## Tag: Math

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On the number of bootstrap samples

The number of possible bootstrap samples for a sample of size N is big. Really big. Recall that the bootstrap method is a powerful way to analyze the variation in a statistic. To implement the standard bootstrap method, you generate B random bootstrap samples. A bootstrap sample is a sample

Programming Tips
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The art of rotations and reflections

Art evokes an emotional response in the viewer, but sometimes art also evokes a cerebral response. When I see patterns and symmetries in art, I think about a related mathematical object or process. Recently, a Twitter user tweeted about a painting called "Phantom’s Shadow, 2018" by the Nigerian-born artist, Odili

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Odani's truism for fractions that are near each other

Odani's truism is a mathematical result that says that if you want to compare the fractions a/b and c/d, it often is sufficient to compare the sums (a+d) and (b+c) rather than the products a*d and b*c. (All of the integers a, b, c, and d are positive.) If you

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Odani's truism: A probabilistic way to compare fractions

Quick! Which fraction is bigger, 40/83 or 27/56? It's not always easy to mentally compare two fractions to determine which is larger. For this example, you can easily see that both fractions are a little less than 1/2, but to compare the numbers you need to compare the products 40*56

Analytics
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The Farey sequence

Here is an interesting math question: How many reduced fractions in the interval (0, 1) have a denominator less than 100? The question is difficult is because of the word "reduced." If we only care about the total number of fractions in (0,1) whose denominator is less than 100, we

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Find a vector that has a specified correlation with another vector

Do you know that you can create a vector that has a specific correlation with another vector? That is, given a vector, x, and a correlation coefficient, ρ, you can find a vector, y, such that corr(x, y) = ρ. The vectors x and y can have an arbitrary number

Analytics
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Intransitive dice

Most games of skill are transitive. If Player A wins against Player B and Player B wins against Player C, then you expect Player A to win against Player C, should they play. Because of this, you can rank the players: A > B > C Interestingly, not all games

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5 things every student should know about data science

SAS has always believed in the power of education, but in today’s data-driven economy, it’s more important than ever to ensure our students are introduced to data science at an early age. We as a company are focusing our resources on creating student experiences in data literacy, computer science and

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The Kullback–Leibler divergence between discrete probability distributions

If you have been learning about machine learning or mathematical statistics, you might have heard about the Kullback–Leibler divergence. The Kullback–Leibler divergence is a measure of dissimilarity between two probability distributions. It measures how much one distribution differs from a reference distribution. This article explains the Kullback–Leibler divergence and shows

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Cubic spline interpolation in SAS

I recently showed how to use linear interpolation in SAS. Linear interpolation is a common way to interpolate between a set of planar points, but the interpolating function (the interpolant) is not smooth. If you want a smoother interpolant, you can use cubic spline interpolation. This article describes how to

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The circular law for eigenvalues

I've previously written about how to generate points that are uniformly distributed in the unit disk. A seemingly unrelated topic is the distribution of eigenvalues (in the complex plane) of various kinds of random matrices. However, I recently learned that these topics are somewhat related! A mathematical result called the

Analytics
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Polygons, pi, and linear approximations

Recently, I saw a graphic on Twitter by @neilrkaye that showed the rapid convergence of a regular polygon to a circle as you increase the number of sides for the polygon. The author remarked that polygons that have 40 or more sides "all look like circles to me." That is,

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Evaluate a function on a linear subspace

This article discusses how to restrict a multivariate function to a linear subspace. This is a useful technique in many situations, including visualizing an objective function that is constrained by linear equalities. For example, the graph to the right is from a previous article about how to evaluate quadratic polynomials.

Programming Tips
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What is a geometric mean?

There are several different kinds of means. They all try to find an average value from among a set of numbers. Although the most popular mean is the arithmetic mean, the geometric mean can be useful for problems in statistics, finance, and biology. A common application of the geometric mean

Analytics
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The math you learned in school: Yes, it’s useful!

What is this math good for, anyway?      –Every student, everywhere I am a professional applied mathematician, yet many of the mathematical and statistical techniques that I use every day are not from advanced university courses but are based on simple ideas taught in high school or even in grade school.

