Statistics and statistical methods are of two basic
Descriptive statistics summarize some facet of a
complete population. They are used when an entire population is small or easy enough to
measure. For example, the average height or weight of everyone in your family is a
descriptive statistic. Because all members of the population are included in the
calculation, the result is a totally accurate, and thus completely reliable, measurement.
Inferential statistics are used to predict or infer
something about a very large population by measuring samples, or subsets, of that
population. This is done when it is virtually impossible, or prohibitively expensive, to
obtain data about all members of a particular population.
Many of the statistics we normally come in contact with while
reading the paper, watching TV, or talking to colleagues are of the inferential variety.
Examples include the number of people projected to carry the HIV virus in 1998, the
average growth rate of maple trees, and the odds of incurring a side effect when taking a
These types of statistics are thus used to make far-reaching
policy decisions regarding everything from the number of street lights needed per city
block, to the level of funding allocated to school lunch programs, to the amount of money
spent to protect the grizzly bear population of the Western United States. Thus, it is
critical that we develop a good understanding of how best to use, and not abuse,
The concept of proportion allows us to compare relative differences in size, quantity,
etc. between or within samples. The emphasis is on relative, because we don't know
the absolute difference or the magnitude of that difference.
Proportion can be measured as a...
...ratio. If there are 10 girls and 5 boys in the
choir, the ratio of girls to boys is 2 to 1.
...percentage. The choir can also be described as 67%
female and 33% male.
Percentages also play a major role when trying to determine
the "odds" that an event might happen. If it is discovered that 7 out of 10 mice
are brown, we can infer that a.) 70% of mice are brown, or that b.) there is a 70% chance
that the next mouse we see will be brown. This idea leads us nicely into the concept
When it comes to statistics, the term probability is used to describe...
...the likelihood that an event will or will not happen.
(There is a 90% probability that it will rain tomorrow.)
...the degree of certainty regarding the relationship of
two or more variables. (90% of a tree's growth rate can be explained by the amount of
rainfall received during the spring and summer months.)
...the level of confidence that what you think is real
actually is real. (We are 90% certain that half of the children that will be
born next year will be female.)
Generally speaking, the larger the sample size, the higher the level of probability that
the statistics actually mean what they say they mean. For example, using a sample of 10
people to draw inferences about the behaviors of a million people is not a smart thing to
do, since there is only a small probability that the sample is representative of the whole
population. However, a sample of 300 may be more than adequate to allow for statistically
Watch out when samples that are of
reasonable size become subdivided. This situation occurs fairly frequently when
analyzing survey results, as there is a tendency to learn as much as possible about very
specific (but usually very small) sub-samples. This type of analysis is usually referred
to as a "cross-tabulation."
Say that out of 300 people asked about their eating habits, 15 indicate that they are
vegetarians. From this we can reasonably infer that about 5% of the population eats no
meat. (So far, so good.)
What if we ask those 15 vegetarians to name their favorite
sport and 60% say "soccer." Does this mean that most vegetarians prefer soccer?
No! We only asked 15 people, a sample size that is far too small to give a reliable
result. To make this claim, we would have to start again by asking the question of enough
vegetarians (at least 100) that we could be reasonably sure that the answer we received
was projectable to all vegetarians.
All samples are not created equal. Generally, the fairest way to generate a sample
is to do so randomly, letting the laws of probability spin their magic. Random samples of
a large enough size will do a surprisingly nice job of modeling a large population.
Sometimes non-random samples make sense, especially when
trying to draw inferences about specific population sub-groups (e.g., Americans of Irish
heritage or fruit flies that were born with an extra pair of wings). But
be careful of their use, especially when reviewing advertising claims. The use of
non-random samples to "stack the deck" is a favorite trick of unscrupulous
car manufacturer will claim in big letters that its new model is preferred over a
competing model. It's only in the small print that we find out that the study was
conducted among a specific sample: First time buyers over the age of 65. Thus, the results
are projectable only among the portion of the population that has never bought a car but
has filed for Social Security.
When working with inferential statistics, it is important to remember that the numbers are
approximations of what the total population is like, not numbers describing the total
population itself (unless, of course, the sample IS the population). They are thus subject
to some amount of error, leading to a need to express their degree of reliability.
One way to do so is through the concept of sampling error.
For example, if a survey is said to include a sampling error of +/- 3 points, it means
that any quoted figure could actually be 3 points above or 3 points below that figure. So
if a study infers that 10% of the population has 12 toes, a more accurate description is
that "the percent of people with 12 toes falls between 7 and 13%." While this
answer is less precise, it is certainly more accurate!
Another way that reliability is presented is as a percentage.
A Confidence Level of 95% indicates that there is a 95% probability that the number quoted
is in fact the actual number that would be found by studying the entire population.
Traditionally, differences that occur at the 95% Confidence Level are considered to be significant,
and those at the 99% level are considered to be very significant.
It is very important to know the
reliability factor when comparing two different pieces of information. Let's say
that you are conducting a long term study designed to measure the male/female proportion
among silver foxes. If the percentage of males changes from 50 to 52%, should you note an
increase? The answer is "yes", but only if differences of 2 or more points are
considered meaningful (statistically significant). Otherwise, you can only claim a
directional increase, which technically is no increase at all.
This concept of statistical significance is rarely discussed
in enough detail when surveys are presented by the media, industry and special interest
groups. Since it is only natural for a group to present data in the best possible light,
it is extremely important to be able to assess the value of both the raw numbers and the
findings drawn from them. Statistical significance is an excellent way to do so.
This is one of the most misunderstood concepts in all of statistics and the reason why
otherwise smart people will consistently bet (and lose) on the lottery. Let's start with a
coin. If you flip it, the odds of having it land heads-up are 50%. So if you flip the coin
and it comes up tails, what are the odds that it will come up heads the next time? It's
fairly obvious that the odds remain at 50% because each coin toss is an independent
event, unaffected by the previous toss.
The independent nature of coin tosses is easy to understand.
But what about the chances of winning while playing the lottery? If the numbers 3,5,7 came
up today, does that mean you should do what most people do and not bet on them tomorrow?
Not at all! Each day's drawing is an independent event, just like a coin toss. Thus, the
odds of drawing 3,5,7 tomorrow are the same as they were today.
Absolute vs. Relative Change
Should you be frightened to learn that the number of airplane accidents doubled between
1995 and 1996? Maybe, but you'd probably feel differently if the number went from only 1
to 2 than if it increased from 500 to 1000.
Or, should you be alarmed if the concentration of a certain
toxic chemical in the water supply tripled? Probably "no" if it went from 1 part
per trillion to 3 parts per trillion, but probably "yes" if it went from 1000
parts per million to 3000 parts per million. Thus, it is always important to view
relative change in an absolute context, and vice versa.
Not looking at both relative and
absolute differences leaves us ripe for being manipulated. For example, if the
incidence of skin cancer in a sample increased from 2% to 3%, optimists might say it only
went up by 1% (absolute), while pessimists would argue that it jumped by 50% (relative). Both
are right! That's why you have to see the actual before and after percentages, not
just the changes.
By the way, relative effects are far more exaggerated at the
small end of the spectrum than in the middle: A two percentage point increase from 2 to 4%
is a 100% relative gain, but a two percentage point increase from 50 to 52% is only a 4%
relative gain! That's another good reason to always look at the numbers that are changing
and not just the changes themselves.
As any carpenter will tell you, it's critical to match the tool to the job. The same goes
for statistics. Before you can choose the right way to measure data, you have to examine
the way that the data distributes itself. There are three frequency patterns that account
for a very large percentage of sample distributions:
A common frequency distribution that characterizes many human and biological phenomena is
called the normal distribution. This type of distribution describes a population in
which the majority of score values occur in a central
"average" range. The frequency of other scores is the same above and below the
average, with fewer and fewer scores occurring as one looks farther from the center. This
type of distribution is the familiar bell curve.
Example: Using the test scores from a
biology test as an example, the horizontal axis represents the actual scores the students
achieved, and the vertical axis represents the frequency with which each particular score
occurred. (By the way, the "curve wrecker" who made all of our lives miserable
in high school had a score at the far right of the horizontal axis!)
Like a normal distribution, a skewed distribution describes a population in which
the majority of score values occur within an average range. However, unlike the normal
curve, the minority scores in a skewed curve are unequally divided among the possible higher and lower values outside the average
range. The curve that's created looks like a mountain that falls off sharply to one side
and more gradually to the other.
Let's say you decide to analyze sales of ball point pens. You would probably find that
most of them were fairly inexpensive, since they were made to be disposable. But there
were still quite a few that were fairly costly (silver plated barrels) and a few that were
downright expensive (gold barrels). By graphing pens by price, you'd find a frequency
distribution that skewed toward the large volume inexpensive items and then tailed off
toward the expensive types.
Some score patterns have two (or more) central clusters, rather than one. When graphed,
they look like two bell curves next to each other. In fact, in many cases, they are!
Example: Measuring the heights of a sample of people will
produce a bimodal distribution -- one cluster for men and one for women. If instead you
measure height separately by sex, the results will be two normal distributions. Thus, a
bimodal frequency distribution may be a sign that what is being measured as one group
should really be reconsidered as two separate populations.
Next, we have to examine the way in which scores cluster within these distributions. A
mathematical device frequently used to do so is the measure of central tendency, commonly
known as the average.
Much of the time we use what's known as the "arithmetic
average," which is found by adding up all of the scores in a sample and dividing by
the number of scores. The statistical term for this commonly used and widely understood
measurement is the mean. However, the mean is only one of five types of average
that may need to be calculated and analyzed. Most of the time, we only need to concern
ourselves with three:
The mean is the arithmetic average, found by adding up
all the scores and dividing by the number of scores. In many cases, the mean is a
perfectly suitable measure. But when there are extreme values in the distribution, the
mean starts to lose its luster.
The median is the value in a sample that cuts the
sample into two equal parts: an equal number of scores are above it, and an equal number
are below it. Unlike the mean, the median is more concerned with the total number of
scores, rather than with their values.
The mode is the score that occurs most often in a
frequency distribution. It is not used much in scientific work, but pops up on TV when you
see "person on the street" interviews: ("We asked five people if they like
spaghetti. Three said yes, two said no. Thus, the average person likes spaghetti.")
When looking at a normal distribution, all three averages
provide the same result: The mean, median and mode all represent the arithmetic average,
the score in the middle of the distribution and the score that occurs the most frequently.
It's when distributions are not bell shaped that we have to be careful.
income distributions tend to be skewed toward the lower end of the scale. However, there
are a relatively few households with ENORMOUS incomes. When looking at the mean, these
very few households will produce an average that is rather high and doesn't really present
a true picture of the typical household.
The median, on the other hand, will produce a more accurate
and somewhat lower income picture, as it examines the distribution of scores without
regard to their value. (This is why you generally hear the term median income, rather than
Thus, beware of the term
"average!" Always try to find out something about the distribution and
whether or not average refers to the mean, median or mode. If the distribution is normal
(bell curve), the mean is generally the preferred measure, although any of the three will
do. If the distribution is skewed, the median is probably the better choice. In the
case of skewed distributions, substitution of the mean for the median should make you
Although it sounds complex, the concept of standard deviation is actually fairly simple:
How far from the mean do the scores actually fall? Obviously, the tighter that a group of
scores clusters around the mean, the easier it is to make accurate predictions about the
value of additional scores. Thus, samples with lower standard deviations provide more
reliable and predictable data than samples with higher standard deviations. For this
reason, the standard deviation is one of the most important concepts in all of statistics.
Start with two numbers: 1 and 99. Their mean score is 50 (1 + 99 divided by 2). The
deviation of each of these numbers from the mean is 49 (1 is 49 less than 50; 99 is 49
greater than 50). This indicates that the range in which you can safely predict where
another score might fall is huge: 98, or 49 units below 50 and 49 units above 50.
On the other hand, start with the numbers 49 and 51. Their
mean score is also 50, but each of their deviations from it is only 1! Thus, the
predictive range is 2 -- 1 unit below 50 to one unit above 50.
In more concrete terms, what if you were William Tell and had
to pick one of two archers to shoot the apple on top or your son's head? By holding a
contest and reviewing the scores, you note that both archers put 4 out of 6 arrows in the
bull's eye. Archer A put the other two at the far end of the target and at opposite sides
from each other. Archer B put the other two just outside the bull's eye, right next to
If you're fairly sane, you'd choose Archer B, because her
arrows all fell in a nice tight cluster. In statistical terms, her scores would have a
lower standard deviation, meaning that her chances of putting an arrow where she wanted to
put it were better than what you could expect from Archer A.
Coefficient of Correlation
This measure helps us determine the strength and direction of a relationship between two
variables. The Coefficient of Correlation ranges between -1.00 (perfect negative
correlation) to 0.00 (no correlation) to +1.00 (perfect positive correlation). An example
of a negative correlation is the relationship of helmets to deaths caused by motorcycle
accidents: the more that helmets are worn, the fewer the resulting accidental deaths. A
positive correlation would be the amount of rainfall in a given year and average tree
Correlations can be powerful tools, but need to be considered
...they don't necessarily indicate causality, but
rather the presence of a relationship. While we know intuitively that rainfall causes tree
growth, we can't say for sure whether depression causes alcoholism or alcoholism causes
depression. But we do know that the two are correlated.
...strong mathematical correlations may not actually mean
a relationship exists. For example, the cost of buying a new home has risen
dramatically over the last few years, as has the cost of breakfast cereal. Would you be
willing to believe that a strong relationship exists between the two?
Remember, correlations are mathematical constructs with
mathematical meanings. Thus, there is always the chance that they do not accurately
describe real-world relationships. When in doubt, fall back on one of the best tools at
your disposal: common sense!
A Few Good Examples...
As you watch the news on TV, the latest election poll results are announced. The reporter
remarks on the fact that "Candidate Brilton is pulling away from his adversary."
Behind her, this chart flashes up on the screen:
|Approval Ratings (%)
|Sampling error is +/- 3
At first glance, the reporter seems right:
Brilton's score looks as if it increased by 2 points, while Mole's declined by 1 point.
But the small print at the bottom of the chart tells a different story. Because the
standard error is +/- 3 points, any change equal to or less than 2 points is really no
change at all. Thus, the candidate's ratings didn't truly change at all.
Canvassing for Donations
As treasurer of Citizens for A Better Society (CABS), you are in charge of neighborhood
fund-raising efforts. Past efforts have shown that the odds of receiving donations go down
to practically zero if home prices fall below $100,000. Further, it doesn't seem to matter
how much more than $100,000 the house is worth, only that it costs more than this
apparently magic number. You send out three trusty lieutenants to find the best
neighborhood. They come back with the following results:
- Neighborhood A: Average (mean) home price: $125,000
- Neighborhood B: Average (mean) home price: $130,000
- Neighborhood C: Average (mean) home price: $140,000
Based on the above, your aides immediately opt to begin
canvassing Neighborhood C. You're not sure, and ask them to show you their data. After
compiling it and doing a few simple calculations, you come up with this chart:
Home Values for Neighborhood:
Mean 1= 70
Mean 2= 210
Now, the situation looks very
different. Neighborhood C is really two very different neighborhoods, with an equal group
of high and low value homes. The total number of homes worth over $100,000 is five.
Next, you look at Neighborhood B, which has the second
highest mean home value. But the results have been skewed by the fact that their are a few
very expensive homes and many more modest ones. Thus, the median tells a truer story for
this group, since at $85,000 it is more representative of the entire neighborhood. In
fact, only three homes are worth over $100,000.
Finally, you examine Neighborhood A. Even though it has the
lowest mean of the three, it has the most number of homes worth over $100,000 (9 out of
10). Since both the mean and the median are the same, the distribution should be more
centrally clustered and less skewed, which in fact turns out to be the case.
Your conclusion? Go with Neighborhood A, which has the lowest
mean but the largest number of homes that fit your criteria.
Buying A New Car
After thoroughly testing a number of cars, you have narrowed the decision down to either a
new Flotsam or Jetsam. You call the dealers and ask about consumer satisfaction ratings
for their specific car.
The Flotsam dealer tells
you that his car has seen the biggest jump in customer satisfaction ever recorded by the
Powerless Company, which specializes in these surveys. In fact, he says, there's no need
to even call the Jetsam dealer. Since the Flotsam satisfaction number jumped by 20 points
and the Jetsam number didn't change in the last year, the choice is obvious.
Not quite convinced, you hang up and call the Jetsam dealer.
Yes, she says, the Flotsam number did go up by 20 points -- from an approval rating
of 23% to one of 43%. And yes, the Jetsam number didn't change -- it stayed the same at
its old level of 96%.
Which car would you choose? Once you can look at both the
absolute and relative numbers, your decision may change.
A basic understanding
of statistics is a critical component of informed decision making.
Statistical concepts are not hard to master, and mastery will help ensure accurate use,
and minimum misuse, of the large quantity of numerical information that confronts us every
Here is a simple table that you can refer to when thinking
Statistics Crib Sheet
Here is a list of 10 questions to always ask when confronted with
- Was the study performed by a reputable, third party research
- When was the study conducted and for whom?
- Is the sample size large enough to be projectable?
- Was the sample selected randomly? If not, is there a good reason?
- If "the average" is discussed, is it clear which one (mean,
median, mode) is being used and why?
- Are confidence limits and sampling errors clearly indicated?
- Can you get a feel for both absolute and relative changes? Over what
- Do the cited correlations make sense?
- Do the conclusions drawn by the study hold up in light of the
questions listed above?
- Are other similar studies cited? Are the results compared and
comparable to the findings from the current study?
References On the Net...
a multi-institutional effort to foster critical thinking about news and current events
at Rice University
References Off the
(Clicking on the link will take you to the
appropriate catalog page of Amazon.com, where you can learn more about the book and/or
The Spread of Excellence from Plato to Darwin, Stephen Jay Gould (Random House,
How to Lie
with Charts, Gerald E. Jones (SYBEX, 1995).
How to Lie
with Statistics, Darrell Huff (W. W. Norton, 1993).
How to Think
About Statistics, John L. Phillips (W. H. Freeman and Company, 1988).
to Probability, John E. Freund (Dover Publications, 1973).
to Statistical Analysis and Inference for Psychology and Education, Sidney J.
Armore (Krieger Publishing Company, 1983).
without Tears, Derek Rowntree (MacMillan, 1982).
Concepts and Controversies, David S. Moore (W H Freeman and Company, 1996)
Deception at Work, John Maura (Lawrence Erlbaum Associates, 1992).
Cynthia Crossen (Simon & Schuster, 1994).


