four numbers on the line using the equation you created.Population growth in squirrels in TX from 1950-1980 (in millions)*

Year (x) 1950 1955 1960 1965 1970 1975 1980

Pop. (y) 12 12.7 13.1 13 13.6 13.7 14

From here the student would create his/her graph with the plotted points, find a line that best fits the points with equal numbers over and under the line. They would then use the data and the line to find an equation that best fits the scatter plot data that they graphed. They would then find the approximate squirrel population for 1985, 1990, 1995, and 2000.

This could be either an assignment or it could turn into a project for students with different sets of data. Students could even collect their own data to formulate the graph and equation.

*not real data, fabricated for this problem specifically.

Culture

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

The approximation of data through trend lines has been used in pop culture since the birth of popular culture in the mid twentieth century. More relevantly, it is used to map certain cultural trends. When a new movie is coming out, statisticians use previous data from people who watched/reviewed the movie before its release to map out how they believe it will be appreciated by the public. A movie that did will before its release will likely have a positive trend line that continues upward at a somewhat steady rate. It will get more tickets at the box office than a movie that was not as well liked that might have a less-steep slope. Statisticians use this same trend approximation with TV shows and whether they should run another season, or in music when it hits the top of the charts. The more people listen to a song, the more likelihood it has to be listened to other people, thus the trend continues upward until is slowly dies off.

Take for instance, Taylor Swift’s “Look What You Made Me Do” that was released August 25th of this year. From its release and popularity, statisticians were able to track the data and predict that the song would be number 1 on the top 100 just a few weeks after its release.

References:

B1: https://www.cio.com/article/2372429/enterprise-architecture/the-anthropology-of-data.html

C1: http://www.billboard.com/articles/news/7949029/taylor-swift-look-what-you-made-me-do-timeline-reputation

*Posted by John Quintanilla on February 23, 2018*

https://meangreenmath.com/2018/02/23/engaging-students-approximating-data-by-a-straight-line-3/

# Paranormal distribution

*Posted by John Quintanilla on October 30, 2017*

https://meangreenmath.com/2017/10/30/paranormal-distribution/

# Finding the Regression Line without Calculus

Last month, my latest professional article, Deriving the Regression Line with Algebra, was published in the April 2017 issue of *Mathematics Teacher *(Vol. 110, Issue 8, pages 594-598). Although linear regression is commonly taught in high school algebra, the usual derivation of the regression line requires multidimensional calculus. Accordingly, algebra students are typically taught the keystrokes for finding the line of best fit on a graphing calculator with little conceptual understanding of how the line can be found.

In my article, I present an alternative way that talented Algebra II students (or, in principle, Algebra I students) can derive the line of best fit for themselves using only techniques that they already know (in particular, without calculus).

For copyright reasons, I’m not allowed to provide the full text of my article here, though subscribers to *Mathematics Teacher* should be able to read the article by clicking the above link. (I imagine that my article can also be obtained via inter-library loan from a local library.) That said, I am allowed to share a macro-enabled Microsoft Excel spreadsheet that I wrote that allows students to experimentally discover the line of best fit:

http://www.math.unt.edu/~johnq/ExploringTheLineofBestFit.xlsm

I created this spreadsheet so that students can explore (which is, after all, the first *E* of the 5-E model) the properties of the line of best fit. In this spreadsheet, students can enter a data set with up to 10 points and then experiment with different slopes and -intercepts. As they experiment, the spreadsheet keeps track of the current sum of the squares of the residuals as well as the best guess attempted so far. After some experimentation, the spreadsheet can also provide the correct answer so that students can see how close they got to the right answer.

*Posted by John Quintanilla on May 24, 2017*

https://meangreenmath.com/2017/05/24/finding-the-regression-line-without-calculus/

# My Favorite One-Liners: Part 95

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Today’s quip is one that I’ll use in a statistics class when we find an extraordinarily small -value. For example:

There is a social theory that states that people tend to postpone their deaths until after some meaningful event… birthdays, anniversaries, the World Series.

In 1978, social scientists investigated obituaries that appeared in a Salt Lake City newspaper. Among the 747 obituaries examined, 60 of the deaths occurred in the three-month period preceding their birth month. However, if the day of death is independent of birthday, we would expect that 25% of these deaths would occur in this three-month period.

Does this study provide statistically significant evidence to support this theory? Use .

It turns out, using a one-tailed hypothesis test for proportions, that the test statistics is and the value is about . After the computations, I’ll then discuss what the numbers mean.

I’ll begin by asking, “Is the null hypothesis [that the proportion of deaths really is 25%] *possible*?” The correct answer is, “Yes, it’s possible.” Even extraordinarily small -values do not prove that the null hypothesis is impossible. To emphasize the point, I’ll say:

After all, I found a woman who agreed to marry me. So extremely unlikely events are still possible.

Once the laughter dies down, I’ll ask the second question, “Is the null hypothesis *plausible*?” Of course, the answer is no, and so we reject the null hypothesis in favor of the alternative.

*Posted by John Quintanilla on May 6, 2017*

https://meangreenmath.com/2017/05/06/my-favorite-one-liners-part-95/