With the Econ exam just two days away, many students out there are wondering what to expect on the new higher level paper 3, the “quantitative methods” paper. I put this post together to share my own understanding of all the calculations students may be asked to do on this section of the exam.
First, let’s review the command terms you can expect to encounter on HL paper 3, and look at what is expected of students in questions of each type.
Command terms to expect on paper 3:
- Calculate: “Obtain a numerical answer showing the relevant stages in the working.”
- Construct: “Display information in a diagrammatic or logical form.”
- Derive: “Manipulate a mathematical relationship to give a new equation or relationship.”
- Determine: “Obtain the only possible answer.”
- Draw: “Represent by means of a labelled, accurate diagram or graph, using a pencil. A ruler (straight edge) should be used for straight lines. Diagrams should be drawn to scale. Graphs should have points correctly plotted (if appropriate) and joined in a straight line or smooth curve.
- Identify: “Provide an answer from a number of possibilities.”
- Label: “Add labels to a diagram.”
- Plot: “Mark the position of points on a diagram.”
- Show: “Give the steps in a calculation or derivation.”
- Show that: “Obtain the required result (possibly using information given) without the formality of proof. “Show that” questions do not generally require the use of a calculator.”
- Sketch: “Represent by means of a diagram or graph (labelled as appropriate). The sketch should give a general idea of the required shape or relationship, and should include relevant features.”
- Solve: “Obtain the answer(s) using algebraic and/or numerical and/or graphical methods.”
Calculations you may be required to make:
Section 1 Microeconomics
1. Explain a demand function (equation) of the form Qd = a – bP.
2. Plot a demand curve from a linear function (eg. Qd = 60 – 5P).
3. Identify the slope of the demand curve as the slope of the demand function Qd = a – bP, that is –b (the coefficient of P).
4. Outline why, if the “a” term changes, there will be a shift of the demand curve.
5. Outline how a change in “b” affects the steepness of the demand curve.
1. Explain a supply function (equation) of the form Qs = c + dP.
2. Plot a supply curve from a linear function (eg, Qs = –30 + 20 P).
3. Identify the slope of the supply curve as the slope of the supply function Qs = c + dP, that is d (the coefficient of P).
4. Outline why, if the “c” term changes, there will be a shift of the supply curve.
5. Outline how a change in “d” affects the steepness of the supply curve.
1. Calculate the equilibrium price and equilibrium quantity from linear demand and supply functions.
HOW TO: Set the demand and supply equations equal to one another and solve for P. Once you know the equilibrium price, plug it into either equation to find the quantity.
2. Plot demand and supply curves from linear functions, and identify the equilibrium price and equilibrium quantity.
Find the q-intercept of demand (this is the ‘a’ variable)
Find the p-intercept of both demand and supply (set Q = 0 and solve for P in both equations)
Draw the demand curve by connecting the q-intercept and the p-intercept of demand
Find the equilibrium price
Plot the supply curve by connecting the p-intercept of supply and the equilibrium price.
Draw dotted lines over to the equilibrium price and down to the equilibrium quantity.
3. State the quantity of excess demand or excess supply in the above diagrams.
HOW TO: There will only be an excess demand or supply if the question asks how a price OTHER THAN the equilibrium would affect the market. See the section on price controls below for more.
1. Calculate PED using the following equation: PED=% change in Qd% change in P
2. Calculate PED between two designated points on a demand curve using the PED equation above.
3. Calculate XED using the following equation: XED = % change in Qd of good X% change in price of good Y
4. Calculate YED using the following equation: YED=% change in Qd% change in income
5. Calculate PES using the following equation: PES=% change in Qs% change in price
1.3 Government Intervention
1. Plot demand and supply curves for a product from linear functions and then illustrate the effects of a specific tax.
2. Calculate the effects of the imposition of a specific tax on the market (on price, quantity, consumer expenditure, producer revenue, government revenue, consumer surplus and producer surplus).
HOW TO: A tax takes money AWAY from the seller of a product. So to caclulate the effect of the tax in a supply equation you must subtract the amount of the tax from the price the seller receives.
For example: assume the supply of pencils is represented by Qs = -10 + 3P, and a $2 per unit tax is places on pencils. The new supply equation is: Qs = -10 + 3(P-2). Simplified, this gives us Qs = -16 + 3P. This is the new supply equation. Graphically, the supply curve has shifted leftwards by 6 units, or “upwards” by $2.
1. Plot demand and supply curves for a product from linear functions and then illustrate and/or calculate the effects of the provision of a subsidy on the market (on price, quantity, consumer expenditure, producer revenue, government expenditure, consumer surplus and producer surplus).
HOW TO: A subsidy is like a “tax in reverse”. It is a payment to the seller ABOVE what consumers pay, so it’s added to the price in the supply equation.
Assume pencil producers receive a $2 subsidy per pencil. With an original supply equation of Qs = -10 + 3P, the new supply is: -10 + 3(P+2). Simplified, this gives us Qs = -4+3P. The supply curve has shifted rightward by 6 units, or down by $2.
1. Calculate possible effects from the price ceiling diagram, including the resulting shortage and the change in consumer expenditure (which is equal to the change in firm revenue).
2. Calculate possible effects from the price floor diagram, including the resulting surplus, the change in consumer expenditure, the change in producer revenue, and government expenditure to purchase the surplus.
HOW TO: Simply plug the price floor or ceiling into the demand and supply equations to find the quantities that would be supplied and demanded. If there’s an effective price floor, the Qs should be greater than the Qd, meaning there’s a surplus. If there’s a price ceiling, Qd should be greater than Qs, meaning there’s a shortage.
1.5 Theory of the Firm
1. Calculate total, average and marginal product from a set of data and/or diagrams.
HOW TO: Total product is the output of workers as the number of workers increases.
Average product is the “output per worker” = TP/# of workers
Marginal product is the “output of the last worker” = Change in total product / change in the number of workers.
2. Calculate total fixed costs, total variable costs, total costs, average fixed costs, average variable costs, average total costs and marginal costs from a set of data and/or diagrams.
HOW TO: TFC is constant as output increases. It is the firm’s total cost when output = 0.
TVC increases as output increase, at first at a decreasing rate (due to increasing returns), and then at an increasing rate (due to diminishing marginal returns).
TC = TVC + TFC. If you know the firm’s fixed costs and its variable costs, TC can easily be calculated.
AFC = TFC / Q of output. AFC falls as output increases as the firm “spreads its overhead”. Graphically, it is the distance between AVC and ATC.
AVC = TVC / Quantity of output. AVC falls at first due to increasing returns and then increases due to diminishing returns. MC and AVC should intersect at the lowest point of AVC
ATC = AFC + AVC, or TC / Quantity of output. ATC lies ABOVE the AVC curve (since it includes the average fixed costs), and will intersect MC at its lowest pont.
MC = the change in TC / the change in output. It is the cost of the last unit produced. MC sloped down when a firm’s workers experience increasing returns and upwards once the firm experiences diminishing marginal returns.