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Football is a sport that requires lots of mathematical skills such as the ability to analyze the different statistics, the ability to recognize trends and the ability to predict what teams will do next. Math skills are required in every other sports but especially football, because football is a game where teams are competing for a goal and where the game can be won or lost in just a few seconds or even by a single point.
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The other problem is that it can be difficult to tell which team will perform better or worse next week. If the teams performing better tend to lose more often than the teams performing worse, we know that there is something wrong with our numbers. If the opposite is true, then the numbers can be used to predict which team will win next week. In either case, though, you’ll need to have the ability to figure out how each team’s statistics compared to its performance during previous games.
- Read More >> Soccer Predictions Tomorrow – Soccer Betting Tips
With this statistical information, you can figure out what each team’s strengths and weaknesses are. This means that you have a better chance of accurately predicting who will win each game. You may also be able to pick out the teams that will struggle most during the season, and thus you’ll be in a better position to choose your lineup for the season.
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You should never let your mathematical football prediction influence your actual decision-making process. You should make the decision as if you know nothing about the statistics or have no real idea of the team’s strengths and weaknesses. You should only rely on your own judgment when making a final decision.
In order to better understand the mathematics behind the mathematics of mathematical football predictions, you should try to read some books or do a research online. that discusses statistics and the mathematics that go into them. It might also help to read the Wikipedia. to learn more about the game.
Football betting is not just about choosing the team that you think will win. it’s also about choosing the team that you think will win next week!
The only way to be able to do that is by learning the statistics of each team. If a team consistently plays well against its opponents, then it can be considered a strong team. On the other hand, if a team consistently plays poorly against its opponents, then it can be considered a weak team.
One of the easiest ways to analyze the performance of a team’s statistics is to use its historical statistics against its competition. By using statistics like points scored, the number of games played and the number of goals scored, you’ll be able to learn more about the strength of each team. and which teams have the best potential.
Of course, if you don’t have any statistics to play with, you can use another kind of analysis: the Pythagorean scale. This scale compares the number of wins to the number of losses a team has against its opponents. In this model, a lower number of wins is always better than a higher number of losses.
By using this type of model, you can see which team is the most likely to win the most games during its season. and how many games each team should be expected to lose.


