We hope you enjoy our free, printable graphing worksheets for kindergarten. You will find pie charts, count and graph, pictographs, dot plot, line, and bar graphs! We recommend using these activity with K-2 math students to build important math skills. These graphing worksheets are great for morning workstations, early finishers, and take-home activities. Are you needing more graphing worksheets? Take a look at our Graphing Activity Bundle!
Count and Graph Graphing Worksheets
Our count and graph worksheets will be a hit in your classroom. These count and graph worksheets visually illustrate raw data in picture form. You and your students will enjoy these graphing worksheets as they get to color while practicing counting, graphing, and collecting data all in one sheet!
Mythical Creatures Count & Graph Worksheet
Our Mythical Creature Count and Graph worksheet will be a fun addition to your math curriculum. Your students will have fun as they add up the number of each creature then fill in the graph.
Underwater Count & Graph Worksheet
Your students will work on comparing, finding, counting and more with this cute, underwater theme count and graph worksheet!
Pirate Count & Graph Worksheet
Diamonds, parrots, treasure and pirates oh my! This fun pirate theme count and graph will delight your kids as they color AND learn about graphs!
Pictograph Graphing Worksheets
What is your favorite water activity? How about your favorite toy? Our pictograph graphing worksheets take fun topics and ask your students to read the raw data and create pictographs. Each picture graph worksheet has clear instructions for your students to follow. When they are done, they will have a complete view of the data.
Favorite Water Activity Pictograph Worksheet
Your students will work on their counting, reading, writing and graphing skills as they complete this fun Favorite Water Activity pictograph.
Favorite Toy Pictograph Worksheet
Pictographs are a fun and creative way to show data in a graphing form. You and your students will enjoy this educational and fun graphing worksheet.
Favorite 3D Shape Pictograph Worksheet
This single-page pictograph worksheet is great for math centers, early finishers and homework and more. Your students will work on their math skills as they complete the graph.
Dot Plot Graphing Worksheets
Candy Bars Dot Plot Graphing Worksheet
Dot plot graphs will teach your students how to show the data in a clear way. They will learn to compare and contrast, find matching numbers, count, number sequence and more!
Coins Collected Dot Plot Graphing Worksheet
This Coins Collected Dot Plot Graphing Worksheet will have your students reading and answering questions about the the completed graph. It is a great way to help your students learn the parts of graphs.
Dolphins Swimming Dot Plot Graphing Worksheet
How many dolphins did the students see on their tour? Your students will take the data and graph it on this fun dot plot graphing worksheet, making it a hit with you and your students as they learn more about graphs.
Line Graph Graphing Worksheets
Line graph worksheets are a great way to show data. Your students will learn to read, make, and understand them throughout their years of schooling. Now is a great time to get them started. Our line graph worksheets are fun and inviting, letting your students enjoy math as their gain knowledge of graphs.
Bugs Line Graph Worksheet
Your students will work on reading, understanding tally marks, writing, creating line graphs and more with this fun Bugs Line Graph worksheet.
Plants Line Graph Worksheet
Our Plants in the Arboretum Lone Graph worksheet will be a great addition to your math centers as well as homework packs.
Occupation Line Graph Worksheet
This Favorite Occupation Line Graph worksheet will help your students learn to read and answer questions from line graphs.
Bar Graph Graphing Worksheets
Graphing is an important tool for mathematics. These bar graph worksheets will help your students learn comparing, sorting, counting, organizing, and analyzing. Our worksheets are great for teaching graphing for kindergarten and helping your students get a head start in math.
Favorite Bird Bar Graph Worksheet
Bar graphs help us look at data in a visual way. This worksheet will help your students learn to fill in and see what data looks like on a bar graph.
Favorite Snack Bar Graph Worksheet
Which snack has the most votes? How many votes were cast in all? Your students will need to read the bar graph to answer these and other questions.
Favorite Costume Bar Graph Worksheet
Now that your students know the pieces to bar graphs, use this graphing worksheet to teach them how to fill them out with the data provided.
Pie Graph Graphing Worksheets
Your students will learn about the parts-to-whole relationship as they work through these pie graphing worksheets. They are a great way to help your students learn about percentages in a visual way. Pie charts help your students see great than, less than, and equal to in a clear way.
Summer Treat Pie Chart Graphing Worksheet
Who doesn’t love a summer treat? Your students will learn how to read pie graphs to answer the questions on this single-page, cute, pie chart graphing worksheet!
Favorite Dinner Pie Chart Graphing Worksheet
This Favorite Dinner Pie Chart will help your students see a visual representation of data, helping them correlate the size of the piece to the portion of the pie chart.
Favorite Drink Pie Chart Graphing Worksheet
Now that your students have learned the pieces that make up a pie graph, they get to try their hand at filling out questions and the chart with the data they have been given.
I can not thank you enough for these Free worksheets. I am a teaching assistant and work in a Special Education classroom. The students love these worksheets. When you offer them for free it makes teaching the students easier and fun. We can not afford to always buy worksheets for the class.
These are great, thank you so much.
Thank you for sharing. I teach first and second grade students. This will help my students and myself.
These are fantastic. Thank you I can now assess which area they struggle with, creating, reading or filling in.
Thanks for sharing these awesome worksheets. I look forward to sharing with my third graders!!
Thank you so much for these worksheets. They are amazing and great for center work practice. I really appreciate you sharing them.
These are absolutely fantastic!! Perfect addition to our homeschool curriculum! Thank you.!!


This IP core provides resource efficient ternary adders, i.e., adders with three inputs performing
s = x + y + z
for the Altera and Xilinx platforms. Resource efficient means that they need exactly the same resources on modern FPGAs as two-input adders, but are slightly slower.
It was used in the following publications:
M. Kumm, M. Garrido, O. Gustafsson and P. Zipf, Optimal Constant Coefficient Multiplication using Ternary Adders, IEEE Transactions on Circuits and Systems II: Express Briefs, accepted for publication (21.10.2016), DOI
M. Kumm, M. Hardieck, J. Willkomm, P. Zipf, and U. Meyer-Baese, Multiple Constant Multiplication with Ternary Adders, IEEE International Conference on Field Programmable Logic and Application (FPL), 2013, pp. 1–8, DOI
The Project is hosted at opencores.org.


Salary Range $105,000-$125,000
Total Benefits Package of up to $35,000
Must be US Citizen with ability to obtain public trust clearance
Windsor Mill, MD
The selected candidate will compile data to address the most complex data concerns, as well as identify and network with appropriate internal sources to
gather and/or exchange data on specialized matters. PhD preferred.
Other key responsibilities may include:
• Determine the nature of analytic problems, evaluate options, and offer
recommendations for resolution;
• Advise on the methods and data needed and/or available to evaluate the
(intelligence or data?) problem;
• Collaborate with data collectors and analysts to identify and close gaps on
complex monitoring problems;
• Provide accurate, timely, complex and sophisticated data analysis;
• Determine and employ the most appropriate research design for data
collection and analysis;
• Analyze, evaluate, and assess quantitative data (using statistical software,
computer models, geospatial models, software languages, and mathematical models)
to contribute to or develop software tools, analytic models, or reports.
• Ph.D. or Masters in a highly analytic field such as Mathematics, Statistics, Computer Science, Physics, Data Science, etc.
• A documented minimum of 3 years' hands-on expert-level experience working with real world data sets in predictive modelling using a scripting language such as Python, MATLAB and/or R. Preference for Python, then R and then
o Specifically, experience doing classification and regression with support
vector machines, random forests, linear and non-linear regression, etc. on real
• Expert knowledge in conducting data analysis and applying advanced statistical concepts and machine learning methods to build, train, test, and evaluate a variety of supervised and unsupervised analytic models.
• Ability to clean and process large amounts of real world data.
• Experience retrieving and manipulating data from a variety of data sources included Db2, Oracle, SQL Server, Hadoop and flat files.
• Experience with database management systems, e.g. MySQL, SQLite, SQL,
• Either experience with, or the ability and willingness to learn distributed
processing via the Hadoop ecosystem, i.e. Spark, Impala and Hive.
• Experience working in a Linux environment via terminal commands.
• Experience presenting data and results to both technical and non-technical
audiences in a clear and effective manner.
• Ability to obtain a Public Trust clearance.
• An in-depth knowledge of Social Security Administration (SSA) protocols
concerning death processing, debt management, data reporting, etc.
• Knowledge of and experience handling SSA master files and related data.
• Natural language processing experience.
• VBA scripting experience
• Experience using markup languages such as LaTeX, HTML, etc.
• Experience using Agile team development.
• Experience teaching and mentoring in any and all of the above
PhD in Mathematics, Statistics, Computer Science, Physics, Data Science, or similar field. A Masters Degree in the similar fields may be acceptable, but PhD is preferred.