1. Calculate total revenue, average revenue and marginal revenue from a set of data and/or diagrams.
HOW TO: Total revenue (TR) = price X quantity.
Average revenue (AR) is simply the price of the good. The demand curve can be labelled “D=AR=P” to help you remember this.
Marginal revenue (MR) = the change in total revenue divided by the change in quantity. This is the change in total revenue resulting from the last unit sold. For a PC firm, MR is constant and equal to the market price (since the firm is a price taker and can sell additional units for the same price.) But for an imperfectly competitive firm, MR is lower than price beyond the first unit of output, since the firm must lower its price to sell additional units of output. MR fall twice as steeply as the D=AR=P curve in an imperfect competitor diagram.
1. Calculate different profit levels from a set of data and/or diagrams.
HOW TO: Economic profit is usually found by the following equation. Profit = (P-ATC)Q. Find the per-unit profit (P-ATC) and multiply it by the quantity of output (Q).
If you are given total revenue (TR) and total cost (TC) data, than economic profit = TR-TC.
If ATC>P or if TC>TR, then the firm’s profit is negative, and it is earning losses.
1. Calculate the short run shutdown price and the breakeven price from a set of data
HOW TO: A firm should shut down if the price in the market is lower than the firm’s minimum average variable cost. At this point, the firm’s total losses are greater than its total fixed costs, so it will LOSE LESS by shutting down!
A firm will break even when the price in the market equals the firm’s minimum ATC, or if the TR = TC (see above). Economic profits = 0.
1. Calculate from a set of data and/or diagrams the revenue maximizing level of output.
HOW TO: Total revenue is maximized when MR=0. If you have a table you can calculate the change in TR at each level of output, and when this equals zero, the firm would be maximizing its total revenues. Anything beyond this level of output, MR will be negative and the firm’s revenues will begin to fall.
Section 2 Macroeconomics
1. Calculate nominal GDP from sets of national income data, using the expenditure approach.
HOW TO: Nominal GDP is the quantity of output in a particular year multiplied by the prices in that year.
2. Calculate GNP/GNI from data
HOW TO: The difference between GDP and GNP is that you must SUBTRACT the value of output produced in a nation by companies based in other nations, but you must ADD the value of output produced in other nations by companies based in the nation you are calculating GNP for.
3. Calculate real GDP, using a price deflator.
HOW TO: Real GDP is the value of a nation’s output in a particular year measured using the prices from a base year. So you must multiply the quantity from the year in question by the prices from the base year (which should be provided).
If you are not given price and quantity data, rather you are given the GDP deflator price index, you can divide the nominal GDP for a particular by the GDP deflator for that year, and multiply by 100 to get the real GDP.
If you know the nominal GDP and the real GDP and are asked to calculate the GDP deflator, you simply divide the nominal by the real and multiply by 100.
If you have two years’ GDP deflators, and are asked to calculate the inflation between those years, you simply find the percentage change in the GDP deflator price indexes between the years given.
1. Calculate the multiplier using either of the following formulae: k=11-MPC or 1MRl=(MPS+MRT+MPM)
2. Use the multiplier to calculate the effect on GDP of a change in an injection in investment, government spending or exports.
HOW TO: If you know the MPC, you can estimate the effect of any change in spending in the economy on total GDP. For example. Assume the MPC = 0.8 and Net exports increase by $100 million. Calculate the total change in GDP.
k = 1/(1-0.8) = 5.
Now multiply the change in net exports by the multiplier: $100 million X 5 = $500 million
GDP will increase by $500 million as a result of the increase in net exports of $100 million.
2.3 Macroeconomic Objectives
1. Calculate the unemployment rate from a set of data.
HOW TO: The unemployment rate is the proportion of the labor force that is unemployed. This means they are actively seeking work but unable to find it.
You may be given a table showing the number of people in different groups, like college students, retirees, people looking for jobs, people who have given up looking for jobs, part time workers, full time workers, etc… You will have to calculate the unemployment rate from this information.
NOTE: People who are working part time but want to work full time ARE EMPLOYED. People who have given up looking for jobs are DISCOURAGED WORKERS and are no longer considered unemployed, rather, they have dropped out of the labor force. Discouraged workers are not accounted for in unemployment data.
2. Construct a weighted price index, using a set of data provided.
HOW TO: Refer to pages 303 – 305 in the textbook (Pearson Baccalaureate Economics) for a worked solution to this type of problem. It’s not difficult math, you just have to know how to do it.
3. Calculate the inflation rate from a set of data.
HOW TO: The inflation rate = (CPI year 2 – CPI year 1)/CPI year 1. It is the rate of change in the CPI between two years. You do NOT always simply take the CPI and calculate the rate of change since it was 100. This would tell you how much inflation there was since the base year, but inflation is usually measured between two years.
4. Calculate the rate of economic growth from a set of data.
HOW TO: The rate of economic growth is the percentage change in the real GDP between two periods. = (GDP year 2 – GDP year 1)/GDP year 1
5. Calculate the marginal rate of tax and the average rate of tax from a set of data.
HOW TO: Refer to pages 361 – 362 in the text. The important thing here is to take your time and do the math very carefully.
The difference between the marginal rate and the average rates is as follows:
Assume the marginal tax rates for Country S are:
0-20,000 – 5%
20,001 – 50,000 – 15%
50,001 – 100,000 – 25%
100,001 – above – 35%
Assume an individual earns $75,000. Calculate:
- The amount of tax he will pay:
- He will pay 5% on the first 20,000. 20,000 x 0.05 = $1,000
- He will pay 15% on the NEXT 30,000 (up to $50,000). 30,000 x 0.15 = $4,500
- He will pay 25% on the NEXT $25,000 (the income he earns between $50,000 and $75,000). 25,000 x 0.25 = $6,250.
- Total tax paid = 1,000+4,500+6,250 = $11,750
- His average tax rate: This is his total tax liability divided by his income = $11,750 / $75,000 = 0.1567 x 100 = 15.67%.
You may also be given data on a INDIRECT taxes, which are taxes on consumption (such as a VAT). You may be told that the same individual consumes $45,000 of his income, of which 8% was paid in consumption taxes. Calculate the percentage of his income paid in indirect taxes:
$45,000 x 0.08 = $3,600 paid in indirect taxes.
$3,600 / $75,000 = 0.048 x 100 = 4.8% of his income paid in indirect taxes.
Section 3 International Economics
3.1 Free trade:
1. Calculate opportunity costs from a set of data in order to identify comparative advantage.
HOW TO: You may be given a production possibilities table OR a production possibilities curve for two countries. With either of these, you can calculate the opportunity costs of the countries for the two goods shown and determine comparative advantage.
For example: Assume Country S and Country I can produce the following amounts of wine and cheese on a single hectare of land
- Country S: 4 barrels of wine and 5 tons of cheese
- Country I: 8 barrels of wine and 8 tons of cheese
If the two countries were to specialize and trade, who should produce what?
- In country S: 4w = 5c so 1w = 1.25c.