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Microsoft 70-980 Brainitdump d IFRS 2 in February 2004, requiring companies that use equity options as compensation to value them at the time of the grant. In fact, IFRS 2 is more expansive 70-980 real questions and answers than FAS 123R in its coverage of equity based compensation. For the most part, though, the two statements agree or more Microsoft 70-980 Brainitdump than they disagree and 70-980 Exam Guide the 70-980 Test Prep differences that 70-980 Certification Braindumps remain are minor. Some of them are listed below Private versus Public entities IFRS 2 applies the same rules about option valuation to both public and non public entities 70-980 Brain Dumps both have to value options at fair value at the time of the grant and treat it as an expense. 70-980 Braindumps While FAS 123R requires nonpublic 70-980 Free Dumps entities to account for options based Recertification for MCSE: Server Infrastructure on their 70-980 Certification Exams fair value, it does 70-980 Cert allow the use of industry average variances in valuing private company options and for the use of intrinsic value exercise value 70-980 Exam Materials when option valuation is difficult to do. Deferred Tax Treatment In tax jurisdictions such as the 70-980 Practice Questions United States, where only the exe.


Notice these interesting things:
Volume and Surface Area
Volume = 5×(3+√5)/12 × (Edge Length)3
Surface Area = 5×√3 × (Edge Length)2
It is called an icosahedron because it is a polyhedron that has 20 faces (from Greek icosa- meaning 20)
When we have more than one icosahedron they are called icosahedra
When we say "icosahedron" we often mean "regular icosahedron" (in other words all faces are the same size and shape), but it doesn't have to be - this is also an icosahedron, even though all faces are not the same.
20-Sided Dice? Yes! An icosahedron that has 20 equal faces has an equal chance of landing on any face.
In fact, you can make fair dice out of all of the Platonic Solids.
The outer shell of this virus (the Human papilloma virus) is like an icosahedron
Courtesy of VirusWorld at virology.wisc.edu
|The head of a bacteriophage (a virus that targets bacteria) is an icosahedron|
|Illustration of bacteriophage||Photo courtesy of Wikpedia user Kuksi|
|Make your own Icosahedron,
cut out the shape and glue it together.
A soccer ball is related to an icosahedron:
It is a truncated icosahedron (truncated means it has bits chopped off it)
It has 12 pentagons and 20 hexagons


Maths Cognito 2018
Vidyatree Modern World College organized its Annual Maths Cognito, on 6th May,2018. The aim of this event was to help students build connection between their classroom learning and real life. It also gave them opportunity to demonstrate their thinking, research, communication and problem solving skills.
With the theme of Maths Is Easy, students from grades Nursery to 12, set up projects and demonstrated the guests, parents and judges models of their understanding.
Chief guest Mr. A.K. Wakhlu(Retired HOD Pediatric, KGMU) and Guest of Honor Dr.Abhishek Shukla MD (Gold Medalist) ,Ms.Rewa Prasad(National Award winner in Mathematics) were amazed at the students knowledge and creativity. Their clarity of concepts while explaining the model was exemplary. It obviously reflected the way learning takes place at VMWC.
More than 200 projects were on display. Some of the amazing projects were Mathematics in Architecture, GST, Math Magician, Maths is Everywhere, Speedometer, Artificial Intelligence, Maths in Cricket etc. The Exhibition was organized under the vision of Founder Mr. Rakesh Kapoor and guidance of Principal Mr. Presenjit Chakraborty. It was a learning and fun filled day for the parents and the students.


Check for understanding about the maths first.
The letter O has an area of 6 squares. Talk to your partner about where are they? Where are the 6 squares? Can someone point out what this is on the paper.
Area: what do we know about area?
- plans of a house
-Manawatü is an area
- space in the floor
- the "footprint" of a shape
Make sure the conversation in the task is about the maths and not the task. If you have a question, ask.
Area and perimeter of things.
- defining what is a 2D and 3D object?
My brother is double glazing the windows in his house. He has got some odd shaped windows in his house. And I was wondering how the window man figures out how much the glass is.
What is the word on here that means perimeter but doesn't say it?
$150, $140, $380, $310, $350, $340.


Möbi is a simple, quick and addictive number game that can go anywhere and be played in as little as five minutes! No pencil, paper, board or elaborate scoring is required .... all you need is a flat surface! No matter how old you are or how many number games you've played, this little blue whale is bound to enchant! Introduce your little ones to numbers or challenge your friends to numerical combat - this game is made for absolutely everyone! Find out why Möbi is becoming an instant classic!
Start out with seven number tiles each and connect them with mathematical equations to form a crossword-like grid. Run out of tiles? Say FLIP to take three more number tiles - and make everyone do the same whether they're ready or not. First one to connect all their tiles wins!
Möbi allows for many variations of play - form teams, play solo, remove tiles to make it easier for little ones, or throw in some new rules to make it trickier.
Möbi promotes the following skills:
WARNING: Choking Hazard - small parts. Not suitable for children under 3 years.