interface MatrixFactory<out T :
A set of constructors that must be implemented by a koma backend. Generates various types of matrices. Generic parameter is the type of element, i.e. T=Matrix or T=Matrix, etc.
Creates a row-vector with the first value of start and the last value of stop, with increment steps between each value.
Creates a row-vector with the first value of start and the last value of stop, with unit steps between each value.
Creates a row-vector with initial values pulled from an int range, e.g. 1..45
Creates a row-vector with initial values pulled from a double array
Creates a matrix from an array of arrays (row-major)
Creates an identity matrix of the requested shape
Creates an identity matrix of the requested shape, with zero padding if the axis lengths arent equal.
Creates a one initialized matrix of the requested shape
Creates a matrix of uniform 0-1 random samples
Creates a matrix of unit-normal random samples
Generate a zero initialized matrix of the requested shape.
Some functionality to help more easily implement double based koma backends. Feel free to not use if your backend has fast implementations of these functions.


Learn all there is to know about betting odds.
One of the first things you need to understand about betting is how sports betting odds operate if you are new to it.
The odds represent the bookmakers’ perception of your selection’s possibility of winning and how much you stand to win if it does.
Although betting odds might be confusing at first, knowing how they operate can help you get the latest sports odds and make betting on your favourite sport simpler.
What Exactly Is Probability?
Probability refers to the likelihood of something occurring.
If the bookmaker predicts a high probability result for an event, it is more likely to occur than not.
If an event is assigned a low probability, it is less likely to occur than not.
British bookmakers usually show probability as fractional odds, which would be two integers separated by a forward slash, such as 1/2.
A coin flip with two potential outcomes, heads or tails, is a basic probability explanation.
The likelihood of obtaining leads is one in two, which may be expressed by odds of 1/1, implying that the result has a 50% chance of winning.
How to Determine Probability Using Sports Betting Odds
Now that you know how odds are two numbers separated by a forward slash, you may use them to calculate sports betting odds.
This will also help you read betting odds and help calculate the probability of an event occurring.
How to Interpret Betting Odds
You can use the betting odds to determine the potential winnings if you bet on an event now that you know how to calculate the possibility of it happening.
The possible rewards are shown as fractional odds of the investment you put into the water.
For instance, if you bet $1 on Liverpool to defeat Burnley at 5/1, you will get $6 if you win, which essentially is a $5 profit plus your $1 wager.
Any chances where the first number is more than the second are considered odds against, while any odds where the first number is less than the second are considered odds on.
Bookmakers believe that odds-on events are more likely to occur than odds-against ones, and vice versa for odds-against events.
What Are Decimal Odds?
The bet is included in the possible winnings when converted to decimals, so multiply the stake by the odds.
Decimal odds are usually used on betting exchanges, such as Neds because the user controls the odds rather than the bookmaker.
The following are some examples of how to understand decimal betting odds.
- 10.00 odds = $1 bet returns $11 plus stake
- 3.5 odds = $1 bet returns $5.50 plus stake
- 1.5 odds = $1 bet returns $1.50 plus stake
The American Odds System
It’s still good to know how the American version of betting works, even though fractional and decimal odds are the most common way that a bookmaker shows their prices on a certain event.
Returning to the coin toss example, the chances for heads and tails are either money or 1/1 (fractional) or 2.0 (decimal), implying that a $1 bet will yield $2, a $1 profit plus the $1 bet.
The heads and tails results would be -100 or +100 in American or moneyline odds.
And 100 is the important number. It is an American football game this time, and the American odds are as follows:
- The New England Patriots are a -145 favourite.
- +120 for the Minnesota Vikings
If you predict the Minnesota Vikings will win, the +120 indicates that for every $100 you bet on them, you will receive $220 ($120 plus your initial bet of $100).
With a $10 bet, you would gain $23 – your £10 investment plus £13 profit (calculated as £10/£100 x 120).
The – sign in front of the New England Patriots odds indicates favourites, and the computation differs.
The -145 suggests that you will gain $100 for every $145 wagered. So, a $10 bet would yield $16.90 – your $10 risk plus $6.90 profit (calculated as $10/$145 x 100).
In both cases, the computation for the $10 investment is:
(Actual stake/American odds stake) multiplied by American odds or Moneyline odds wins.
When it comes to sports betting, it is critical to understand how betting odds work.
Now that you have mastered the art of reading sports, odds get out there and start making money at your favourite sports betting sites!
Sharp bettors are well-informed bettors, and not every sport is the same.


- This event has passed.
Grades K-4 STEM-inar
Miss Jessica from the Sylvan learning center is bringing the STEM world to the library! Featuring fun stories, activities and experiments, this program takes a closer look at one of the topics in the areas of Science, Technology, Engineering, and Mathematics (STEM).
STEM programming is all the rage right now, and the Sylvan Learning Center provides this FREE curriculum based program to encourage children to develop their skills in science, technology, engineering, and math in a fun, approachable way.
No registration required. Bring a friend!


Currently a beta version of the library is available for download.
If you use the code for your research, please cite some of the following papers:
- Zapletal, J., Of, G., Merta, M. Parallel and vectorized implementation of analytic evaluation of boundary integral operators. Engineering Analysis with Boundary Elements 96, 2018. DOI: 10.1016/j.enganabound.2018.08.015.
- Kravcenko, M., Merta, M., Zapletal, J. Distributed fast boundary element method for Helmholtz problems. Applied Mathematics and Computation 362, 2019. DOI: 10.1016/j.amc.2019.06.017.
- Zapletal, J., Merta, M., Maly, L. Boundary element quadrature schemes for multi- and many-core architectures. Computers and Mathematics with Applications 74, 2017. DOI: 10.1016/j.camwa.2017.01.018.


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Quantitative Analyst, Rates
- Singapur, Singapore
- Festanstellung, Vollzeit
- Argyll Scott Singapore Pte Ltd , EA Lizenz Nr.: 11C3721
- 11 Jan 17
2:1 Advanced degrees (MSc/PhD level or equivalent) in quantitative or numerate subjects
A top tier international bank is looking to hire 2 Quantitative Analyst to validate the front office Interest Rates and Commodity Derivative models. You can expect to join a great environment with open culture and team of high calibre quants.
This is a challenging role where you will be responsible for independent review and analysis of pricing models used for valuation and risk across the bank.
- Understand the mathematical models and implementation methods used, products traded in these markets, and the associated risks inherent from trading these products.
- Where appropriate, independently verify models and numerical schemes for pricing of products in the validation library (predominantly written in C++).
- Communicate outcomes of your review and analysis with key model stakeholders including front office trading, front office quants, market risk manager and finance.
- Actively engage with the due diligence aspects of the new product approval process and bank wide strategic initiatives.
- 2:1 Advanced degrees (MSc/PhD level or equivalent) in quantitative or numerate subjects such as Mathematics, Physics or Engineering. Fresh graduates from top ranking universities may apply.
- Preferably 1 - 3 years experience in a model validation or front office quant role and exposure to more than one asset class in that capacity. (Flow/Exotic Rates preferred)
- Excellent mathematical abilities and an understanding of stochastic calculus, partial differential equations, Monte-Carlo methods, finite difference methods, and numerical algorithms.
- High competency in object-oriented C++ programming to contribute to a shared library
- Interested in financial markets (especially derivative pricing) demonstrated by qualifications and experience.
- Diligent in your attention to details.
- An excellent communicator - both written and oral.
If you are interested in exploring this role further, please send your resume to Noel Huang at NHuang@argyllscott.com or call directly on +65 3157 2131 for a confidential discussion (EA Licence No. R1330171)
More exciting opportunities may be found on the Argyll Scott website at www.argyllscott.com
Licence: Argyll Scott Singapore Pte Ltd (16S8105)