- 5c = 4w, so 1c = 0.8w
- In country I: 8w = 8c, so 1w=1c
In Country S, 1 ton of cheese costs only 0.8 ton of wine, while 1 barrel of wine costs 1.25 cheeses. In Country I cheese “costs” more (1 ton of wine), while wine “costs” less (only 1 ton of cheese).
So Country S should specialize in cheese, because it is cheaper, and Country I in wine. The two countries can then trade and enjoy the foreign produced good at a lower opportunity cost than they could have achieved domestically.
1. Calculate from diagrams the effects of imposing a tariff on imported goods on different stakeholders, including domestic producers, foreign producers, consumers and the government.
HOW TO: If you are comfortable with the tariff diagram and can identify the different areas identified above, then you simply find the AREA of those shapes on the graph to determine the costs, benefits, and revenues necessary.
2. Calculate from diagrams the effects of setting a quota on foreign producers on different stakeholders, including domestic producers, foreign producers, consumers and the government.
HOW TO: Same as for tariffs. Once you are familiar with the graph, simply find the areas representing the impact on each of the stakeholders.
3. Calculate from diagrams the effects of giving a subsidy to domestic producers on different stakeholders, including domestic producers, foreign producers, consumers and the government.
HOW TO: Calculate the areas on a graph! EASY!
3.3. Exchange Rates:
1. Calculate the value of one currency in terms of another currency.
HOW TO: If you know the price of one currency in terms of another, you can quickly find the price of the other.
For example. If 1 euro = 1.2 CHF, then 1 CHF = 1/1.2 euro, or 0.83 Euro. The price of one currency is the inverse of the price of the other.
2. Calculate the exchange rate for linear demand and supply functions.
HOW TO: If you can calculate equilibrium price and quantity for a good using linear equations, then finding the equilibrium exchange rate is easy. Given the equations for two currencies, simply set them equal to each other and find the equilibrium.
Example: Assume the demand for Euros in Switzerland is represented by the equation Qd = 10 -2P, where P is the exchange rate of the Euro in CHF.
Supply of Euros in Switzerland is represented by the equation Qs = -3 + 4P where P is the exchange rate of the Euro in CHF.
3. Calculate the equilibrium exchange rate of the Euro in Switzerland.
Set the demand and supply equal to one another: 8 – 4P = -4 + 6P, and solve for P.
12 = 10P.
P = 12/10 = 1.2 CHF / Euro
4. Plot demand and supply curves for a currency from linear functions and identify the equilibrium exchange rate.
HOW TO: This is the same as plotting linear demand and supply equations for any good.
- Find the q-intercept of demand (this is the ‘a’ variable)
- Find the p-intercept of both demand and supply (set Q = 0 and solve for P in both equations)
- Draw the demand curve by connecting the q-intercept and the p-intercept
- Find the equilibrium price (or exchange rate in this case).
- Plot the supply curve by connecting the p-intercept of supply and the equilibrium exchange rate.
- Draw dotted lines over to the equilibrium exchange rate and down to the equilibrium quantity.
5. Using exchange rates, calculate the price of a good in different currencies.
HOW TO: If you know the price of a good in one currency and you know the exchange rate between that currency and another, you can always find the price of the good in the other currency.
For example: A hotel room in London costs 250 British Pounds per night. The British Pound exchange rate in Switzerland is 1.5 CHF / pound. How much does the London hotel room cost in CHF?
Convert the room’s price to CHF. So, 250 x 1.5 = 375 CHF per night.
6. Calculate the changes in the value of a currency from a set of data.
HOW TO: Consider the following. A London hotel room that costs 250 pounds per night used to cost 375 CHF. Following a change in the exchange rate, the same room now costs 400 CHF. Calculate the new exchange rate of British Pounds in Switzerland.
The old exchange rate was: 375 / 250 = 1.5 CHF / GBP. The new exchange rate, therefore, is 400 / 250 = 1.6 CHF / GBP
3.4 Balance of Payments:
1. Calculate elements of the balance of payments from a set of data.
HOW TO: You must know the components of a nation’s current and financial accounts, and be able to determine from a set of payments data which type falls into which category, and whether it counts as a credit or a debit.
For practice, complete the HL exercise on page 491 (with answers in the back of the book).


Part 1: The Chocolate Factory which covered the regrouping or trading aspect of place value and explored regrouping in base ten and other bases.
Part 2: Base Ten for Young Students which introduced several games and trading activities to help young children acquire a solid foundation in place value.
Part 3: The Bake Sale demonstrated the role of place value in long division.
Part 4: Geometry of Place Value demonstrated the dimensions of a cube in terms of place value and explored the geometric representation of the quadratic equation.
In the conclusion of the series, Part 5, Applications of Place Value, I will show you a sampling of interesting applications of place value to some real-life situations.
It is sometimes easy to add and subtract fractions by putting fractions in place-value-type columns. The usual addition and subtraction with regrouping requires students to conceptually pack or unpack “boxes” (See Part 1: The Chocolate Factory). What if I want to add 3 5/7 + 2 4/7?
The denominator, 7, means that the “whole” has been divided into seven equal pieces. Notice that instead of packing in boxes, the student creates “wholes.” With nine-sevenths, there are enough pieces to make one whole with 2 pieces or two-sevenths left over. There is that word, “leftover” again. In The Chocolate Factory, “leftover” was used to label the ones. Here we are using it to label the pieces, helping students to transfer and integrate concepts.
If students need to “borrow” (the current term is “exchange”) in order to subtract fractions, then they would have to cut up a whole into the necessary number of equal pieces. In the case of 4 2/5 – 1 3/5, the student creates two place-value columns, the place value of one being the “wholes” and the place value of the other being the “pieces.”
With fractions of unlike denominators, the procedure is essentially the same with the preliminary step of finding common denominators.
The Olympics and Time
The odometer of a car is obviously a place value representation, but have you noticed that the clock at the bottom of an Olympic race is formatted in terms of place value with colons marking the separation between hours, minutes, seconds, and hundredths of a second? As the race progresses, the clock gathers hundredths into seconds, then seconds into minutes, minutes into hours. The expression of a runner's time might be 2:14:52:27.
Thinking of time in terms of place value columns simplifies addition and subtraction.
Here's a (sort of) real life problem that occurred in my house recently. My son was invited to watch Lord of the Rings at a friend's house. If the DVD starts playing at 4:17:48 pm and the movie lasts 3 hours 44 minutes and 25 seconds, will my son be able to catch the 8:15 bus home?
Starting with the seconds column, 48 seconds plus 25 seconds equals 73 seconds which is 60 plus 13 seconds. Since 60 seconds equals one minute, cross off the “60” and carry the one to the top of the minutes column. One minute plus 17 minutes plus 44 minutes equals 62 minutes which is 60 plus 2 minutes. Since 60 minutes equals one hour, cross off the “60” and carry the one to the top of the hours column. One plus 4 plus 3 equals 8. The solution: The DVD will end at 8:02:13 pm. The answer: The DVD will end at 8:03 pm, in time to make the bus. “Packing” seconds into minutes and minutes into hours is just like packing chocolates into boxes and boxes into cases.