. Enter Graph Equations Look below to see them all. They are mostly standard functions written as you might expect In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers.. . The graphs of the original and inverse functions are symmetric about the line y=x. . Composite function Suppose that a function y=f(u). depends on an intermediate variable u
Here graphs of numerous mathematical functions can be drawn, including their derivatives and integrals. Draw Function Graphs. Mathematics / Analysis - Plotter - Calculator 4.0 Function Grapher is a full featured Graphing Utility that supports graphing two functions together. It has the unique feature that you can save your work as a URL (website link) The graph of a function f is the set of all points in the plane of the form (x, f(x)). We could also Characteristics of Graphs. Consider the function f(x) = 2 x + 1. We recognize the equation y = 2 x + 1.. The graph of a function is the set of all points whose co-ordinates (x, y) satisfy the function `y = f Since there is no limit to the possible number of points for the graph of the function, we will follow..
Plot the graphs of functions and their inverses by interchanging the roles of x and y. Find the relationship between the graph of a function and its inverse. Which functions inverses are also.. A graph of a function is a visual representation of a function's behavior on an x-y plane. Graphs help us understand different aspects of the function, which would be difficult to understand by just..
Sal is given a graph of a logarithmic function with four possible formulas, and finds the appropriate one Graphical Function Explorer (GFE). Operating instructions. GFE is a free online function graphing tool that allows you to plot up to three functions on the same set of axes
Everything you're expected to graph on your own is based on a more basic graph (parent function) that you NEED to memorize. Look at the image above and check with your teacher to see which you.. . Includes all the functions and options you might need. Easy to use and 100% Free
Functions and Graphs. 1. Introduction to Functions. 2. Functions from Verbal Statements. The graph of a function is really useful if we are trying to model a real-world problem In this section we graph seven basic functions that will be used throughout this course. The resulting curved graph is called a parabolaThe curved graph formed by the squaring function. Online Graph draw: plot function, plot parametric curves,plot polar curves. The online curve plotting software, also known as a graph plotter, is an online curve plotter that allows you to plot functions..
A simple to use online function plotter with a lot of options for calculating and drawing graphs or charts of mathematical functions and their score tables A free graphing calculator - graph function, examine intersection points, find maximum and minimum and much more
Graph::Function. This gem's goal is to make it easy to compare the asymptotic performance of two or more functions via graphing. When I work on katas and exercises I found I often wanted to compare.. Graphing Exponential Functions. What is an Exponential Function? To begin graphing exponential functions we will start with two examples. We will graph the two exponential functions by making a..
Algebra. When we look at a function such as. we call the variable that we are changing—in this case. --the independent variable. We assign the value of the function to a variable we call the dependent variable Graph of the function f(x) = x4 − 4x over the interval [−2,+3]. Also shown are its two real roots and global minimum over the same interval. In mathematics, the graph of a function f is, formally, the set.. Functions and Their Graphs. What you should learn. GOAL 2 Graph and evaluate linear functions, as applied in Exs. 55 and 56. Why you should learn it © 2020 GeoGebra. Graph of function of two variables. Autho I'm implementing a simple Graph library for my uni project and since this is the first time I'm dealing with graphs, I would like to know which functions you guys consider to be most common and useful for..
Graph is an open source application used to draw mathematical graphs in a coordinate system. Anyone who wants to draw graphs of functions will find this program useful Graphing Quadratic Functions. The term quadratic comes from the word quadrate meaning square or rectangular. Similarly, one of the definitions of the term quadratic is a square graphing-functions. asked 20 hours ago. Samuel Martinez. ordinary-differential-equations physics graphing-functions mathematical-physics interpolation Graph Individual (x,y) Points. The most basic plotting skill it to be able to plot x,y points. This page will help you to do that
Polynomial functions of the form f ( x) = x n (where n is a positive integer) form one of two basic graphs, shown in Figure 1 The first derivative of a function can be interpreted as the slope of a graph of that function. There are a few very important things that can be noticed about the derivative of a function based on its.. The lines( ) function adds information to a graph. It can not produce a graph on its own. Usually it follows a plot(x, y) command that produces a graph. By default, plot( ) plots the (x,y) points How igraph functions handle attributes when the graph changes. The functions find cliques, ie. complete subgraphs in a graph
In this mode, there is a gravitation pull that acts on the nodes and keeps them in the center of the drawing area. Also, the nodes exert a force on each other, making the whole graph look and act like.. A graph is a pictorial representation of a set of objects where some pairs of objects are connected by Mathematical graphs can be represented in data structure. We can represent a graph using an.. Graph Panel The main panel in Grafana is simply named Graph. It provides a very rich set of graphing options. Clicking the title for a panel exposes a Module 2: How to graph functions and linear equations. Graph functions and relations. Study Notes. Save
Provided is a method for creating a cardiac function graph of an atrial fibrillation or sinus arrhythmia patient and an analysis evaluation method for a pathophysiologic mechanism from thoracic.. The graph of a function f is the set of all ordered pairs ( x, f(x) ) where x is in the domain of f. A function is increasing on an open interval if the function rises (positive slope) on the interval as you.. self.graph = [None] * self.V. # Function to add an edge in an undirected graph. def add_edge(self, src, dest Plotting and graphing are methods of visualizing the behavior of mathematical functions. Use Wolfram|Alpha to generate plots of functions, equations and inequalities in one.. Graphviz is open source graph visualization software. Graph visualization is a way of representing structural information as diagrams of abstract graphs and networks
This tool visualizes any complex-valued function as a conformal map by assigning a color to each point in the complex plane according to the function's value at that point We can forget about the original function f and view f as a function in its own right. The graph of f is pictured on the right above My program allows you to build a graph of a function given a string. In the program the user himself defines the boundaries within which there is a graph of the construction and function of the color line
This chapter describes various functions on a graph. This chapter describes various functions on a graph. A lot of these accept a vertex (or edge) example as parameter as defined in the next section Reflections of Functions: -f (x) and f (-x). Reflection over the x-axis. -f (x) reflects f (x) over the Think of folding the graph over the x-axis. On a grid, you used the formula (x,y) → (x,-y) for a reflection in.. Basic Graphing. Plotting Mathematical Functions. 2D Function Plot. Working with Microsoft Excel. Graphing. Customizing Your Graph. Graphical Exploration of Data
. From LearnZillion. Created by Brenda Campbell. In this lesson you will learn how to graph piecewise functions by using your knowledge of graphing other functions Graph points on the function at the following x locations: (use a comma separated list ex. When you enter a function, the calculator will begin by expanding (simplifying) it
2. Graphical Transformations of Functions. In this section we will discuss how the graph of a function may be transformed either by shifting, stretching or compressing, or reflection .x and higher. Create a new function, and while in the template selection screen, choose the Microsoft Graph scenario diagrams.net is free online diagram software for making flowcharts, process diagrams, org charts, UML, ER and network diagrams.. Graph Theory is a relatively new area of mathematics, first studied by the super famous mathematician Leonhard Euler in 1735. Since then it has blossomed in to a powerful tool used in nearly every branch.. The graph of a function on real numbers is identical to the graphic representation of the function. For general functions, the graphic representation cannot be applied and the formal definition of the..
View graphs and track agent performance. Customizable Is there a function that you need and that is not integrated? Our development team will study the best way to integrate it into the system Download Function graph stock vectors at the best vector graphic agency with millions of premium high quality, royalty-free stock vectors, illustrations and cliparts at reasonable prices Functions and their graphs. Given two sets X and Y, a function from X to Y is a rule, or law, that associates to every element x ∈ X (the independent variable) an element y ∈ Y (the dependent variable) Generic function for plotting of R objects. For more details about the graphical parameter However, there are plot methods for many R objects, including function s, data.frame s, density objects, etc
Graphing a function based on the derivative and the double derivative. Here is the graph of the function. Can we see how they correspond The graph can be zoomed in by scrolling with your mouse, and rotated by dragging around. The table below lists which functions can be entered in the expression box Graphs of exponential functions. By Nancy Marcus. Example 1: Graph the function and graph the function on the same rectangular coordinate system function-graph. 0.1.2 • Public • Published 6 years ago. $ cd node_modules/function-graph. $ npm test Graphing a recursive function (self.math). submitted 5 years ago by [deleted]. I have a recursive piecewise function that I'd like to graph, but I can't find any graphers that support recursion
Graph-theory type graphs (nodes/edges layout; see the igraph package). ggplot2 offers two functions for creating small multiples: facet_wrap(): define subsets as the levels of a single grouping.. Keras provides utility functions to plot a Keras model (using graphviz). This will plot a graph of the You can also directly obtain the pydot.Graph object and render it yourself, for example to show it in an..
Graphing Functions: Trigonometric Functions. Graphs of the sine, cosine, and tangent functions, including definitions of periodicity and the general sinusoidal wave, with examples Function and graphs by Rione Drevale 3660 views. Limits of functions by Louie Joy Rosit 1674 views. WordPress Shortcode. Link. Graph of functions. 2,497 views. Share Functions can always be graphed and different kinds of functions will produce different looking This means that a function graph can have potentially many x -intercepts, but only one y -intercept In this theme is described the application developing of drawing the graph of two variables function. Using the given example, you can create own programs to drawing the graphs of other functions Graph title: Horizontal axis label To print graph, press the print button and print from browser menu or press Ctrl+P
Graph of a function — In mathematics, the graph of a function f is the collection of all ordered pairs (x, f(x)). In particular, if x is a real number, graph means the graphical representation of this collection.. Linear functions are among the easiest to graph; each is simply a straight line. To plot a linear function, calculate and mark two points on the graph, and then draw a straight line that passes.. Different Functions and their graphs. Modulus Function. Last updated at July 12, 2018 by Teachoo
When the graph of a function is changed in appearance and/or location we call it a transformation. Now plot the points and compare the graphs of the functions g and h to the basic graph of. f(x)=x2 It records a graph of all the operations performed on a gradient enabled tensor and creates an Neural networks are nothing more than composite mathematical functions that are delicately tweaked.. A list of functions available in the Node Graph can be found in the table below. Function. How to access it. What it does The chart is interactive: you can mouse over to highlight individual values. You can click and drag to zoom. Double-clicking will zoom you back out. Shift-drag will pan. You can change the number and hit.. Graphing with Desmos. Equations. Preliminaries. Defining Functions. Piecewise Functions. Other Equations. Inequalities. Graph Segmentation. Simultaneous Graphing
It explains when logarithmic graphs with base 2 are preferred to logarithmic graphs with base 10. It also explains several advantages that dot plots have over bar charts Download Graph - Graph is a program for drawing graphs of mathematical functions in a coordinate system. Graphs may be added with different color and line styles Analyze functions using different representations. CCSS.Math.Content.HSF.IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using.. Such functions, which we will refer to as modular graph functions, arise, for example, in the low energy expansion of genus-one Type II superstring amplitudes. We here introduce a class of.. Chapter 5: Functions. Learners must be able to determine the equation of a function from a given graph
This Linear Functions Worksheet will produce problems for practicing graphing lines in slope-intercept form EXAMPLE 1. Graph a function. STEP 2. Plot a point for each ordered pair ( x , y ). ANSWER. for Example 1. GUIDED PRACTICE... Exponential functions are functions written in the form , where a is the base and is positive and , and x is a real number. Graphs of Exponential Functions. Example 1: Consider the following exampl