Introduction. this is a 4 th article on the series of articles on analysis of algorithms. in the first article, we learned about the running time of an algorithm and how to compute the asymptotic bounds. we learned the concept of upper bound, tight bound and lower bound. in the second article, we learned the concept of best, average and worst analysis. in the third article, we learned about the. Google map pedometer gmaps pedometer to map and compute running, walking, cycling, and hiking distances. These usually have considerably worse running times than o(n 2) algorithms even if they don’t have a different name! a quadratic running time is common when you have nested loops. to calculate the running time, find the maximum number of nested loops that go through a significant portion of the input. 1 loop (not nested) = o(n) 2 loops = o(n 2). Google map pedometer gmaps pedometer to map and compute running, walking, cycling, and hiking distances.
Calculator use. find your running pace per mile, kilometer, yard or meter. calculate the distance you can run in a given amount of time, time required to run a specific distance, or your running speed. use this calculator to find average pace for running, biking, swimming or walking. how to calculate running pace. divide your run time by your. This technique works on the web interface on a computer, as well as on iphone and android. however, it doesn’t work in the web interface in a mobile browser. with that said, let’s begin. in my case, i’m going to be measuring the distance of a big trail loop that goes around the local ymca.
Calculating The Running Time Of Algorithms Algorithm Tutor
Google map pedometer / gmaps pedometer for running, walking.
Fitness training made easy with mapmyrun. com. create, discover, and save your favorites — from your neighborhood to around the world. On the go map is a route planner for running, walking, biking, or driving. it lets you create and share routes, calculate distances, and view elevation profiles for those routes. plan a route. click or tap points on the map to create a route. drag points to move them, or drag segments to insert a new point. you can also undo or redo changes to. The total running time is $$theta(max(n, n^2 = theta(n^2)$$ for loops. it is relatively easier to compute the running loop calculator running time of for loop than any other loops. all we need to compute the running time is how many times the statement inside the loop body is executed. consider a simple for loop in c. printable quarter fold cards budget for kitchen remodel calculator headlight running light relay wiring diagrams hr generalist sample resume
Running Time And Bigo Learneroo
Sport distance calculator is a distance calculator tool. it enables you to map your runs, cycles and walks and calculate the distances and elevation profiles of your routes. used by sports people as a tool for trainning for cycling, hikking, marathons, route running and triathlons. Mapometer. com is an online map-basedroute planner for sports people. it enables you to map your runs, cycles and walks and calculate the distances and elevation profiles of your routes. it also provides training logs so you can track your progress running loop calculator and monitor the calories you burn. The running calculator for run/walk pace. this tool allows you to figure out how fast you’re going when using a run/walk plan. by the editors of runner’s world and kiera carter. Routeloops is the site that creates free, custom routes for running and biking that begin and end at the same location. just enter a starting location and a distance, and you’ll receive a custom route, of whatever distance you specify, starting and ending at your location.
Ohm’s law calculator rapidtables. com.
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This is by far the best calculator online! ” — jelle “thanks for the site. i use it running loop calculator every time i am preparing for a half or full marathon. ” — daniel “i really love runcalc. net, it’s my go-to tool and helps me regularly. ” — nick. other resources /r/running, the reddit running community. Running record template story starter generator south australian spelling test age calculator easy grader calculator. download 20 picture prompts free!. Calculator use. find your running pace per mile, kilometer, yard or meter. calculate the distance you can run in a given amount of time, time required to run a specific distance, or your running speed. use this calculator to find average pace for running, biking, swimming or walking. Find your running pace calculator here. from age-grading to pace wristbands, we have it all.
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lesson, so that you can have multiple background loops running seemingly simultaneously: youtube/teq3-p0gwgi rtos Capacitance calculator (capacitor design) to use the calculator: 1. choose the units of measurement, english or metric. 2. enter the length of the antenna conductor, which is the distance around the loop. That’s because, as n gets bigger, the fact that x is keeping the loop from running exactly n times is going to make less of a difference. if x is 2 and n is 1,000, the runtime’s going to be about 1,000. computer scientists won’t mind if the actual number’s 998 iterations. Equivalent race times metric distances; 800m 1000m 1200m 1500m 1600m 2000m 2400m 3000m 3200m 4km; time: 2:04. 8: 2:40. 8: 3:17. 7: 4:14. 6: 4:33. 9: 5:51. 5: 7:10. 1: 9:04.


I’M SO SORRY I JUST COULDN’T STARE AT THIS GIF ANYMORE AND ENDLESSLY HAVE MY DREAMS SHATTERED I HAD TO MAKE THINGS RIGHT
rebageling for people in other timezones who might need some relief too
Seeing this made me stop being an atheist for 45 seconds.
Thank you kinetic-squirrel.
Marble artist Mike Gong creates beautifully detailed designs inside of small, handmade glass marbles. He pays close attention to detail to get every single element just right within the round interior of glass, and each of his creative concepts tells a crazy little story.
Gong produces mesmerizing clouds of color, texture, and patterns that seem out of this world. No two pieces are alike and, as you turn each marble around, the various details and unusual elements change color and sparkle in the reflecting light.
In particular, his Acid Eaters collection features wacky little faces that have big smiles, swirling eyes, and tongues sticking out of their mouths. Inside of the thick glass, hallucinations come alive and the happy guys appear distorted as viewers spin the glass around in their hands.
h o w
36 sided polygon made with corel draw
The intersecting lines give the illusion of concentric circles in the center, but that is an illusion.
The original file is 36 x 36 inches, so that meant I had to re-size it for posting on the internet, but I’m thinking of getting a poster printed.
Also, no, I did not connect all of these dots by hand. I did connect one point to 35 other points, and then copy/paste/rotated until it went all the way around.
Thanks for stopping by.
The Beauty of Math in Science - Lissajous Curve
Lissajous curve, also known as Lissajous figure or Bowditch curve, is the graph of a system of parametric equations: x = A.sin(a.t + δ) and y = B.cos(bt)
The appearance of the figure is highly sensitive to the ratio a/b - Image 3 (3/2, 3/4 and 5/4). For a ratio of 1, the figure is an ellipse, with special cases including circles (A = B, δ = π/2 radians) and lines (δ = 0). Another simple Lissajous figure is the parabola (a/b = 2, δ = π/4). Other ratios produce more complicated curves, which are closed only if a/b is rational. The visual form of these curves is often suggestive of a three-dimensional knot, and indeed many kinds of knots, including those known as Lissajous knots, project to the plane as Lissajous figures.
Visually, the ratio a/b determines the number of “lobes” of the figure. For example, a ratio of 3/1 or 1/3 produces a figure with three major lobes (see image). The ratio A/B determines the relative width-to-height ratio of the curve. For example, a ratio of 2/1 produces a figure that is twice as wide as it is high. Finally, the value of δ determines the apparent “rotation” angle of the figure, viewed as if it were actually a three-dimensional curve. For example, δ=0 produces x and y components that are exactly in phase, so the resulting figure appears as an apparent three-dimensional figure viewed from straight on (0°). In contrast, any non-zero δ produces a figure that appears to be rotated, either as a left/right or an up/down rotation (depending on the ratio a/b).
See more at source: Lissajous curve.