Subtraction is a matter of unpacking then. Suppose we want to work backwards. If the DVD must end by 8:10 pm to make the bus, what is that latest time it can start. The problem is 8:10:00 minus 3:44:25.
We must subtract 25 seconds from zero seconds. So we just unpack one minute or dump a minute into the seconds columns. Since we unpacked one of the 10 minutes, there are 9 minutes in the minutes column and 60 seconds in the seconds column which still represents the original ten minutes but in a slightly different form. Now it is easy to subtract 25 seconds from 60 seconds. Now we have to subtract 44 minutes from 9 minutes. Simple. Just unpack one hour into minutes, leaving 7 hours and giving 69 minutes. Now it is easy to subtract the 44 minutes. The solution: the latest the DVD can start playing is 4:25:35 pm. The answer: the latest the DVD can start playing is 4:25 pm.
(An small digression: I have twice made a distinction between the solution and the answer with the movie problems. I did not make this distinction with the fraction problems. The fraction problems did not require the distinction because they were context-free problems.
The movie problems were word problems or story problems. They have context. I teach students to find a solution, then interpret the solution to find the best or most reasonable answer to the question. Home clocks are not like Olympic clocks, and rounding by the rules does not always produce the best answer. In the case of the second movie problem, rounding to 4:26 pm might cause my son to either miss the bus or run like the dickens to catch it.
In real life, people cannot stop when they have found the solution. They must then apply the solution to the situation, or context, in the most reasonable way. We are failing to teach our children critical thinking when we allow them, even encourage them to conflate solution with answer).
Number base clocks are a manipulative that use the student's internalized understanding of clock time as a peg to hang the concept of bases. A base-7 clock means that a rotation of 7 “minutes” equals one base-7 “hour.” Using a variety of base clocks as aids, students can become quite adept at adding and subtracting in different bases. It is even possible to extend the skill to multiplication and division in bases as well.
Calendars can also be used to teach in place value especially if you confine yourself to hours, days, weeks, years because the months are not uniform “packages” of days or weeks. In calendar math, the number 14 would be 1 week, 4 days. The number 305 would be 3 months, 0 weeks and 5 days.
You can use the calendar to explore interesting patterns numerically as Michael Naylor does. Although he does not specifically target place value, the algebraic pattern partly follows from the place value of a calendar.
Fast box addition (Grades 6-8)
Have a student choose a 2 x 2 box and demonstrate how you are able to quickly give the total. Tell your students the secret: Add 4 to the first number and multiply the result by 4. Have the class test the result on several boxes.
The secret is algebra; if the first number is x, the other numbers are x + 1, x + 7 and x + 8. The total is 4x + 16, which is the same as 4(x + 4).
Have your students outline a 3 x 3 box and ask them which is greater – the sum of all of the numbers or 9 times the center number? Relabel the center number as x and write all the other numbers in terms of x as shown here:
When adding all of those terms, the constants cancel (–8 + 8 = 0, –7 + 7 = 0, etc.) so all that is left 9x. The total of all nine numbers, then, is 9 times the center number.
One teacher considers the calendar as probably the one math essential for her kindergarten class even though she may not necessarily focus on place value. Her website lists activities and links.
What we see today is that although regular number bases maintain column values in terms of powers of the base, place value is way more flexible. Each place, or column, can be whatever it needs to be as long as it is in terms of groups of the preceding place.
Fractions: Wholes, parts
Olympic Time: Hours, Minutes, Seconds, Hundredths of a Second
Calendars: Years, Months, Weeks, Days
Gettysburg Time: Score, Days
You can have more fun working in multiple places with
Historical Time: Millenia, Centuries, Decades, Years, Months, Weeks, Days, Hours, Minutes, Seconds, Hundredths of a Second. That's eleven places. Once the students understand grouping and ungrouping, they do not worry about the number of places. The more places, the more fun.
Maybe you can think of some more ways we use place value everyday.
O'Block Books sells manipulatives.
So does Creative Teaching Press.
Base ten packing set from Digi-Block.
"Great Source" for calendar math.


Frequency Stability & Precision Timing Toolset™Perform allan deviation, power oscillation, phase noise measurement, and more.
EZL’s Time Domain Frequency Stability and Precision Timing Toolset™ includes the most widely used statistical analysis tools for frequency stability and precise time and time interval (PTTI) data.
The time domain stability of a frequency source is measured as a statistic of its phase or frequency fluctuations, as a function of time. Because the noise types commonly associated with time and frequency sources are divergent, common metrics – such as the standard variance, which assumes a constant mean – cannot be applied. Therefore, other variances have been developed for the time and frequency metrology, which provide far better characterization.
The Allan Variance (typically presented as its square-root – the Allan Deviation) is the most common time domain measure of frequency stability.
The National Institute of Standards and Technology (NIST) provides a great report of time domain frequency stability analysis in their Handbook of Frequency Stability Analysis.
- Allan Deviation1 (ADEV) – Allan Variance (AVAR)
- Modified (MDEV)
- Hadamard (HDEV)
- Time (TDEV)
- Power Oscillation
- Phase noise measurement
- Built-in Noise generator
- Generate statistically accurate noise data sets of unlimited size
- Model your noise sets with one or more weighted noise types, including
- Random Walk Frequency Modulated (FM)
- Flicker FM
- White FM
- Flicker Phase Modulated (PM)
- White PM
1 Visit the wikipedia page on Allan Variances for some great details on this very useful statistic!


When? Tuesday May 28 2019, 08:30 - 12:15
We are proud to present a high-quality program with speakers from different communities. Speakers who want to publish their talks after the satellite can send their slides via E-Mail.
|08:30 - 08:45||Ingo Scholtes
|Session chair:||Ingo Scholtes|
|08:45 - 09:10||Bailey Fosdick (Department of Statistics, Colorado State University, CO, USA)
Invited Talk: Inferring Influence Networks from Longitudinal Bipartite Relational Data Longitudinal bipartite relational data characterize the evolution of relations between pairs of actors, where actors are of two distinct types and relations exist only between disparate types. A common goal is to understand the temporal dependencies, specifically which actor relations incite later actor relations. There are two primary approaches to this problem. The first projects the bipartite data in each time period to a unipartite network and uses existing unipartite network models. Unfortunately, information is lost in calculating the projection and generative models for networks obtained through this process are scarce. The second approach represents dependencies using two unipartite influence networks, corresponding to the two actor types. Existing models taking this approach are bilinear in the influence networks, creating challenges in computation and interpretation. In this talk, we introduce a novel generative model that permits estimation of weighted, directed influence networks and does not suffer from these shortcomings.