1. Apple has announced that the new MacBook lasts 10 hours. Do the results from the battery life tests agree with the company’s statement? (10)
2. Compare the ages of employees at Microsoft and Tesla. What can you conclude? (10)
3. Evaluate if undergrad major has an impact on the careers chosen. (10)
4. Using the data provided in “Assignment 3.xlsx (Car prices)”, answer the following questions.
4.1. Which independent variable has the most significance for predicting price? (20)
4.2. Develop the best possible multiple linear regression model for the data. (30)
4.3. Use hold-out validation (100 -training, 50 – validation) and discuss your finding based on the value of the residuals. (10)
4.4. Is developing piecewise linear regression model for the independent variable, mileage and dependent variable, price, useful? Why? (


We want our children to develop a love for mathematics from the moment they enter into our school. Our curriculum is based on a ‘teaching maths for mastery’ approach where children develop their understanding and fluency by experiencing concrete, pictorial and abstract representations of concepts. We give children time to practice key skills and processes; encourage them to look for patterns, make connections and reason; and enable them to move onto applying their skills to solve more complex problems. We use the Maths — No Problem! programme which is endorsed by the Department for Education, NCETM and Ofsted.
A new way of thinking and teaching
Whole class moves through content at the same pace
When teaching maths for mastery, the whole class moves through topics at broadly the same pace. Each topic is studied in depth and the teacher does not move to the next stage until all children demonstrate that they have a secure understanding of mathematical concepts.
Time to think deeply about the maths
Students are given time to think deeply about the maths and really understand concepts at a relational level rather than as a set of rules or procedures. This slower pace leads to greater progress because it ensures that students are secure in their understanding and teachers don’t need to revisit topics once they’ve been covered in depth.
Builds self-confidence in learners
In a traditional primary school maths lesson, children are put in different groups and given different content based on their anticipated ability. This means that from an early age children are classed as those who can and can’t “do maths”. Teaching maths for mastery is different because it offers all pupils access to the full maths curriculum. This inclusive approach, and its emphasis on promoting multiple methods of solving a problem, builds self-confidence and resilience in pupils.
Differentiates through depth rather than acceleration
Though the whole class goes through the same content at the same pace, there is still plenty of opportunity for differentiation. Unlike the old model, where advanced learners are accelerated through new content, those pupils who grasp concepts quickly are challenged with rich and sophisticated problems within the topic. Those children who are not sufficiently fluent are provided additional support to consolidate their understanding before moving on.
“Children’s chances of success are maximised if they develop deep and lasting understanding of mathematical procedures and concepts.” NCETM
Home Learning – children can consolidate skills and processes through a set task on Education City or a simple worksheet. Children can practice times tables with the help of a school pack which they are given at the beginning of Year 3.
Multiplication tables: For anyone wanting a singing approach to ‘chanting’, teacher Martyn Yeo has helpfully made a YouTube playlist putting sing-along videos in one place.


Answered By: Reference Librarian Last Updated: 2018.Aug.09 Views: 77
If you choose to take the math test, you are not eligible to receive course credit for testing into the respective Advanced level.
If your math SAT is between 540 and 605 and you wish to be eligible for Calculus I, you must take the optional test and test into Math Level 6. In addition, for certain majors (generally B.A. degrees), testing into the Math Level will satisfy the university's math competency. When you satisfy the competency with the placement test, those 3 credits can be used for any college class. In essence, it can count as an additional free elective. If you are majoring in Teacher Education, taking the optional math test is unnecessary because this major requires a minimum of 2 college level math courses. Source: Placement Office. Contact: 717-477-1014


In investments, we all wish we had a crystal ball. Unfortunately, many people need to plan their lives around how much their investments will grow. So, in order to calculate the answer, many financial advisors, insurance salespeople and budget gurus use a simple investment growth calculator such as the one below. The calculator below takes an annual percentage return and applies it to your investments year after year.
Unfortunately, these calculators are useless. Why?
• They don’t factor in inflation. Inflation is a phenomenon that causes money to be worth less with the passage of time. So, pretend you want to retire in 25 years and you want to have all the things a million dollars can buy you in the year 2015. In 25 years, things will cost a lot more because of inflation.
So, let’s say inflation will average 3.4% over the next 25 years. Let’s also say you manage to grow your investments to one million dollars by 2040. Unfortunately, you will find that your million dollars in 2040 can only buy as much as $420,000 bought in 2015. You would actually have to save $2,307,000 if you wanted to have all the things a million dollars would have bought you in 2015. Unfortunately, even if you try to subtract 3.4% from your returns to account for inflation, you can’t fix the calculator. That is because….
•The entire premise is flawed. This calculator shows a fundamental misunderstanding of mathematics. In such a calculator, highly variable, compounding factors (investment returns, inflation) are averaged and applied linearly (every single year). This does not work. For instance, pretend you start with $100,000 and you have an average investment return of 25% over 2 years. This sort of calculator will tell you you have $156,000 in your investment account. In actuality, you might only have $100,000. Why? See the two year scenario below for an example:
Year 1: In 2015, you start investing with $100,000. Your investments decline 50% over the course of the year. You now have $50,000.
Year 2: At the beginning of 2016, you have $50,000. Over the course of the year, your investments go up 100%, meaning they double. You are back where you started at $100,000.
Now, you had two years of returns: -50% and +100%. To find the average, add those numbers together (-50%+100%=50%) and divide that 50% by 2 (50%/2=25%). As you can see, you have an average yearly return of 25% and nothing to show for it. The calculator is clearly flawed, yet many continue to use it. So, how do we do it differently?
• Endress Capital Management uses Monte Carlo simulation, a much more sophisticated approach. Monte Carlo is a mathematical modeling tool that takes historical data and randomizes that data to generate yearly investment returns. For instance, under our Monte Carlo calculator, you might have a -5% return in year one and a 15% return in year two. If you want to know how much money you will have in twenty years, we generate those random numbers for twenty years, ending with one final value.
However, it goes even further. If we generated twenty annual returns and called it a day, you might have gotten very unlucky a couple of years. Maybe you had two years in a row where the stock market declined by 40%. It isn’t likely, but if we just ran the numbers once, very lucky or unlucky scenarios could give you an inaccurate result. So, we run the simulation thousands of times. In this way, we can provide a good estimate of what your return will be over time. Do not let bad math guide your planning; your future is too important.