Louisa used to hate math. Then we discovered The Life of Fred and now she LOVES math!
I was highly suspicious as I listened to Louisa laugh through her math lessons and totally enjoy doing them. I was stunned to watch her go straight for math when she started her homeschool work each morning. I picked up the book and read an outlandish story about a miniature boy named Fred and all his adventures. I couldn't imagine she was really learning math, so I asked her to tell me what she had learned so far in the 16 lessons she had completed. She quickly jotted down this list for me:
infinite sets natural numbers integers set notation ratios coefficients . . . (need I go on?)
I was impressed to say the least! How could a girl who hates math actually be enjoying Algebra and really learning the terminology as well as how to do the operations? I contacted Stan Schmidt, the author of The Life of Fred, and a retired college math professor of 28 years whose self-proclaimed mission is to help kids love math. I asked him, "How do I know a student who does Life of Fred will retain it?," to which he replied, "How much of your school algebra did you retain?" Ouch!
The idea behind Life of Fred is that if students have an enjoyable experience and have fun doing their math, they will remember it and use it and have a good taste in their math about the whole subject. The sub-title on each of his math books is "As Serious As it Needs to Be". And that is just the point: math doesn't need to be horrid and dry. This is math—just as serious as it needs to be—and I can tell you from Louisa's smiles and chuckles, that's not very serious at all!
As a bonus, these books are very low priced compared to any other math books on the market. (Compare to Saxon at $50-70)
If you have a creative child who is languishing on traditional math programs, try Life of Fred!
Math, As Serious As it Needs to Be!
Beware: This is not a traditional math book. This is a child-directed course. The student reads the adventure story, does the math problems that occur as a natural part of the story, and checks their answers (the solutions are right there for the looking.) And learns to love math in the process! You will not get the detailed formula explanations that you get in a traditional math book. I am still amazed that kids can read the story and learn the concepts, but they do!
Begin when Math-it or the addition and multiplication facts, plus long division have been mastered—about 5th grade.
Contents: Less Than, Billion, Cardinal and Ordinal Numbers, Diameter and Radius, Savings and Expenses, Definition of a Fraction, Sectors, Comparing Fractions, Reducing Fractions, Adding and Subtracting Fractions, Common Denominators, Roman Numerals, Least Common Multiples, Improper Fractions, Lines of Symmetry, Division by Zero, Circumference, Multiplying Mixed Numbers, Commutative Law, Canceling, Definition of a Function, Area, Unit Analysis, Division of Fractions.
Sample questions from Fractions:
1. Write 99 in Roman numerals. 2. What is the square of two and five-eighths? 3. What is the inverse function to "multiply by six and then add twenty-four"?