|09:10 - 09:35||Austin R. Benson (Department of Computer Science, Cornell University, USA)
Invited Talk: Graph-based Semi-Supervised Learning for Edge Flows We develop a graph-based semi-supervised learning method for learning edge flows defined on a graph. Specifically, we are given flow measurements on a subset of edges and want to predict the flows on the remaining edges. To this end, we develop a computational framework that imposes certain constraints on the overall flows, such as (approximate) flow conservation. These constraints render our approach different from classical graph-based semi-supervised learning for vertex labels, which posits that tightly connected nodes share similar labels and leverages the graph structure accordingly to extrapolate from a few vertex labels to the unlabeled vertices. We demonstrate our method on traffic flow data on road networks.
|09:35 - 10:00||Mikko Kivelä (Department of Computer Science, Aalto University, FI)
Invited Talk: Microcanonical randomized reference models for temporal networks Temporal networks have turned out to be a useful and popular generalization of a simple static graph. The higher-order interactions in temporal networks have lead to a surge of various ways of constructing randomized reference models which are implemented as procedures of shuffling these interactions in empirical data. These models have turned out to be a versatile toolbox for studying temporal networks, but at the same time these models and their relationships to each other have been poorly understood, rendering their use non-trivial and susceptible to misinterpretation. I will talk about the framework of microcanonical randomized reference models (MRRMs), which are defined as ensembles of random networks with given features constrained to match those of an input (empirical) network. MRRMs can be used as a unified framework for classifying and understanding of various reference models in the literature. This framework can be used to build a taxonomy of such model that proposes a canonical naming convention, classifies them, and deduces their effects on a range of important network features. Furthermore certain classes of compatible MRRMs may be applied in sequential composition to generate over a hundred new MRRMs from the existing ones we have surveyed. I will give examples of how to use these models to systematically analyse a range of features and how to add new models to this framework. The modeling framework itself is not dependent on temporal networks as object of study, but could potentially be applied to any other higher-order network structure in which there are a large number of meaningful ways of randomizing the observed data.
|10:00 - 10:30||Coffee Break and Poster Session|
|Session chair:||Ingo Scholtes|
|10:30 - 10:55||Heather Harrington (Mathematical Institute, University of Oxford, UK)
Invited Talk: Topological data analysis for investigation of dynamics and networks
Persistent homology (PH) is a technique in topological data analysis that allows one to examine features in data across multiple scales in a robust and mathematically principled manner, and it is being applied to an increasingly diverse set of applications. We investigate applications of PH to dynamics and networks, focusing on two settings: dynamics on a network and dynamics of a network.
Dynamics on a network: a contagion spreading on a network is influenced by the spatial embeddedness of the network. In modern day, contagions can spread as a wave and through the appearance of new clusters via long-range edges, such as international flights. We study contagions by constructing ‘contagion maps’ that use multiple contagions on a network to map the nodes as a point cloud. By analyzing the topology, geometry, and dimensionality of manifold structure in such point clouds, we reveal insights to aid in the modelling, forecast, and control of spreading processes.
Dynamics of a network: one can construct static graph snapshots to represent a network that changes in time (e.g. edges are added/removed). We show that partitionings of a network of random-graph ensembles into snapshots using existing methods often lack meaningful temporal structure that corresponds to features of the underlying system. We apply persistent homology to track the topology of a network over time and distinguish important temporal features from trivial ones. We define two types of topological spaces derived from temporal networks and use persistent homology to generate a temporal profile for a network. We show that the methods we apply from computational topology can distinguish temporal distributions and provide a high-level summary of temporal structure.
|10:55 - 11:20||Timothy LaRock (Network Science Institute, Northeastern University, USA)
Invited Talk: Detecting Path Anomalies in Time Series Data on Networks Abstract: We are often interested in studying the mechanisms behind sequential movement through networks, such as humans moving through transportation networks or seeking information by navigating the web. Understanding sequential data on networks requires the development of appropriate representations and null models to compare observed data against. In this talk, we present a null model for higher-order de Bruijn graphs based on the Generalized Hypergeometric Ensemble of Graphs, a generalization of the configuration model. We show how this null model can be applied to discover pathways that occur statistically significantly more or less often in the observed data than expected at random, which we name path anomalies. We then analyze path anomalies detected in synthetic and real-world data, demonstrating how these anomalies inform our understanding of the underlying system.
|11:20 - 11:45||Alexandre Bovet (ICTEAM, UC Louvain, BE)
Invited Talk: Community detection in non-stationary temporal networks
Many temporal networks exhibit non-stationary dynamics, such as cyclical patterns due to daily, weekly, seasonal or yearly cycles, increase or decrease in population size or drastic change of dynamical regime. Several works have generalized existing community detection methods for static networks to temporal networks, but they usually rely on the assumption of an underlying stationary process, or sequences of different stationary epochs, and a null model corresponding to the stationary state of the process. Here, we propose first-principle method allowing to take into account continuous time temporal networks, interactions that may have a duration and systems that non-necessarily reach a steady state, or follow a sequence of stationary states. Our approach is based on the concept of the stability of a network partition generalized to temporal networks with non-Markovian and non-stationary dynamics. This is achieved by looking at the autocorrelation of scalar signals on the network nodes as observed by continuous time random walkers respecting the activation time of edges. In our case, the null model of random walk transition probabilities does not assume the existence of a stationary state of the random walks. This approach opens the doors for the definition of new concepts of spatio-temporal network structures to detect cyclical or transient structures. We show how our method, applied to datasets of contact networks, allows to recover the community dynamics.
|11:45 - 12:00||Christoph Gote (ETH Zürich, Zurich, CH)
Talk: Multi-order network models based on path data In this talk we propose a generalised the multi-order graph model considering both inter-layer and intra-layer transitions between higher-order models up to a maximum model order. Resulting multi-order models are fully described by a supra-transition matrix motivated by the concept of supra-adjacency matrix used in the domain of multi-layer models. We provide a method for model selection, allowing to determine the most suitable maximum model order based on AIC. To this end, we show how AIC computation is adjusted to model paths in sparse networks. Finally, we show preliminary results demonstrating how the resulting multi-order models can be used for path generation as well as path length prediction.
|12:00 - 12:15||Luka Petrović (Department of Computer Science, University of Zürich, CH)
Talk: Counting Causal Paths in Temporal Networks Static network models do not give a complete picture of paths in a temporal network, because the arrow of time prohibits many paths that exist in the time-aggregated, static network. Higher-order models fitted to the statistics of paths in temporal networks can mend this deficiency. They can be used to improve model selection of higher order models, diffusion models, centrality rankings, and clustering in temporal networks. Contributing to the focus of this satellite, in this talk I will show how to efficiently obtain statistics of paths of finite length from a temporal network.


Carroll Financial Associates, Inc.
4201 Congress St Suite 210, Charlotte, NC 28209
6 Contact hours available
Math in the Real World-for Middle and HS Math Teachers
A Program of the NCCEE
Facilitated by Mary Richards, Veteran High School Math Teacher
What is included?