This article features top scholarships to study Mathematics in the US, Canada and UK.
Do you want to study Mathematics but are worried about the cost of tuition? You’ve come to the right place. Read on!
A Bachelor of Mathematics (B.Math) is an undergraduate academic degree awarded for successful completion of a program of study in mathematics or related fields, such as applied mathematics, mathematical physics, pure mathematics, financial mathematics, computational science, data analytics, actuarial science, operations research or statistics.
The Bachelor of Mathematics is awarded to high achieving students looking for how to develop comprehensive specialized knowledge in a discipline of mathematics or a high level of sophistication in the applications of mathematics.
Maths is generally considered to be one of the hardest courses to study at a degree level and is also a course which generally leads a higher-paid graduate job.
It is no surprise then that universities offer scholarships to make sure that they can not only attract academically bright students but also assist those who have the ability and flair but may come from a disadvantaged background or an under-represented group.
Below are some examples of some of the scholarships to study Mathematics being offered to students across the world. Some of the deadlines for these scholarships may have passed for this year, but they are often offered every year and it will give you an insight into the sort of thing you might expect to find.
We’ll start with USA, then Canada and United Kingdom.
Scholarships To Study Mathematics In the USA
The following are some of the scholarships offered to study Math or Math related courses:
ACF Barnes W. Rose Jr. And Eva Rose Nickol Scholarship Program
The Barnes W. Rose, Jr. and Eva Rose Nickol Scholarship Fund was founded in 2003 in honor of Eva and in memory of Mr. Rose. Barnes was a graduate of set “1951” at Albuquerque High-School and a graduate of set “1957” at West Point.
Eva is a long-time residence of New Mexico. The program will award a scholarship to an Albuquerque High-school graduate aiming to achieve a bachelor’s degree in one of the STEM majors: Science, Technology, Engineering, and Mathematics (STEM).
ACF Woodcock Family Education Scholarship Program
This scholarship program was established in 1993 by a businessman known as Woodcock and his wife, Dixie, to assist students who are academically bright in the fields of mathematics and science.
Scholarships are awarded to Albuquerque metro area high-school graduating seniors (includes Sandoval, Valencia, Bernalillo and Torrance Counties). Scholarships are payable over 4-years.
African American Network of the Carolinas
The African American Network of the Carolinas provides scholarships to college-bound students from South and North Carolina who are pursuing a major in mathematics, science, computer science, engineering, accounting, finance or business administration.
Alaska Space Grant Program For Undergraduates
The Alaska Space Grant Program offers undergraduate students scholarships of up to $1,000. The scholarships are open to freshers and continuing undergraduate students majoring in the fields of Science, Technology, Engineering and/or Mathematics (STEM).
Boston University’s Noyce Scholarship Program In Math
Boston University offers scholarships to students in need of financial aid through the Robert Noyce Program. This program is for talented STEM (Science, Technology, Engineering and/or Mathematics) majors and STEM (Science, Technology, Engineering and/or Mathematics) professionals to become mathematics teachers.
Scholarships To Study Mathematics In Canada
A.W. Chisholm Bursary
This award will be awarded to a full-time student pursuing a first degree in Mathematics, the student must be a resident of Heatherton or St. Andrew’s, Antigonish County, with good academic records, and who has shown financial need.
Anne Molson Scholarship
The Ann Molson Scholarship is open to only to students in the Physical Sciences proceeding from the second to the last year into the final year. Awarded on the recommendation of the Departments of Mathematics, Chemistry and Physics.
Bertha and Bernard Harrison Scholarships
This scholarship is awarded yearly to students with excellent academic achievement, entering each of the faculties of Humanities; Science and Mathematics; Applied Health Sciences; Goodman School of Business; and Social Sciences.
Applicants for this Scholarship must meet Ontario residency requirements and demonstrate financial need.
CAST Scholarship Foundation
The CAST Foundation helps students to minimize the cost of self-rescue swim lessons for those in need by granting scholarships in different amounts to families swimming with CAST Affiliated Instructors. CAST offers three main scholarships which are subject to verification, this scholarship includes scholarships for mathematics students.
Dr. James C.S. Wong Memorial Bursary
This award was established in memory of James C.S. Wong, a professor in the Department of Statistics and Mathematics.
The program offers annual awards to a continuing undergraduate students in the Department of Statistics and Mathematics.
Scholarships To Study Mathematics In UK
This award spans over a 3 or 4-year course at the University of Nottingham. To be qualified you must: be enrolled on the first year of a first full-time undergraduate course in Mathematics or Mathematics joint honours, and have a residual household income of less than £42,600
Maths & Statistics Scholarship
Every 1st stage home entrant who firmly accepts an offer at the University of Plymouth for a Mathematics/Statistics degree program will automatically receive a Mathematics and Statistics Enrollment Scholarship of £500 per A grade in A level Mathematics plus £500 for an A in Further-Mathematics up to a sum of £1,000
The Annie Currie Williamson Scholarship
The Annie Currie Williamson Scholarship program offers a sum of £1,500 per year at the University of York. The scholarship is given on the basis of academic-potential and targeted students with an assessed household income of between £25,001 – £30,000.
Mathematics Excellence Scholarship
This scholarship program offers up to £1,000 to students at the University of Birmingham who achieve A*A*A* in Mathematics and two other subjects apart from General Studies, at A level.
Gillian Stead Maths Bursary
This fund is open to UK/EU undergraduate students who enter the first-year of a B.Sc or M.Math degree at the University of Hull in Maths and who have gained 128 UCAS Point system from 3 A-Levels including an A in A-level Maths.
Final Words On Scholarships To Study Mathematics
Mathematics is an exciting and varied discipline that can open up series of opportunities for students. The study of mathematics makes students better at solving problems. It gives students skills that they can use across other courses and apply in other job roles.
Have you been searching online for scholarships to study mathematics in USA, Canada or UK?
I hope you find the information here useful. Also read other recommended posts for you below.


Now we know how many degrees there are in an Ingraham angle. 90 degrees is a right angle. Less than 90 degrees is an acute angle. Laura Ingraham, with her comments on David Hogg, has demonstrated that the angle is obtuse, between 90 and 180 degrees. She is now on “vacation.” Will she return to FOX? Probably, but not for long. Donald Trump might hire her to replace Sarah Huckabee Sanders.
Trump must go


The Algorithms Group is looking for self-motivated PhD students with interests in Algorithms for Big Data, in particular, streaming algorithms; sublinear algorithms, algorithms on distributed data, and related areas, with strong motivations from databases, data mining, bioinformatics and network applications. One with decent mathematical maturity and strong programming skills is preferred.
Here is a motivating problem: You are posted to a very long stream of numbers, one by one. Since your memory is limited, you can only memorize a few bits of information about the steam, then how can you extract useful information of the stream and “write” into your memory so that you can answer questions like what is the number of distinct numbers that you have seen at the end of the streaming process? Such solutions are very useful in today’s world which is immersed in a flood of data.
If you are interested in working with us as a PhD student in this frontier, you can contact us directly by sending your CV, undergraduate transcripts and a statement of purposes/interests to firstname.lastname@example.org or email@example.com.