International Journal of Industrial and Operations Research
Volume 3, Issue 1
A Statistical Comparison between Different Multicriteria Scaling and Weighting Combinations
Sean Harris 1, Luisa Nino2* and David Claudio2
Table of Content
Figure 1: Example of a single clear....
Example of a single clear winner.
Figure 2: Example of complete ambiguity.....
Example of complete ambiguity.
Table 1: Summary of weighting-scaling combinations.
Table 2: Numeric value example for single clear winner, Problem 5 IV + Rating.
Table 3: Summary of combination scores.
Table 4: Average score by scaling method.
Table 5: Results breakdown by problem.
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- Jurgita Antucheviciene, Zdeněk Kala, Mohamed Marzouk, Egidijus Rytas Vaidogas (2015) Solving civil engineering problems by means of fuzzy and stochastic MCDM methods: Current state and future research. Mathematical Problems in Engineering 2015.
- Athanasios Kolios, Varvara Mytilinou, Estivaliz Lozano-Minguez, Konstantinos Salonitis (2016) A comparative study of multiple-criteria decision-making methods under stochastic inputs. Energies 9: 1-21.
- Saaty TL (2008) Decision making with the analytic hierarchy process. Int J Services Sciences 1: 83-98.
- Triantaphyllou E, Mann SH (1989) An examination of the effectiveness of multi-dimensional decision-making methods: A decision-making paradox. Decision Support Systems 5: 303-312.
- Stelios HnZanakis, Anthony Solomon, Nicole Wishart, Sandipa Dublish (1998) Multi-attribute decision making: A simulation comparison of select methods. European Journal of Operational Research 107: 507-529.
Sean Harris 1, Luisa Nino2* and David Claudio2
1Jake Jabs College of Business and Entrepreneurship at Montana State University, USA
2Department of Mechanical and Industrial Engineering, Montana State University, Bozeman, USA
Luisa Nino, Department of Mechanical and Industrial Engineering, Montana State University, 420 Roberts Hall, Bozeman, MT 59717-3800, USA, Tel: 406-994-7144, Fax: 406-994-6292.
Accepted: May 16, 2020 | Published Online: May 18, 2020
Citation: Harris S, Nino L, Claudio D (2020) A Statistical Comparison between Different Multicriteria Scaling and Weighting Combinations. Int J Ind Operations Res 3:006.
Copyright: © 2020 Harris S, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Multicriteria decision making presents several challenges for researchers. These include selecting a technique that will produce accurate results but not require too much time or resources. Researchers have been comparing different techniques for several years, though a comprehensive study of using many different techniques for the same set of problems is relatively new. Additionally, knowing if the difference in scores between alternatives is significant or not presents another challenge. The use of confidence intervals has recently been employed by researchers to examine whether results are actually statistically different from one another. This study uses 21 different scaling-weighting combinations and confidence intervals on six different decision-making problems to measure their ability to produce unambiguous results. Linear normalization as a scaling technique tends to be the best at identifying one or two clear winners while avoiding complete ambiguity. Despite the variety of combinations used, several common themes emerge across the decision problems.
Multicriteria decision making, Confidence intervals, Scaling-weighting techniques
Multicriteria Decision Making (MCDM) involves deciding on the best alternative from a set of potential candidates. These alternatives often contain multiple and conflicting criteria that can be qualitative or quantitative [1,2]. MCDM methods have been used to aid in decision making for a variety of contexts, such as river basin planning, water resource planning, equipment selection, environmental problems, and other problems involving resource management [3-7]. Evaluating tradeoffs between alternatives can be quite difficult, but MCDM can aid decision-makers in their quest to choose the best alternative .
Numerous MCDM techniques exist for solving decision problems. However, these methods often represent radically different approaches and the selection of which technique to use depends on multiple factors such as the specific problem, the decision maker's familiarity with the technique, and computational efficiency, among others. In fact, selecting the most appropriate MCDM method for a given problem could actually be considered a MCDM problem on its own .
Most MCDM methods involve determining the importance of each criterion and assigning weights based on the Decision Maker's (DM) preferences . There are many different techniques for weighting criteria [9-11]. The scaling of criteria presents another challenge because different criteria are rarely measured with the same units. Consequently, a number of different scaling techniques exist to address the best way to convert the criteria into like units . Recently, researchers have been to experiment with different weighting and scaling combinations in an attempt to gain insights into which combinations tend to produce the best results .
A major concern with decision making is that different MCDM methods can provide different results for the same problem. In fact, the best alternative according to one method may be the worst alternative according to another [1,9]. It can be difficult to discern if there really is a significant difference between the highest scoring and second-highest scoring alternatives. Recent research has explored this concept of looking at how different weighting and scaling combinations produce different results for the same problem . Another important consideration is what method yields results that are closest to the DM's naturalistic choice . In looking at how different MCDM methods yield different results in both orders of alternatives and magnitude of difference between those alternatives, an interesting research question presents itself: Do certain weighting and scaling combination in MCDM tend to produce statistically similar results? This would be of great use to DM's because if two different MCDM techniques produce statistically similar results, then the DM can select the technique that is easiest for them to use, whether it is because of available data, computational complexity, or time needed, among others.
Until recently, little research has been done to address this question. However, the use of confidence intervals can help since they give the range in which a result is valid . The idea of using confidence intervals in decision-making techniques is relatively new. Statistical tests can then be used to verify if the mean scores are significantly different from each other, which in this case means that the alternatives can be ranked based on the value of the mean score. Otherwise, if the difference between the mean scores of the alternatives is not significantly different, then the ranking of the alternatives overlap .
Combining the ideas from  and  can then provide an answer to the proposed question. We conduct a comprehensive study where 21 different scaling-weighting combinations are applied to six different problems and then confidence intervals of the mean scores are calculated for those results. Tukey's HSD test is then used to test whether the alternatives are statistically different from one another. The combination of weighting and scaling methods is one of the unique aspects of our research because, to our knowledge, only  have studied this in the past. The other unique aspect is the use of confidence intervals to evaluate and compare the outcomes of the different methods, which, to our knowledge, only  have also done.
We use the data from  that has the results for six participants giving criteria preferences for six different MCDM problems, where each problem was composed of 6 different criteria. The scaling methods used are ideal values, linear normalization and vector scaling (IV, LN, L1 Norm, L2 Norm, L3 Norm, and L∞ Norm). These scaling methods were paired with the following weighting methods: rating, ranking, analytic hierarchy process (AHP), L1 Metric, L2 Metric, L3 Metric, and L∞ Metric when applicable. Then, we calculate 95% confidence intervals of the mean score of the alternatives and analyze how often the results are actually statistically different as was proposed for . We could then see if certain combinations tend to produce the same results and what these combination families look like. We evaluate these combinations according to their ability to clearly distinguish the best alternative and to avoid complete ambiguity.
MCDM helps identify and assess a set of conflicting criteria and alternatives in decision making. In classic multicriteria decision environments, there is no universal agreement as to what constitutes the most preferred or satisfying solution . The general objective of MCDM methods is to provide a framework and technique to order the set of possible alternatives in accordance with the DM's preferences. The specific objectives to be reached by several MCDM methods need to be identified; in addition to what degree the attributes meet these objectives .
Numerous studies have been carried out to compare various MCDM methodologies [13,15-18]. However, the main question that emerges is why outcomes differ when different techniques are applied to solve the same decision-making problem, even under the same assumptions [1,9,18]. These differences have appeared in the literature, but should not be viewed as inconsistencies [9,18]. Rather, the results of these studies indicate that different algorithms, scaling methods, and uses of weights lead to different outcomes. These differences illustrate the need to know how weights, scales, and other factors are used before applying a technique [9,18]. Therefore, it is vital that users of such techniques be aware of the manner in which results are reached and how their weightings and scaling factors influence these outcomes . Though it is known that different MCDM methods can yield different results, the analyst must still decide which one to use. Often, the method is one that was developed by the DM, one the DM has the most faith in or a method that the DM has used in the past .
In addition to not knowing what MCDM method is best for a particular problem, there is also a stochastic element to the criteria and alternatives that further complicates the matter. Researchers have studied the uncertainly surrounding MCDM through simulation . For instance, by using a Monte Carlo simulation,  studied the flexibility to vary all of the weights of MCDM method while at the same time changing the functional form of aggregation. The authors analyzed the changes using statistical parameters such as mean, mode, standard deviation, and quartiles to provide recommendations about the correct procedure to assign and assess the weights of the criteria. Also, they worked with the weights of the criteria and used statistic distributions to analyze and compare the data.  demonstrated that the functional form of aggregation (multiplicative or additive) had no effect on the outcome of the model.  Worked with six deterministic MCDM methods doing a comparative analysis to develop a new methodology that allows the use of those methods in decision-making problems with stochastic inputs. The main objective was to assign confidence levels on the use of certain methods based on the resulting outputs.
Regardless of the MCDM method selected or the uncertainty surrounding the problem, most methods require the establishment of criteria weights that capture the DM's preferences for how important each criterion is relative to the others . Though there are numerous weighting methods, this paper uses the same as . The first method is the rating method. Here, the DM rates from 1 to 10 each criterion independently, usually with 1 being the least important and 10 the most important. The second method is the ranking method, in which pair wise comparisons of criteria are performed and the weights are obtained using Borda Count . The third method is the Analytic Hierarchic Process (AHP). AHP assigns a magnitude of the preference of one criterion over another. AHP uses a standard scale number from 1 to 9, where 1 means no preference and 9 means that criterion is extremely preferred over another . Then, ratio scale priorities are established based on the sets of pair wise comparisons performed between the criteria and alternatives.
Another aspect that plays a role in MCDM problems is the type of decision problem. In single-dimensional MCDM problems where all the units are the same, calculating scores for the alternatives is relatively straightforward. However, difficulties arise in multi-dimensional cases where the additivity utility assumption is violated because it combines different dimensions and thus different units so the result is comparable to adding apples and oranges . Therefore, scaling methods are needed to preserve the additivity utility assumption to solve multi-dimensional MCDM cases.
Several methods exist for scaling. These include ideal values, linear normalization, and vector scaling. Ideal values use the best criteria values among the alternatives. Ideal values denote the best criteria values that are possible to obtain among the alternatives. In linear normalization, the maximum and minimum criteria values are used to scale the data . Vector scaling uses LP Norm to scale the data. LP Norms quantify the lengths of vectors and LP metrics calculate the distance between points. The LP metric can be used as a method to obtain the best alternative once the data is normalized by using the vector scaling method. Velazquez, Claudio  presented the LP metric between two vectors x, y as follows:
Where x, y ε Rn
If one of these points is the ideal solution, then the distance denotes how close each alternative is respect to the ideal solution. Therefore, the smaller the distance, an alternative will be closer to the ideal solution .
Once the weights of the criteria have been established the next step to solve a MCDM problem is to find a way to process the numerical values to compare and determine the ranking of the different alternatives. One of the most commonly used approaches is the Weighted Sum Model (WSM), which is governed by the addition utility assumption . The WSM is presented by  as follows:
In a MCDM problem with m alternatives and P criteria. Let the matrix represent the criteria values to be maximized where fij is the value of criterion j for alternative i. If wj is the normalized weight of criterion j assessed subjectively such that wj ≥ 0 and ∑wj = 1, then the weighted sum model computes a score Si for each alternative i, as follows:
Then, the ranking of the alternatives is created based on the Si scores and the DM selects the best alternative.
Researchers have evaluated the accuracy and efficiency of weighting and scaling methods independently [9,11,12,16,24]. However, little research has been conducted to identify the combination of scaling-weighting methods that will better match the real preferences of the DMs .  Experimented with various weighting and scaling combinations for six participants conducting six different decision-making problems. The results were compared to the DM's naturalistic choice in addition to the majority of the combination methods.
 Compared different weighting-scaling combinations, however, no testing was done to determine if the methods were significantly different from a statistical perspective. In fact, using confidence intervals to compare different MCDM techniques is relatively new.  Applied confidence intervals to the fuzzy AHP and fuzzy Analytical Network Process (ANP) methods by using the mean obtained to calculate the arithmetic standard deviation. Then, the mean and standard deviation were used to calculate the 95% confidence interval . The confidence intervals help to identify any overlaps between the alternatives and therefore provide insight as to whether they are statistically different, which allows for a more informed ranking of the alternatives. Observing if there is a statistical difference between alternatives is the main advantage of confidence intervals because conversations can be initiated regarding tradeoff among the available alternatives. This also helps guarantee the selection of the best alternative since its performance presents a statistically significant difference against the other alternatives.
This paper combines the primary ideas from  and  to statistically compare different MCDM weighting and scaling combinations. While previous researchers have used stochastic inputs or simulation to examine the uncertainly surrounding MCDM problems, this paper uses confidence intervals calculated directly from DM preferences. While  compared different MCDM weighting and scaling techniques, it was hard to draw meaningful conclusions from the results because there was no statistical testing to determine if two different techniques truly were different from one another. Similarly,  provides the framework to compare different MCDM techniques, but this paper will use a more comprehensive set of MCDM techniques.
Material and Methods
This paper used the data from , in which six participants, half female and half male all between ages 24-27, were independently given six different MCDM problems. The problems included buying a sport utility vehicle (SUV), buying a house, selecting a restaurant, buying a laptop computer, buying airplane tickets for a trip to Italy, and selecting a job. Each problem had five different alternatives and six different criteria within each problem. See Appendix A for more details on these problems.
Seven different weighting techniques and six different scaling techniques were used in different combinations to determine the optimal alternative(s) for each problem. The weighting techniques were rating, ranking, AHP, L1-L3 metric, and the L∞ metric. The scaling techniques were ideal values, linear normalization, L1-L3 norms, and L∞ norm. These techniques were calculated as previously described in the literature review. This paper used the same 18 combinations as . In addition, three other combinations were examined for a total of 21 combinations. These additional combinations were included after an initial analysis of the data suggested that linear normalization was the best scaling method since it yielded the highest total scores regardless of the weighting method used. Table 1 summarizes the combinations used in the study. New combinations, or those that were not used in , are denoted with an asterisk.
For each problem, the total score using the weighted sum technique was estimated for each alternative for each participant. It was decided to use the weighted sum technique since, as it was proved by , the form of aggregation (multiplicative or additive) had no effect on the outcome of the model. Anderson-Darling tests were conducted for each of the 21 scaling-weighting combinations, per MCDM problem, per alternative to determine whether the six participants' scores had a normal distribution. The tests yielded p-values greater than 0.05. Therefore, normality was assumed to calculate confidence intervals. Then, a 95% confidence interval of the mean score of the alternatives was calculated using the arithmetic mean and standard deviation of the six individual participant scores. The ranked order of alternatives according to the weighted sum calculations was done for each participant for each problem, with the highest scoring alternative being one and the lowest being five. The same methodology was applied to the mean of the ranked order of alternatives. The corresponding 95% confidence interval using the geometric mean was then calculated for each alternative, for each problem, and each of the weighting-scaling methods.
As  noted in their paper, the confidence intervals of the differences between alternatives for a given problem often included zero. This was the same case for our study. Therefore, a particular weighting-scaling combination's efficacy was measured by its ability to distinguish a clear winner(s) among the alternatives. Ideally, a MCDM technique yields one clear winner that is statistically significantly better than all other alternatives, an example is shown visually in Figure 1 and then by the corresponding t and p values in Table 2. Conversely, a MCDM technique where the highest scoring alternative is not statistically significantly different than the lowest-scoring alternative (complete ambiguity) is the least preferred result because no alternatives can be eliminated from consideration. Figure 2 shows an example of this complete ambiguity.
The particular weighting-scaling combination's efficacy measure was obtained through the following procedure: Once each weighting-scaling combination produced the results for the six different problems. Three metrics were recorded. The first was the number of times the combination produced a single winning alternative that was significantly different from the other four alternatives (for example, Figure 1). The second metric was the number of times the combination produced two winning alternatives that were significantly different than the other three alternatives (but the two winners were not significantly different from each other). The third and final metric was the number of times a combination produced a completely ambiguous result where the highest scoring alternative was not significantly different than the lowest-scoring alternative (for example, Figure 2). A clear winner was awarded three points, two clear winners were awarded 1 point, and a completely ambiguous result was given negative three points. These point totals reflect how effective each result is because a single clear winner is much preferred to two clear winners, where additional analysis is required to select among the two remaining alternatives. Similarly, a completely ambiguous result is fairly useless and requires the DM to redo the entire process. The total number of points or total score for each weighting-scaling combination were then calculated and compared to one another.
Three other scoring methodologies were considered to determine the robustness of the proposed 3, 1, -3 system. First, a binary system of one point for one clear winner, zero points for two clear winners, and negative one point for complete ambiguity. Second, a quadratic system rewarding clear winners where four points were given for one clear winner, two points for two clear winners, and negative one points for complete ambiguity. Finally, a reverse quadratic system penalizing complete ambiguity with two points for one clear winner, one point for two clear winners, and negative four points for complete ambiguity.
Results and Discussion
Table 3 summarizes the results of the combinations scores. When looking at the weighted sum scores, the linear normalization with L1 metric achieved the highest score, followed by linear normalization with L2 metric and linear normalization with rating. L3 Norm with ranking, L2 Norm with ranking, L∞ Norm with ranking, and L1 Norm with ranking all tied for the lowest score. These results contrast with the results from , where L∞ Norm with ranking was found to be the best method. For the mean of the alternative rankings, the linear normalization with L1 metric and linear normalization with L2 metric were first and second, respectively, while L1 norm with L1 metric had the lowest score.  Did not have as clearly defined results when evaluating the combinations in a group context, where 7, 12, and 13 of the methods tied for first according to their three performance metrics.
Table 3 also illustrates that the scaling trends are relatively robust to the scoring system. Linear normalization remains the best scaling method in five of the six other scenarios, with it being second best to L∞ norm for the mean of alternative rankings with the reverse quadratic scoring methodology. This suggests that the chosen scoring methodology does not have much effect on determining the best scaling method.
When looking at the combinations by scaling method, a few trends emerge. Table 4 summarizes these results. Linear normalization clearly performs the best among the six scaling methods used with the proposed evaluation method for both weighted sum and ranked alternatives. The L1, L2, and L3 norms all perform in a similar manner. These three norms all have values between -3.67 and -4.67 for the calculated score. The L∞ norm follows a similar trend, but with slightly higher numbers. Linear normalization having the highest scores makes intuitive sense because the method assigns values between zero and one, with the lowest alternative having zero and the highest having one for each criterion. Therefore, this method would better separate alternatives than ideal values, where the alternatives may have very similar scaled values if the raw data is close to one another.
Trends by problem also appear in the data. Table 5 summarizes the result breakdown by problem. Certain problems tended to have similar themes throughout the different combination methods. For example, problem 1 (buying a house) and problem 3 (selecting a job) rarely had a single or two clear winners. Problem 1, however, had complete ambiguity more than twice as often as problem 3 for weighted sum scores. Conversely, problems 5 and 6 had a single or two clear winners the vast majority of the time, with problem 6 having a single or two clear winners 21/21 times for ranked alternatives. Interestingly, the same two alternatives (1 and 6) were always chosen for problems 5 and 6, whether it was one of those alternatives being a single clear winner or the two alternatives being the two clear winners. For problem 4, on all six occasions that the weighted sum produced a single clear winner, it was the same alternative (alternative 2).
These results suggest that the type of problem, or particular details of the problem, do play a role in whether MCDM methods will be able to clearly distinguish the top alternatives for group decisions. Furthermore, there was never a single instance in which one alternative was the statistically worst choice under one method and the statistically best choice according to another method. This is somewhat encouraging, as previous research has sometimes found this dynamic to occur. This previous research, however, did not include whether the choice reversals were statistically significant or not.
There are some differences when considering the weighted sum score or a ranking of the alternatives by the mean. The weighted sum score tends to produce complete statistical ambiguity more often than when the alternatives are ranked and aggregated according to the mean. This finding aligns with the previously mentioned finding that the different methods tend to select the same alternatives. This produces similar rankings across the board, which then makes it less likely to have ambiguity because there is a bigger distinction among the alternatives. For example, the weighted sum score will often only have a difference of a few tenths between all the alternatives. Therefore, even if the methods tend to find similar results, the small numerical differences can result in ambiguity. Conversely, when using the rankings of alternatives, the top alternatives tends to have a value closer to one and the worst alternative close to four or five, making ambiguity less likely.
Researchers have been comparing different MCDM techniques with one another for many years. Comparing techniques' results against a statistical backdrop to determine if the results are significantly different is relatively new. This study used 21 different scaling-weighting combinations of common MCDM techniques and applied them to six different MCDM problems with six participants. Both the weighted sum scores and the rankings of the alternatives were aggregated into a group score and the corresponding confidence intervals were calculated to test if the results were statistically significantly different. The scaling-weighting combinations' ability to distinguish a clear winner or two clear winners and to avoid complete ambiguity in the results were measured.
Among scaling techniques, linear normalization performed best at distinguishing clear winners. This is most likely because of how the technique spreads out the values all the way from zero to one amongst alternatives, whereas other techniques do not. Trends emerged from the results by both problem and the type of aggregation, where the weighted sum scores had much more ambiguity than the ranking of alternatives with mean.
There are some opportunities for future research surrounding this issue. Several of the most common MCDM techniques were used; however, others such as TOPSIS, Delphi, and SPECTRE were omitted and could be included. The experiment could be replicated with a larger sample size to see if any trends by participant demographic emerge. Likewise, the type of problem could be examined in more depth. For example, the topic (purchasing a product, selecting a type of service, life decision such as moving to a new city) or structure (number of alternatives and criterion) could be systematically varied to determine if any trends emerge. Finally, the results of the study provided information that helps to evaluate the positive and negative characteristics of different MCDM techniques. This information could also be useful to face the challenges that are emerging in areas such as automation, social sciences, environmental sciences, modern construction management, and software packages, modeling and recent advances in intelligent decision making.
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.