Students & Educators
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Elementary Spotlight: Math

Early math skills—such as number sense, number relations, pattern identification, estimation, and measurement—are the foundation for future math achievement. By mastering these basic concepts, young students can engage in more complex cognitive processing and develop higher-level math skills. Also, early math achievement has been linked to an increased likelihood to take

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Gershgorin discs and the location of eigenvalues

The eigenvalues of a matrix are not easy to compute. It is remarkable, therefore, that with relatively simple mental arithmetic, you can obtain bounds for the eigenvalues of a matrix of any size. The bounds are provided by using a marvelous mathematical result known as Gershgorin's Disc Theorem. For certain

Students & Educators
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How blended learning helped solve an algebra challenge

(Editor's note: This guest blog post was written by Buhl High School, Idaho math and science teacher Pat Van Patten. When asked why she stays in the classroom after 30 years, Pat replied,  “I still love teaching. I love watching students learn; when the light turns on for them and they

Students & Educators
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Black History Month: Honoring Math Pioneers

In honor of Black History Month, let’s showcase a few African American mathematicians who have made their mark in the teaching profession.  Elbert Frank Cox (1895-1969) was the first black person in the world to receive a Ph.D. in mathematics. In 1917, Cox earned his undergraduate degree from the University of

Students & Educators
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In June, we launched our free, online lesson-building tool, Crio. Since then many teachers have created and shared their lessons. But for those yet to jump in, I’ll bet there are at least three questions you've asked yourself. So, let me answer them! Why should I create Crio lessons? Because with

Programming Tips
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How many perfect riffle shuffles are required to restore a deck to its initial order?

Last week I compared the overhand shuffle to the riffle shuffle. I used random operations to simulate both kinds of shuffles and then compared how well they mix cards. The article caused one my colleague and fellow blogger, Rob Pratt, to ask if I was familiar with a bit of

Programming Tips
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Distances on rectangular grids

Given a rectangular grid with unit spacing, what is the expected distance between two random vertices, where distance is measured in the L1 metric? (Here "random" means "uniformly at random.") I recently needed this answer for some small grids, such as the one to the right, which is a 7 x 6

Programming Tips
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The continued fraction representation of a rational number

Continued fractions show up in surprising places. They are used in the numerical approximations of certain functions, including the evaluation of the normal cumulative distribution function (normal CDF) for large values of x (El-bolkiny, 1995, p. 75-77) and in approximating the Lambert W function, which has applications in the modeling

Students & Educators
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Three ways a free online course can help you blend algebra instruction

Did you know Curriculum Pathways resources include an entire online Algebra 1 Course? And it's much more than a typical math textbook. The Algebra 1 Course consists of 46 lessons organized into 10 units. In addition, students can work through two pre-algebra units (7 lessons) in preparation for the course.

Students & Educators
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Back to School: Math

Kicking off the new school year, you may be looking for ways to ease your students back into math. Here are a few ways that SAS can help. And remember, all of these resources, including apps, are free! CodeSnaps Have some fun coding a Sphero robot using CodeSnaps! With the

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Offset regions: Find all points within a specified distance from a polygon

My colleague Robert Allison recently blogged about using the diameter of Texas as a unit of measurement. The largest distance across Texas is about 801 miles, so Robert wanted to find the set of all points such that the distance from the point to Texas is less than or equal

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The intersection of two line segments

Back in high school, you probably learned to find the intersection of two lines in the plane. The intersection requires solving a system of two linear equations. There are three cases: (1) the lines intersect in a unique point, (2) the lines are parallel and do not intersect, or (3)

Students & Educators
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That exciting #edtech day is finally here! The new iPads are unboxed, charged, and configured. The only question left is where to begin? Of course the best answer involves ongoing professional development and a good school-wide education technology plan. And that's no secret. Well-planned and resourced professional development is the

Students & Educators
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Medal with these Engaging Sports-themed Lessons

Tap into student interest in athletic competitions with these sports themed lessons. Creating and Annotating Maps Our Interactive Atlas (also available in Spanish) is a great tool for project-based learning. Using our map and annotation tools, students demonstrate mastery of content while engaging their research and writing skills. Consider the following sports related activities:

Students & Educators
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Teaching Middle School Math? Need Data Analysis/Statistics Resources?

A new Star Wars movie debuts in just a few weeks! Star Wars: The Last Jedi will be the next chapter in the saga we’ve all come to love. The first film, Star Wars: Return of the Jedi, debuted in 1977. So, math teachers, how old were you when that