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â€“ Cite this article: We present our data set of. Combinatorics and Graph Theory. Kostka numbers, the measures of symmetry, and other number theory. Graphical representation of the number of graphs with given symmetry. Main ideas.. T. F. Kojima, E. Saitoh: Forbidden index and analog of â€œgraphical matchingâ€ with applications to perfect matchings.
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1. If a researcher takes a large enough sample, he/she will almost always obtain:
a. virtually significant results
b. practically significant results
c. consequentially significant results
d. statistically significant results
2. The null and alternative hypotheses divide all possibilities into:
a. two sets that overlap
b. two non-overlapping sets
c. two sets that may or may not overlap
d. as many sets as necessary to cover all possibilities
3. Which of the following is true of the null and alternative hypotheses?
a. Exactly one hypothesis must be true
b. both hypotheses must be true
c. It is possible for both hypotheses to be true
d. It is possible for neither hypothesis to be true
4. One-tailed alternatives are phrased in terms of:
b. < or >
c. » or =
5. The chi-square goodness-of-fit test can be used to test for:
a. significance of sample statistics
b. difference between population means
6. A type II error occurs when:
a. the null hypothesis is incorrectly accepted when it is false
b. the null hypothesis is incorrectly rejected when it is true
c. the sample mean differs from the population mean
d. the test is biased
7. Of type I and type II error, which is traditionally regarded as more serious?
a. Type I
b. Type II
c. They are equally serious
d. Neither is serious
8. You conduct a hypothesis test and you observe values for the sample mean and sample standard deviation when n = 25 that do not lead to the rejection of .png”>. You calculate a p-value of 0.0667. What will happen to the p-value if you observe the same sample mean and standard deviation for a sample > 25?
c. Stay the same
d. May either increase or decrease
9. The form of the alternative hypothesis can be:
c. neither one nor two-tailed
d. one or two-tailed
10. A two-tailed test is one where:
a. results in only one direction can lead to rejection of the null hypothesis
b. negative sample means lead to rejection of the null hypothesis
c. results in either of two directions can lead to rejection of the null hypothesis
d. no results lead to the rejection of the null hypothesis
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Much in the same way that statistical models often compete with one another for relevancy and efficiency, these two works on paper strive for emotional poignancy in displaying the exact same data set in different fashions. Each composition is derived from 720 data points, ranging from 0-10, which describe my overall emotion state (as subjectively assigned by me), once per minute, for 12 hours.
In the first piece, the data reads from left to right, in 30 rows of 24 columns, representative of the hours in a day, the days in a month. It is meant to allow the data to create its own narrative, as well as an emotional plot line which shifts according to real life occurrences, for me, but may very nearly be an abstract representation of emotion reactions that have taken place in the personal life of the viewer.
The second work is much more contrived and draws from multiple cooperating mathematical principles. Firstly, it splits the data from the initial set into 12 separate hours, instead of a single flow of points from start to finish. Next, the number of similar occurrences (i.e. all of the “4”s) within each hour is totaled. Based on the mathematical principle that any number can be written as the sum of 4 square numbers, each of these totals was split into its 4-square factorizations.
An example, the numbers 7 and 10 can be written:
7 = 4 + 1 + 1 + 1, or 7 = 2^2 + 1^2 + 1^2 + 1^2
10 = 9 + 1 + 0 + 0, or 10 = 3^2 + 1^2 + 0^2 + 0^2
The largeness of these 4 squares dictated the largeness of the shapes represented in the actual work. Reading from top left to bottom right the 12-hour span runs its course a second time.
As the data proceeds throughout the day, some specific values which may not be closely representative of the entire hour’s data have been placed further away from the general streamline of the main set, thus representing a statistical outlier. Finally, as the the day reaches its end, the set seems to deviate from its relatively stagnant correlation and spreads out in a funnel shape, which, in statistics, is descriptive of a property known as heteroscedasticity. Heteroscedasticity implies that although parts of the data seem to show a reliable statistical trend, the set as a whole contains some confounding factor that discredits the sanctity findings. Even with the utmost transparency, we can never obtain full understanding of another person’s mind, or for that matter, our own.


I am a lecturer in mathematics at the University of Hull. I obtained my PhD from Durham University under the supervision of Andrew Lobb, and was a postdoctoral fellow at McMaster and Syracuse Universities before coming to Hull. My full CV is available here.
My mathematical interests are in low-dimensional topology, including link homology and concordance.
I also have a blog about text book covers.
All comments very welcome.
- On knots that divide ribbon knotted surfaces with Hans U. Boden, Ceyhun Elmacioglu, Anshul Guha, Homayun Karimi, Yun-chi Tang, and Bryan Wang Peng Jun. Submitted. Preprint available here.
- Minimal crossing number implies minimal supporting genus with Hans U. Boden. Bull. Lond. Math. Soc. 53, 2021. Preprint available here.
- On ribbon graphs and virtual links with Scott Baldridge and Louis Kauffman. European J. Combin. 103, 2022. Preprint available here.
- Generalized Fishburn numbers and torus knots with Colin Bijaoui, Hans U. Boden, Beckham Myers, Robert Osburn, Aaron Tronsgard, and Shaoyang Zhou. J. Combin. Theory Ser. A 178, 2021. Preprint available here.
- Ascent concordance, Algebr. Geom. Topol. 21, no. 6, 2021. Preprint available here.
- A parity for 2-colourable links, Osaka J. Math. 58, no. 4, 2021. Preprint available here.
- Additional gradings on generalisations of Khovanov homology and invariants of embedded surfaces with Vassily Manturov, J. Knot Theory Ramifications 27 (9), 2018. Preprint available here.
- Computations of the slice genus of virtual knots, Topology Appl. 253, 2019. Preprint available here.
- Doubled Khovanov homology, Canad. J. Math. 70 (5), 2018. Preprint available here.
All course information distributed via Canvas.
- Spring 2023: Applied Complex Analysis.
- Autumn 2022: Differential Geometry.
At Syracuse University
- Spring 2022: MAT331 Linear Algebra I.
- Fall 2021: MAT296 Calculus II.
At McMaster University
- Winter 2021: 3T03 Inquiry in Topology.
- Fall 2020: 1B03 Linear Algebra I.
- Winter 2020: 3T03 Inquiry in Topology
- Winter 2020: 1ZC3 Engineering Mathematics II
- Winter 2019: 1B03 Linear Algebra 1/ 1ZC3 Engineering Mathematics II.
- Fall 2018: 2Z03 Engineering Mathematics III.


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Napkin Math: Average Life on a Napkin
April 24, 2019
The latest entry in our Napkin Math video series demonstrates how to quickly understand average life on a napkin. Like the previous videos in this series, this short video shows the simple math that credit union leaders can literally do on the back of a napkin to understand ballpark financial performance.
When you have 4 to 5 minutes, please watch the new video and let us know what you think. To view other Napkin Math videos, visit our videos page.


room puzzle variants
old popular puzzle, called “Five Room House
puzzle” (also known as “Walls
and Lines puzzle”, or “Cross
the Network puzzle”), is canonically represented
as a rectangular diagram divided into five rooms, as
shown opposite. The object of the puzzle is to draw a
continuous path through the walls of all 5 rooms, without
going through any wall twice, and without crossing any
path. The path can, of course, end in any room, not necessarily
in the room from where it started. Some puzzle diagrams
represent the rooms with openings supposed to be doors.
In this instance, the challenge is to visit every room
of the apartment by walking through every door exactly
with 16 openings/doors
rendering of the puzzle
Whether starting and ending in the same room, or starting
in one room and ending in another one, every other
room of the diagram/apartment must have an even number
of doors... That is, pair(s) of ‘in’ and ‘out’ doors (as
doors CANNOT be used TWICE, we have then to use an
even number of doors as we ENTER and LEAVE those rooms).
suppose we start in a room with an odd number of
doors, then it is possible to visit all the 5 rooms
of the apartment if and ONLY if another room has
an odd number of doors - representing the
departure and the arrival points of the continuous
path - , and all the other rooms have an
even number of doors. In a few words, for this topological
puzzle to be solvable, there may NOT be more than
TWO rooms with an odd number of doors. Since the
puzzle has THREE rooms with an odd number of openings/doors,
it is mathematically impossible to complete a circuit
a continuous line that enters and leaves one of the
rooms crosses two walls. Since the THREE contiguous
larger rooms each have an odd number of walls to
be crossed, it follows that an END of a line must
be inside each of them if all the 16 walls are crossed.
But a unicursal line has only TWO ends, this
contradiction makes the 5 Room House puzzle unsolvable.
if we close a door or add an extra room to the puzzle
(see fig. a and b below), then
it becomes solvable. Now, you can easily draw one
continuous line that passes through every opening
exactly once... Try it! (the five-room variant on
the left (fig. a) is just a little harder to solve,
because you have to figure out where to start)
Five Room House is actually a classic example of
an impossible puzzle — one that bears no positive
solution. In this particular case, the solution
consists in finding that the problem has no solution!
(remember: puzzles always have one, several or no solutions;
to puzzle solving)
insolubility of the 5 Room House problem can be proved
using a graph
theory approach, with each room being a vertex and
each wall being an edge of the graph (see
image opposite). In fact, this puzzle is similar to
the famous “seven
bridges of Königsberg” problem thanks
to which the eminent Swiss mathematician Leonhard
Euler laid the foundations of graph theory.
Euler wondered whether there was a way of traversing
each of the 7 bridges over the river Pregel at Königsberg (now
Kaliningrad) once and only once, starting and returning
at the same point in the town. He finally realized
that the problem had no solutions!
to 'solve' the puzzle
As you experienced, this puzzle is impossible to solve
on paper... But ‘impossible’ puzzles sometimes
have out-of-the-box solutions, as the non-standard
solution depicted below.
neat out-of-the box solution...
Everything down to this point has been in 2 dimensions,
either a diagram drawn on paper, or a five room apartment
on a flat surface. In order to draw a continuous path
that goes from one room to another without crossing
a line or going through a door twice, you have to reproduce
the 5 room house puzzle onto a surface that is not
topologically equivalent to a sheet of paper. The solid
that may help you is a torus, a kind of ring-shaped
solid resembling a doughnut or a bagel. The puzzle
diagram should be reproduced so that the hole of the
torus is inside one of the 3 larger rooms, as shown
in the example below.
the Most Wanted Puzzle Solutions in a look!