We standardize each measure within each state to the average of counties in that state. Recall that our measures are in a number of different scales—some are percentages, some are rates, some are averages of survey responses, or other metrics. Standardizing each of these measures transforms them to the same metric—a mean (average) value of 0 and a standard deviation (measure of spread) of 1. We refer to these as Z-scores where:
Z = (County Value) - (Average of Counties in State) (Standard Deviation of Counties in State)
Each Z-score is relative to the other counties in that state—not compared to an absolute standard—and shown in the metric of standard deviations. A positive Z-score indicates a value higher than the average of counties in that state; a negative Z-score indicates a value for that county lower than the average of counties in that state. For example, if a county has a Z-score on a measure of 1.2 that means the county is 1.2 standard deviations above the state average of counties for that measure. For counties with a population of 20,000 or less, any z-score that is < -3.0 or > 3.0 is truncated to -3.0 or 3.0, respectively.
For most of the measures, a higher Z-score score indicates poorer health (e.g., years of potential life lost before age 75). However, for some of our measures (e.g., high school completion) a higher score indicates better health or a more desirable value. We have to take this into account before computing summary scores. For these measures we compute the Z-score as usual but multiply it by -1, so that higher scores indicate poorer health. The measures that we reverse code in this manner are:
- Food environment index
- Access to exercise opportunities
- Flu Vaccinations
- Mammography screening
- High school completion
- Some college (post-secondary education)
- Social associations
The scores we compute are weighted composites of the Z-scores for individual measures where the weights represent relative importance of the different measures. A weighted composite is computed by multiplying each Z-score by its weight and adding them up. Below is the formula we use for our weighted composite scores:
In this formula the Zi values are the Z-scores of the measures used to compute the summary score. The wi values are the weights applied to each Z-score. The ∑ sign simply means to add up all the Z-scores multiplied by their weights.
All of the summary scores we compute use the formula above, standardized Z-scores for each measure (reverse coded when necessary), and the weights described in previous sections. Remember that we always compute composite scores separately by state.
After we compute composite scores we sort them from lowest to highest within each state. The lowest score (best health) gets a rank of #1 for that state and the highest score (worst health) gets whatever rank corresponds to the number of units we rank in that state.
It is important to note that we do not suggest that the rankings themselves represent statistically significant differences from county to county. That is, the top ranked county in a state (#1) is not necessarily significantly healthier than the second ranked county (#2).
Beginning in 2021, we have displayed data in quartiles on the snapshot to support comparison of a county to a similar grouping in your state, instead of comparing difference in individual ranks which might not be statistically meaningful. Looking at the underlying data and how those data have changed over time provides a better picture of your county's progress.
Health Outcome and Health Factor rankings are grouped into four equally sized groups (quartiles), ranging from the least healthy to healthiest counties (Lowest 0-25%, Lower 25-50%, Higher 50-75%, or Highest 75-100%) within each state. A county with a rank of #1 lies in the Healthiest (Highest 75-100%) quartile.
For additional information about the rationale behind our rankings, see our publications.