- A light breakfast and lunch will be provided
- All participants receive VE 4.5 valued at $160
- Pedagogy featuring math through the lens of personal finance and featuring lessons on Compound Interest, Tax Math, Credit Card Interest and Fixed Payments, the Math of Saving and Earning Credit
- Introduction to the SMG Program including Math Behind the Market Curriculum and instructions on how to incorporate the program into your math classes at no cost
- A Certificate for 6 hours of continuing education in math and financial literacy
- Teachers from Title 1 schools may have their substitute teacher pay reimbursed
Read what other teachers said about NCCEE workshops:
- "Best workshop I have ever attended. You guys really rocked it!"
- "It was so great to be treated like a professional!"
- "The hands-on activities make the day go by quickly and provide excellent opportunity for collaboration."
This workshop and the accompanying materials are made available to teachers through the generous support of State Farm and the Council for Economic Education. Additional funding was provided by the SIFMA Foundation for Investor Education. Special thanks to Carroll Financial for hosting this workshop.


ISBN: 978-1-937585-22-8 (print)
Ebook: 978-1-9375-85-33-4 ($39.95)
Pub Date: March 2016
Specs:8 x 10 Paperback (4 color) with CD
Designed for the person who needs to learn algebra as a prerequisite for further study or for a refresher course before moving on, the book covers all of the basic algebra concepts such as variables, equations, quadratic equations, factoring algebraic expressions, exponents, roots, radicals, and more. It includes numerous step by step examples and practice exercises that help the reader to understand the topics in a “self-study” format, designed for those who are uncomfortable with mathematics. The companion disc includes self-correcting exercises and all the figures from the text. Instructor resources available for use in course adoptions.
BRIEF TABLE OF CONTENTS
Part I. First Impressions. Why Study Algebra? Part II. Arithmetic Review. Arithmetic Operations. Part III. Algebraic Operations. Addition and Subtraction. Multiplication. Division. Part IV. Equations. Linear Equations in One Variable Expressions. Quadratic Equations in One Variable Expressions. Part V. Inequalities. Linear Inequalities. Appendices. Review Exercises. Answers Keys. Theorems, Corollaries, & Proofs. Key Words. Index.
- Presents basic concepts in an easy to understand style, designed for those who are uncomfortable with mathematics
- Provides hundreds of step by step examples and practice exercises that help the reader to understand the topics in a “self-study” format
- Includes a companion disc with self-correcting exercises and all the figures from the text
ABOUT THE AUTHOR
DAVID A. SANTOS (late) held a PhD from the University of Michigan and was an instructor at the Community College of Philadelphia. OLGHA B. DAVIS has served as an adjunct instructor at Wake Technical Community College (NC). She holds degrees in biomedical engineering from Boston University and the University of North Carolina at Chapel Hill.


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When there is a high-definition video version available, these are preferable, as the writing is easier to read. Videos average about 8 to 15 minutes each.
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Wolfram Alpha: The company which makes Mathematica has a free knowledge base covering a wide range of topics. This is a very helpful site for exploration, particularly for math and the sciences. You can also input functions and have them graphed, but it does a lot more than that---your function will be analyzed and put into alternative representations, it will find the roots, derivative, indefinite integral, maxima and minima, and more. The analysis clearly shows each step of the process. Wolfram also has an extensive reference section for mathematics.


Mobile Math Coach
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Schaum's outline of theory and problems of tensor calculus. David Kay
ISBN: 0070334846,9780070334847 | 238 pages | 6 Mb
Schaum's outline of theory and problems of tensor calculus David Kay
This Schaum's Outline gives you Practice problems with full explanations that reinforce knowledge Coverage of the most up-to-date developments in your course field In-depth review of practices and applications Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten He is the author of more than 30 articles in the areas of distance geometry, convexity theory, and related functional analysis. Students of the physical sciences and of physics, mechanics, electromagnetic theory, aerodynamics and a number of other fields will find this a rewarding and practical treatment of vector analysis. Mathematical methods for physicists - Arfken Webe r Contenido: Vector Analysis Curved Coordinates, Tensors Determinants and Matrices Elements of Real Analysis - R. Based on Schaum's critically acclaimed pedagogy of concise theory illustrated by solved problems, Schaum's Outline of College Mathematics features: Mathematical modeling throughout Modernized graphs Graphing and scientific calculator Partial Differential Equations. SpiegelMediafire link download Math book and Math softwareSchaum's Outlines Vector Analysis (And An Introduction to Tensor Analysis) by Murray R. Schaum's Outlines Vector Analysis (And An Introduction to Tensor Analysis) by Murray R. Bartle Contenido Introduction: A glimpse at set Theory I - The real numbers II - The Topology Of Cartesi Composición de textos científicos con LaTex - Gabriel Valiente . Review of Elementary Mathematics, 2nd Ed. Probability and Statistics, 2nd Ed. Set Theory and Related Topics, 2nd Ed.
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[List of Quiz
is the missing energy ?
A capacitor (capacity C) is charged at the voltage U
and then disconnected from the power supply. It
contains the electric charge Q = C·U and the energy
W = 0.5·C·U2.
Now, the capacitor is connected (parallel) to the same
uncharged capacitor (total capacity now C+C=2·C),
and the voltage will drop to U1 = Q /
(2·C) = U / 2.
Each of the capacitors has the energy W1 =
0.5·C·(U/2)2 = W / 4 and both
capacitors together have the energy W2 =
2·W1 = W / 2, which is
half of the energy W at
start, even if the wires are very thick and have very
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Year 11 is the first full year of SACE and, as such, students should be aware that while they have more choice of subjects, some areas are compulsory and must be completed satisfactorily.
Students study 20 credits of English, at least 10 credits of Mathematics, 10 credits of CRAVE, and then choose 8 elective subjects.
Specialist Mathematics must be taken in conjunction with Mathematical Methods. Both subjects are to be taken as full year subjects.
Please be aware that students must pass English and Mathematics with a ‘C’ Grade or higher to complete their SACE.
One week of additional school attendance after November examinations is compulsory for students who are yet to reach these targets in Year 11.


डीपी बॉस मटका फाइनल अंक ट्रिक गणित is a popular term in the world of online matka gambling. Matka is a form of lottery that originated in India and has gained immense popularity over the years. In this article, we will delve into the depths of DP Boss Matka and explore the final ank trick, the role of mathematics, popular strategies, advantages of playing online, and common mistakes to avoid. So, let’s get started!
Understanding DP Boss Matka
Basics of DP Boss Matka
DP Boss Matka is a game of numbers and luck. It involves betting on a combination of numbers, ranging from 0 to 9, that are derived from various sources. The game consists of two parts: the open result and the close result. Players place their bets on these results and wait for the winning numbers to be announced. It’s a thrilling game that has captivated millions of players worldwide.
History of DP Boss Matka
DP Boss Matka has a rich history that dates back to the 1960s when it was introduced by Kalyanji Bhagat. Over the years, the game evolved and became popular among the masses. Today, DP Boss Matka is played both offline and online, providing players with an opportunity to test their luck and win big.
Final Ank Trick
What is the Final Ank Trick?
The final ank trick is a technique used by DP Boss Matka players to increase their chances of winning. In this trick, players analyze previous results and calculate the final ank, which is a single-digit number that holds the key to winning. By using this trick, players can make informed decisions and place their bets strategically.