How Much Square Feet Does A Bundle Of Shingles Cover
To properly calculate how many bundles of shingles are needed to roof your home, you need to calculate the total square footage of the roof and convert it to squares. Thereof, how many square feet are in a bundle of cedar shingles?
Specialty shingles, such as iko armourshake, are heavier and therefore packaged to cover 20 square feet per bundle, or five bundles per square.
How much square feet does a bundle of shingles cover. To find the amount of shingles you will need, multiply the number of squares by three. The average bundle of shingles covers 33.3 ft2, so three bundles of shingles are needed per square. The average bundle of shingles covers 33.3 ft2, so three bundles of shingles are needed per square.
(a square of shingles covers 100 square feet of roof.) step 3 look at the bundle (package) of cedar shingles to see how many bundles it takes to make up one square of shingles. The average bundle of shingles covers 33.3 ft 2, so three bundles of shingles are needed per square. So, each bundle provides about 33 square feet of coverage.
By 36 in.) in each bundle. Four bundles to a square. Singles are often sold in bundles, with three bundles equaling one roof square.
4” 4” installation the following is a general summary of the installation methods. The average bundle of shingles covers 33.3 ft2, so three bundles of shingles are needed per square. However, shingles are packaged in a way that they are light enough for the average person to carry around.
A bundle is around 1/3 of a square and costs an average of $30 to $600. Each square will need three bundles of shingles to cover it. On the low end, asphalt averages $100 and $150 per square while slate averages about $1,200.
The average bundle of shingles covers 33.3 ft 2, so three bundles of shingles are needed per square. Or 4 bundles for a roof square which is 100 sq. As you prepare for your roofing project, you'll use squares to determine the amount of roofing materials you need, from asphalt shingles down to the underlayment.
One bundle of shingles covers roughly 33 square feet of roof. Detailed installation instructions are supplied on each bundle of presidential shake tl and presidential shake shingles and must be followed. Ft.) what about the waste factor?
Normal architectural shingles are approximately 3 bundles per square. A square of shingles is the number of shingles needed to cover 100 sq feet of roof area. 3 bundles = 1 roofing square (98.4 sq.
So if its a 2100 sq.ft roof you need 63 bundles plus ridge and starter which can be done with shingles or they make ridge caps and s. The square footage of a roof is normally measured in squares of 100 feet. There are 26 standard shingles in a bundle.
Generally with standard fibreglass reinforced asphalt shingles there is 33.3 square feet in one bundle. This unit of measurement determines how many shingles you'll need for every 100 sq. Each bundle covers 33.3 sq.
Remember, a good wood shingle roof can last from 25 to 30 years, even up to 50 years in some cases. How much does a bundle of shingles cover? You want to order extra material to account for waste.
You can also salvage waste shingle pieces and portions of damaged shingles for use as caps. Each bundle of cedar shingles should cover about 33.3 sq. So just order one bundle for each sheet of roof sheathing.
Separate application sheets may also be obtained from certainteed. Of roof area—close enough to the 32 sq. In roofing, one square equals 10 x 10 ft., or 100 sq.
Should i order extra roofing materials? How many bundles of shingles in a square you can’t merely go by the number of shingles in a bundle since shingle size and coverage vary from one brand to the next. Some manufacturer bundles will cover slightly less and will range from 28 up to 33 square feet.
A square represents 100 square feet of area covered by shingles. That is 3 bundles covers 100 square. A bundle of shingles is a package that includes about 33.3 square feet of shingles.
How many square feet does a bundle of shingles cover? The average bundle of shingles covers 33.3 ft 2, so three bundles of shingles are needed per square. Find the total square footage of your roof by multiplying the width by the length and then divide that number by 100 and you will get the number of squares you need to cover.
A package of shingles is called a bundle. Most shingles come 3 bundles to a square — which is equal to 100 square feet. For some brands and types, the square feet per bundle may vary.
Standard 3 tab shingles its 3 bundles to a square and architectural shingles its 4 bundles to a square on most. One bundle of shingles covers approximately 33 square feet of roof, depending on the size and weight of the shingles. Furthermore, how many bundles of shingles do i need for 800 square feet?
A square is 10' x 10' or 100 sq.ft. If the shingles you are using come three bundles to a square, calculating the number of bundles you’ll need is simple. A square is 100sq feet, if your using some of the heavier roofing, most one bundle of shingle covers about 33 square feet of roof hence around 850 bundles of shingles will be needed for 28000 square feet roof!
Shingles are sold by the bundle and by the square. A square is equal to 100 square feet and will cost anywhere between $100 and $1,800.
bundle, does, square


I am a research associate in computational neuroscience at University College London in the Cortical Processing Laboratory, led by Kenneth Harris and Matteo Carandini. I am developing mathematical tools and software for analyzing extracellular recordings from high-density multi-electrode probes.
I hold a Ph.D. in computational neuroscience, obtained under the supervision of Romain Brette in the École Normale Supérieure (ENS), Paris. I worked on the computational role of correlations in neural coding and in intracellular data analysis. Previously, I graduated from the Mathematics and Computer Science departments of the ENS, and I made research internships at Princeton University and Collège de France.
I’m interested in spike-based computation, temporal coding, neuronal correlations and synchronization, coincidence detection, analysis of intracellular and extracellular electrophysiological recordings. I use mathematical tools (dynamical systems, probability theory and stochastic processes, statistics and machine learning) and computational tools (spiking neural network simulations, numerical optimization, parallel/distributed computing and general-purpose computing on graphics processing units (GPGPU)).
I’ve worked on coincidence detection in noisy neurons and in electrode compensation techniques for current clamp in-vitro recordings. I’ve also developed a model fitting toolbox for spiking neuron models with both CPU and GPU implementations, which is now integrated in the neural simulator Brian. The toolbox is based on Playdoh, a Python library for parallel and distributed scientific computing that I’ve created with Bertrand Fontaine and Dan Goodman.
I’ve created a high-performance visualization package in Python named Galry, and a graphical software for the manual stage of spike sorting, adapted to recordings with large multi-electrode arrays, named KlustaViewa. I’m now involved in the successor of Galry, Vispy, in collaboration with other developers of visualization toolkits. More details are available on the Research page.
- Rossant C, Fontaine B, Magnusson AK, Brette R (2012). A calibration-free electrode compensation method. Journal of Neurophysiology, 108(9): 2629-2639; doi: 10.1152/jn.01122.2011.
- Rossant C, Leijon S, Magnusson AK, Brette R (2011). Sensitivity of noisy neurons to coincident inputs. Journal of Neuroscience, 31(47): 17193-17206; doi: 10.1523/JNEUROSCI.2482-11.2011.
- Rossant C, Goodman DF, Fontaine B, Platkiewicz J, Magnusson A, Brette B (2011). Fitting neuron models to spike trains. Frontiers in Neuroscience, 5:9. doi:10.3389/fnins.2011.00009.
- Rossant C, Fontaine B, Goodman DF (2011). Playdoh: a lightweight Python library for distributed computing and optimisation. Journal of Computational Science. doi:10.1016/j.jocs.2011.06.002.


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