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National Mathematics Day is celebrated on 22 December every year to mark the birth anniversary of legendary Indian mathematician, Srinivasa Ramanujan.
December 22 was designated as National Mathematics Day (NMD) by Government of India in 2012.
The celebrations aim at spreading awareness about the importance of mathematics and the contributions made by Srinivasa Ramanujan, in the field of mathematics.
Srinivasa Ramanujan was born on 22nd December, 1887 in Erode, Tamil Nadu. At the age of 12, he had gained knowledge in Trignometry. At the age of 15, he obtained a copy of George Shoobridge Carr's Synopsis of Elementary Results in Pure and Applied Mathematics. In 1911 Ramanujan published the first of his papers in the Journal of the Indian Mathematical Society. He was awarded in 1916 the B.A. Degree by research of the Cambridge University. He was elected a Fellow of the Royal Society of London in Feb. 1918.
He had spent his childhood days in great difficulty and poverty. He used to borrow books and read books from friends in school. When he was young, he took up the job of clerk to meet the economic needs of the house. During his leisure time, he used to solve mathematics questions and worked on various types of theorems. He died on 26th April, 1920 at a very young age of 32.
Srinivasa Ramanujan hailed as an all-time great mathematician, for his natural genius, has left behind 4000 original theorems, despite his lack of formal education and a short life-span.
1729 is also known as the Ramanujan Number. When the great Indian mathematician, Ramanujan was in England, he once became very ill. Prof. Hardy met Ramanujan in the hospital and stated Ramanujan that 1729 seemed to be a very dull number, and hoped it doesn’t turn out to be an unfavourable omen. But Ramanujan in contrary said it is a very interesting number, 1729 is the smallest number which can be written in the form of sum of cubes of two numbers in two ways, i.e. 1729=1³+12³=9³+10³. Since than the number 1729 is called Ramanujan’s number.
Last Modified : 7/4/2023


Because I am too busy blowing out candles today, my tall, dark, handsome, and incredibly geeky fabulous boyfriend took it upon himself to “guest blog” for me. Here is the product, wherein he demonstrates how I am in the square prime of my life. Enjoy!
* * *
While lying in bed with my sexy girlfriend on the morning of her birthday, I noted that her age is now the product of a prime number and a square. For a review for those who haven’t thought about math since Mr. Johnson kicked you out of Analytical Geometry for being drunk (oh wait, that was me), a prime is a number only divisible by one and itself (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 …), and a square is the product of a number times itself (4, 9, 16, 25, 36 …)
Of course, knowing her own age, she was able to determine the numbers in question. Let’s call them p1 and s1. Without having figured out the answer yet, I postulated, “When is the next time that will happen?” Giving up all too quickly to befit her geek creds, she said she didn’t know. When I told her it would happen next year, she thought about it for a few seconds and said, “You mean p2 and s2?”
Now p2 is a prime, and s2 is a square, but the product of them is not her age next year. The sum of them is! “Wow!” I thought, that’s really interesting, next year is her age not only the product of a square and a prime, but also the sum! (So there is also p3 and s3, which were the ones I thought of that work if you multiply them.)
So after a reasonable amount of birthday canoodling, I get up to take a shower. And of course the brain is still cranking away with squares and primes, when I realize that next year, there will be another Square Prime set where the sum will be her age, AND this year there is a set where the sum is her age! When will the madness end?
So for those who always thought the hardest part of the story problems was putting them into equations, I’ll do that part for you:
p1 * s1 = Age this year
p2 + s2 = Age next year
p3 * s3 = Age next year
p4 + s4 = Age next year
p5 + s5 = Age this year
One of the really neat things about this is that it works for you too! That is, when you are Jodi’s age, all of the figures will still work. But it is unlikely that any of us will work our figures the way Jodi does, nor have a figure that works so well.
Please have your answers submitted by the beginning of class next Friday.


|Luogo di studio
|Laurea Magistrale, a tempo pieno
|2 years. For each year of enrollment, students may choose between full-time or part-time status. Tuition fees vary accordingly. (120 ECTS)
|Lingua di studio
|2.800 € all'anno
For the academic year 2023/2024 the tuition fees ranged from 156€ to 2.800€ per year, depending on the student’s financial situation.
For more information on the tuition fees for the a.y. 2024/2025, please refer to the dedicated webpage on our University portal. At the same page you’ll also be able to verify if you are eligible for a fee reduction or exemption.
|Qualifiche di accesso
Diploma di Laurea (laurea di primo livello)
You must hold a first-cycle degree (i.e. bachelor’s degree), gained after at least 3 years of University education, in either Mathematics, Statistics, or Physics. You will also need to meet specific admission requirements and demonstrate a suitable academic preparation and a solid background.
See Admissions for details and for a suggested syllabus.
I documenti di studio sono accettati nelle seguenti lingue: inglese / francese / italiano / spagnolo.
Se i documenti sono rilasciati in una lingua diversa da quelle precedentemente elencate, oltre all’originale è necessario allegare la traduzione ufficiale certificata.
|Requisiti in base al territorio
This degree program has 50 available places for non-European applicants residing abroad.
Your knowledge of the English language (at least B2 level) will be verified during the admission procedure.
È necessario aggiungere una lettera di motivazione alla propria candidatura.
The Master’s Degree in Stochastics and Data Science aims at preparing students with a solid and modern education in probabilistic and statistical methods. Core topics will be advanced methodologies and techniques that are nowadays required tools for performing mathematical modelling and analysis under uncertainty (Stochastics) and for analysing and being able to interpret high-dimensional and complex data structures (Data Science). The emphasis of the course topics will be both theoretical and oriented towards computation and applications, providing not a “book of recipes” but the capability of discerning, combining and applying the most useful and modern methods available in Probability and Statistics.
The Master’s Degree in Stochastics and Data Science draws inspiration from the best graduate programs in Data Science recently started in the USA and Europe, and is currently the only offer of this type in Italy.
Gaining solid skills related to the above outlined topics will enable to deal with and model data collected in a variety of disciplinary fields by exploiting a deep mathematical understanding of the underneath structures. Graduates in Stochastics and Data Science will have a solid knowledge in key topics from Applied Mathematics, Probability and Statistics combined with computational skills essential for modern interdisciplinary applications. Such preparation allows to use this theoretical knowledge for concrete tasks, for example for: autonomously formulating complex probabilistic models for describing static and dynamic phenomena of interest; developing the necessary mathematical and statistical tools for their analysis; using these models together with sets of available data to perform estimation, forecasting and uncertainty quantification of phenomena under study; designing and implementing computational strategies in the form of algorithms for concretely carrying out the statistical procedures.
Such skills are nowadays highly demanded in a variety of professional environments, both in the private and in the public sector.
Alternatively, those more interested in proceeding with PhD studies, will have gained a solid background which allows to enter programs in Statistics, Mathematics, Applied Mathematics, Operations Research, Computer Science, Economics and Mathematical Finance, among other topics.