How to use the Final Ank Trick in DP Boss Matka
To use the final ank trick effectively, players need to analyze the previous results and identify patterns. They can then calculate the final ank using mathematical formulas and algorithms. By considering factors such as the frequency of a particular number appearing in the results, players can make intelligent choices and increase their odds of winning.
Tips and strategies for using the Final Ank Trick
- Keep track of previous results and analyze patterns.
- Understand the probability of different numbers appearing.
- Use mathematical formulas and algorithms to calculate the final ank.
- Combine the final ank trick with other strategies for better results.
- Practice patience and discipline while applying the final ank trick.
Mathematics in DP Boss Matka
The game of DP Boss Matka is not purely based on luck; mathematics also plays a significant role. Understanding the mathematical concepts involved can give players an edge over others. Let’s explore the key mathematical aspects of DP Boss Matka.
The role of mathematics in DP Boss Matka
Mathematics provides a foundation for analyzing and predicting the outcomes in DP Boss Matka. Players can use mathematical concepts such as probability, permutations, and combinations to assess the likelihood of certain numbers appearing in the results. By applying mathematical principles, players can make informed decisions and improve their chances of winning.
Probability and odds in DP Boss Matka
Probability is a crucial element in DP Boss Matka. By understanding the probability of specific numbers or combinations appearing in the results, players can make calculated bets. Additionally, knowing the odds associated with different types of bets can help players determine their potential winnings and risks involved.
Mathematical approaches to increase winning chances
Several mathematical approaches can be used to enhance the winning chances in DP Boss Matka. Strategies like number tracking, trend analysis, and statistical modeling can help players identify patterns and make accurate predictions. However, it’s important to remember that no strategy can guarantee consistent wins, as luck still plays a significant role.
Popular DP Boss Matka Strategies
DP Boss Matka enthusiasts employ various strategies to maximize their chances of winning. Here are three popular strategies that players often use:
Single Patti strategy
The Single Patti strategy involves betting on a single-digit number from 0 to 9. Players choose a specific number they believe will appear in either the open or close result. By focusing on a single number, players can increase their odds of winning but should also be cautious of potential losses.
Jodi Patti strategy
The Jodi Patti strategy revolves around betting on a pair of two-digit numbers. Players select a combination of two numbers, and if both numbers appear in the results, they win. This strategy offers higher payouts but requires greater accuracy in predicting the outcome.
Panel chart strategy
The Panel chart strategy involves analyzing the panel charts, which display the results of multiple games. By studying these charts and identifying patterns, players can make informed bets and potentially increase their winnings. The panel chart strategy requires attention to detail and a good understanding of previous results.
Advantages of Playing DP Boss Matka Online
With the advent of technology, playing DP Boss Matka online has become increasingly popular. Here are some advantages of opting for online play:
Convenience of online play
Playing DP Boss Matka online offers unparalleled convenience. Players can participate in the game from the comfort of their homes or on the go using their smartphones. Online platforms provide 24/7 access to the game, allowing players to bet whenever they want.
Wide range of betting options
Online platforms offer a wide range of betting options in Madhur Bazaar. Players can choose from various markets, such as Kalyan Matka, Milan Day, Rajdhani Night, and more. This diverse selection allows players to explore different games and find the ones that suit their preferences.
Enhanced privacy and security
Online DP Boss Matka platforms prioritize the privacy and security of their players. They employ robust encryption and security measures to protect personal and financial information. Players can enjoy a safe and secure gambling experience without worrying about their data being compromised.
Common Mistakes to Avoid in DP Boss Matka
While playing DP Boss Matka, it’s crucial to avoid common mistakes that can lead to losses. Here are three common mistakes to steer clear of:
One common mistake is chasing losses. When players experience consecutive losses, they often try to recover their losses by placing bigger bets. This approach can be risky and lead to further financial losses. It’s important to play responsibly and avoid chasing losses.
Not managing bankroll properly
Proper bankroll management is essential in DP Boss Matka. Players should set a budget for their bets and stick to it. It’s crucial to avoid wagering more than one can afford to lose. By managing their bankroll wisely, players can enjoy the game without jeopardizing their financial stability.
Overlooking previous results
Previous results hold valuable information in DP Boss Matka. Ignoring the patterns and trends in previous outcomes can hinder players’ ability to make informed decisions. It’s important to analyze past results and use them as a reference to improve future bets.
The Future of DP Boss Matka
The world of DP Boss Matka is constantly evolving, driven by technological advancements and regulatory changes. Here are two factors that shape the future of this game:
Advancements in technology have transformed the way DP Boss Matka is played. Online platforms, mobile applications, and digital payment methods have made the game more accessible to a wider audience. In the future, we can expect further technological innovations that will enhance the overall gaming experience.
Regulatory changes also play a crucial role in shaping the future of DP Boss Matka. Governments and regulatory bodies are continually reviewing and updating laws related to gambling. These changes aim to ensure fair play, protect players’ interests, and prevent illegal activities. It’s important for players to stay updated on the regulatory landscape to engage in lawful and secure gameplay.
डीपी बॉस मटका फाइनल अंक ट्रिक गणित is a fascinating world that combines luck, mathematics, and strategic thinking. By understanding the final ank trick, leveraging mathematical concepts, employing popular strategies, and avoiding common mistakes, players can enhance their chances of success in DP Boss Matka. With the convenience of online play and the continuous evolution of the game, DP Boss Matka promises an exciting future for enthusiasts. So, embrace the thrill, apply the strategies wisely, and may luck be on your side!
Can I use the final ank trick to guarantee wins in DP Boss Matka?
No, the final ank trick is not a foolproof method to guarantee wins. It is a strategy that can increase your chances of winning, but luck still plays a significant role in the game.
Are there any legal concerns associated with playing DP Boss Matka?
The legality of DP Boss Matka varies from country to country. It’s essential to understand the legal regulations in your jurisdiction and engage in lawful gameplay.
Can I apply mathematical formulas to consistently win in DP Boss Matka?
While mathematics can improve your understanding and decision-making in DP Boss Matka, there is no formula or strategy that can guarantee consistent wins. Luck and chance are inherent in the game.
Is playing DP Boss Matka online secure? Reputable online platforms prioritize the security and privacy of their players. They employ advanced encryption and security measures to ensure a safe gambling experience.


|Why some students needs to spend more than 50% time for learning MATH?|
Math most of the question looking for student's previous cumulative MATH Knowledge! Not like the others Subjects. (Unlike a lot of other subjects, there is no room for error when it comes to math; There’s only right and wrong)
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Note that if student going to study Engineering at University, in Grade 12 he/she has to take 3 MATH subjects out of 6 subjects. (Physics upto 80% Math and Chemistry 40-60% Math) Again more than 50% of study is MATH.
Math is not difficulties to learn; but you must follow the right way to do step by step it takes time. There are No Shortcuts to learn Math. Anybody can study MATH, only if he/she spends time with focus!