Should you give customers prompt-pay discounts? When suppliers offer discounts to you, should you take them even if you must borrow? Here’s how to do some quick math with your own real numbers to get the answers.
Customer Discounts – When cash flow is tight and receivables are slow, offering a discount to your customers for prompt payment may appear to be an attractive solution to the problem. In most cases, however, discounts are an economically unwise choice. Here’s the math.
To make the calculations simple, use a nice round sales figure such as $100,000. Do the following:
1) Calculate the cost of the discount by multiplying the discount percentage times the sales amount. A 1% discount on $100,000 equals $1,000.
2) If you borrow money on a line of credit, calculate the daily cost of borrowing money. In this case, we would need to borrow $100,000 and we’ll assume a 9.5% borrowing rate of $9,500 per year. To get the daily interest cost, divide $9,500 by 365 days for a daily cost of $26.03.
3) Calculate the number of days the customer would need to withhold payment to equal the discount. In this case, we divide $1,000 (the discount) by $26.03 (daily interest cost) for a result of 38 days.
What this calculation tells us is that the customer must delay payment for more than 38 days for our discount to be cost effective! In all likelihood, most of your customers are not that late and the ones who are probably wouldn’t have the cash to be able to take advantage of your discount!
If your company is in the fortunate position of not borrowing money, perform the same calculation, but use your savings rate of interest in lieu of a borrowing rate. For instance, at a 5% savings rate, the daily lost interest is $13.70 ($100,000 x 5% divided by 365). If you give a $1,000 discount, now a customer would have to be 73 days late for you to break even!
Sometimes it’s easier to think of discounts in terms of annualized equivalent interest costs. Here’s how two popular discounts stack up:
1%/10, net 30 = 18.4%
2%/10, net 30 = 37.2%
Notice that a 2% net 10 discount is equivalent to paying 37.2% in interest. Ouch!
Before you rule out discounts entirely, however, there is one time when they can make economic sense. If you can raise the price of your products by the discount amount or more, and then get your money in the door faster with the perceived discount, you have a winner!
Supplier Discounts – As you may suspect by now, your company is almost always ahead of the game by paying suppliers early to gain their discount. We use the same calculations to check out our savings:
1) Calculate discount amount.
2) Calculate daily cost to borrow the funds (or lost savings rate).
3) Divide discount amount by daily cost of borrowing/lost savings to determine breakeven point.
4) Pay supplier early if breakeven days are in excess of supplier non-discount terms. (For instance, with a $100,000 purchase, a supplier discount of 1% net 10 and a 9.5% borrowing rate, as long as supplier normal terms are shorter than 38 days, you should take advantage of the 1% discount.
This example of a clean and simple net 10 is not what stumps marketers. It’s when the terms get a little trickier that the math gets a bit more difficult. For instance, if you sell home heating oil burners, lubricants or tires, you may see some interesting terms out there. There may be a 30/60/90 day payment option with a 2% net 10 or 1% net 30 offer. Don’t let the terms deter you from doing the math.
Again, start with calculating each discount. Then calculate your cost of borrowing if you took advantage of the shorter terms.
The math is a bit more complicated in this example because the normal supplier terms call for a $33,333 payment at the end of each month. Therefore, you need to calculate the daily interest cost for $100,000 and for $66,666 and for $33,333. You’ll also need to be careful that you have the right number of days calculated at each dollar amount.
After you do one or two of these calculations, you’ll find that even the most complex supplier payment offerings take you less than 10 minutes to calculate and decide. That little bit of time spent may save your company tens of thousands of dollars each year!
Once you see the cash and profit benefit of trade discounts, get a little more aggressive with requesting discounts from vendors. The worst they can do is say no. Nothing ventured, nothing gained!
So, in summary, the general rule of thumb with discounts is take them, but don’t give them.


NOW ENROLLING - Holy Rosary School is happy to announce its new program, the Little Dominicans Preschool Learning Center, for the 2017-2018 school year. This program is available to preschool-aged children (ages 2-3 years old).
As we look forward to summer vacation, we want to remind you how important it is for students to keep up with their Math and Reading skills. Check out the summer Reading and Math packets available for students to complete over their summer vacation.
Read More »
Is your child interested in learning how to play a band instrument? If so, MUSE Band is accepting 2017-2018 enrollment applications for children in grades 3-8. Register now to reserve a space.


To study the vibration of an acoustic string under tension.
Primary language: Fortran.
We considered a string of length L and a known linear density ρ(x), which is fixed at both ends and subjected to a tension of intensity σ. The following differential equation allows us to describe the vibration of the string:
The Dirichlet boundary conditions are: y(0) = y(L) = 0 and y(x) represents the position of a material point of the string on the y-axis.
To solve this problem, we must look for solutions of this differential equation, which are the proper pulsations ωk and the eigenmodes yk(x).
The numerical approach that we use is the finite difference method which first discretizes the field of study in N regular intervals with a step of h = L and then it approaches the N derivatives of y(x) with linear combinations. The second derivative of y(x) can therefore be rewritten:
With these two equations and the Dirichlet boundary conditions, we can write our system of N equations in matrix form:
where [M] is a positive definite symmetric matrix and [I] is the identity matrix.
Programmed Methods :
- Matrix Generator of [M]
- Descent Method
- Ascent Method
- LU Decomposition Method
- Cholesky Decomposition Method
- Inverse Power Method - Cholesky Decomposition
- SOR Method
- Inverse Power Method - SOR
- Power Iteration Method
- Deflation Method


So the puzzler was to determine what circumference pine tree would be required to obtain a 12" square Cant? As with all technology I told my students that they should get credit for taking a foreign language. I mean after all if we learn "this language" won't we communicate, working and doing business with new people in a different culture and become productive citizens?
Cant is a sawmill term used to describe the first timber piece that is sawed from the log as "slabs" are cut off. You have heard of the saying, "You can't put a round peg into a square hole?" Sounds like an Odyssey of the Mind (OM) challenge if you ask me. If you think about it all you have to do is cut off the sides of the round peg (log) to make it a nice square and the problem is solved! Great care needs to be taken when positioning a tapered 10, 12, 14... foot long log on the sawmill to get the largest square Cant from each log. Then the sawyer must start critically thinking of how to turn the log and make the best cuts to get the desired boards from each Cant. More complex than it looks. All done with computers by the "real" lumber companies. Maybe I'll promote the foreign language thing by adding a technical word or two to each new post.
It would require a 37.7" circumference log to yield a 12" square Cant.
It took me years to realize that the "check mark" on my Casio calculator was for square root. I am a D- math person and I have the grades to prove it! The only thing I am worse at is spelling.
Some formulas and short cuts that could be used for the calculation -
- Square Cant size wanted * 3.14 = required tree circumference needed
- Diameter of Log you have * .707 = Maximum square Cant size that can be cut from this log
Puzzler - Calculate the total number of Bd ft of wood (not just the Cant) in a log 8" in diameter and 10' long?
Tech Word - Wane


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parachain_bond_fund = total_reward * 0.3 # Amount of PONT that goes to Parachain Bond Fund.
collators_reward = (total_reward) - parachain_bond_fund # Amount of rewards that would be shared between collators.
collator_reward = collators_reward * generated_blocks_fraction # How much reward goes to the collator based on the amount of blocks generated by the collator per total blocks generated in that round. This calculation is done for each collator.
commission = collator_reward * 0.2 # How much commission collator earned.
reward = collator_reward - commission # How much reward is going to be distributed between collator and nominators.
pay_to_collator = commission + (reward * stake_fraction) # How much PONT to pay to the collator.
pay_to_nominator = (reward * stake_fraction) # How much PONT to pay to each collator nominator.


How many gigabytes are you? Are us? This guy did the math and says that our genetic code only takes 1.5 gigabytes of data, while our complete body takes about 60 zettabytes of data. But since we share about 99.9% of our genetic code with everyone else, and each of our cells contain our entire genetic code, what separates us from another comes down to a measly megabyte of data. Watch the video, it’s pretty mind blowing.