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A major drawback is that the inductive method cannot be used for all topics. Unfortunately, deductive method encourages blind memorization which leads to lack of creativity and originality.
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Wild Kratts Creature Math
Creature Math helps children practice addition and subtraction as they create their very own animal habitat, filled with cool creature pals!More +
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Using Slots to increase Your Likelihood of Winning
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Just about the most popular casino game types is the standard casino game. This modern casino game is named after the popular William Benton McDonald who created the standard deviation. Standard deviation is a mathematical concept that determines the likelihood of a specific event to occur. There are three main types of standard deviation games: binomial lottery games, online games, and standard slot games. Binomial lottery games cope with a finite group of numbers you need to include spins with equal chance elements.
Another casino game type may be the slot machine game game. Slots are common because you can find always consistent winning combinations. Additionally it is among the easiest games to learn. Slot machines are available in a wide variety of casino hotels everywhere.
Another popular kind of casino game is the house edge. A house edge is the percentage of a casino’s expenditure, which is continuously created by an investor. Residence edges vary widely depending on gaming casinos. For example, in a progressive modern casino where jackpots are won often, a house edge using one single combination is likely. The home edge on multi-layaway machines is even lower, because the probability of hitting more jackpots simultaneously is slender.
All types of gambling are characterized by some type of betting in terms of the amount wagered and the number of combinations possible. There exists a relationship between these two variables, referred to as the Blackjack Breakdown. The basic idea behind the Blackjack breakdown is certainly that a casino will decrease the probability of winning a combination if more folks are at that casino, or the gambling house decreases the odds of hitting a particular sequence of numbers, like a specific amount of cards dealt.
Just about the most popular casino video games is poker. There are many different variations of the card game. One of the most popular variations is holdem. Although the rules for each variation of poker varies slightly, they all follow the same pattern of betting, raising the 인터넷 카지노 amount wagered, and taking the amount of money from the player before the match begins. In older video games, the house advantage is the players’ advantage, or the amount of money they might potentially win without spending some of their own funds.
Among the simplest forms of casino games is the sport of blackjack. Blackjack makes use of the “house” benefit to calculate the expected damage and the quantity of chips it is possible to possibly lose. The typical deviation is used to measure the random variables, or variables which are away from control. Blackjack, like holdem, also offers an assumption of what the internet casino will do, namely what the typical deviation would be if the overall game were to keep on its training. This assumes that the internet casino will stay in the same hands more often than not, which it often will.
Slot machines are popular game titles in casinos since they allow members to bet on smaller amounts, but have the biggest jackpot payouts. To calculate the best times to place these wagers, developers use statistical research. They acquire the random variables of a slot machine and apply them to the forms of reels that will be used, plus the odds of a specific outcome. They then determine the best times to put bets, and where you can place them to increase your likelihood of hitting more tickets. Although you might not personally manage to determine the very best times to play slots, the slots that pay the most are the most popular places to play, in order that is where you need to put your wagers.


Understanding the Probability of Hitting a Slot Jackpot
In ice and field hockey, the slot is a rectangular area, extending toward the blue line. Its name comes from the Greek word slotia (which means “slot”) and the Latin verb sleutana, which means to “slip.” The word slot is cognate with the German word Schloss.
Probability of hitting a particular combination of symbols on a slot machine
Probability of hitting a particular combination of symbol on a slot machine is an important aspect for slot players. It is a simple mathematical calculation that is used to determine the odds of hitting a particular symbol or combination. This calculation will help players determine the amount of money they should bet and what combination of symbols they should bet on. Knowing the odds of hitting a combination on a slot machine can help players make informed decisions about the amount they spend on a specific spin.
The probability of hitting a particular combination of symbols on slot machines varies widely. This is due to the variance between machines and the amount of money they pay out. For example, if you bet on a machine that pays out seven hundred and twenty-nine Bar symbols, the probability of hitting such a combination is 2%. By contrast, if you bet on a slot machine that pays out 80 Bar symbols, your chances are 6%. While this does not sound like a lot, it is important to note that the average slot machine will pay out more than $1000 per spin. So, if you play a machine with a higher pay table than another, you’ll have a higher chance of hitting the desired symbol combination.
Variations in the original slot machine concept
Slot machines are a type of casino gambling machine that has become wildly popular over the years. They offer the promise of instant wealth and have become an integral part of the casino gambling industry. The early versions of these machines were simple machines with three spinning reels and small prizes. They had little replay value and were viewed by many as distractions.
The original slot machine concept was developed in 1895. The machine featured pictures of hearts, diamonds, and spades. The machine spawned an industry of mechanical gaming devices. These early machines also gave out fruit flavoured chewing gum. Many of today’s fruit symbols are descended from these early machines, including the “BAR” symbol, which derives from the logo of the Bell-Fruit Gum Company.
As the technology of slot machines advanced, new types of machines were created, allowing players to play for larger amounts. The earliest slot machines used nickels, but over the years, they have evolved to offer a variety of denominations. The most popular of these games is the penny slot, which allows players to play with one cent. The low stakes of penny slots make them a convenient option for players with a small budget and limited time to spend playing.
Ways to calculate the odds of hitting a jackpot
When playing slot machines, players may wonder about the probability of hitting a jackpot. It is important to understand that the slot jackpot probability varies based on the game’s RTP (return to player) rate and variance. While the RTP rate is the chance of winning a prize, the variance is the likelihood of the jackpot being hit. In some games, the variance can be as high as 600 million to one.
In some games, it is possible to calculate the odds of hitting a jackpot by multiplying the number of winning combinations. For instance, in a three-reel slot, there are 1,000 possible combinations, so the probability of hitting a jackpot is 83.2 percent. The same is true for a four-reel slot. The same method works for calculating the odds of hitting a jackpot with three mango symbols on each reel.


The intention is to determine the variability one should expect when reading our interpolated water table map. The water table map is a long-term average with uncertainty associated with the date measurements were made. Water table depths vary seasonally by more than 1m in many locations and must be taken into account when trying to estimate seasonal high water levels.
Total expected variability is assumed described by 2 components: seasonal variability + inter-annual variability.
- seasonal variability is modelled using a statistical model (below)
- inter-annual variability is assessed by the confidence intervals from the statistical model
Generalized Additive Model
The Generalized Additive Model (GAM) is a flexible means of fitting a model to time series data. It is a generalization of the linear models we’ve all fitted to point data, which means it retains the ability to efficiently compute confidence intervals and prediction intervals and whose behaviour can be controlled intuitively.
Locations with >34 water level measurements are queried from our database. At every location:
- water level data are re-sampled to daily averages
- outliers removed using the ±1.5 * IQR (inter-quartile range) method
- values are mapped to their ‘day-of-year’ as the data’s sole degrees of variability
- values are fitted to a linear Generalized additive model (GAM), using 12 (monthly) degrees of freedom
- seasonal variability is taken as the amplitude of the GAM
- 90% prediction intervals are then computed around the GAM, assumed the inter-annual (long-term) variability
- Total expected variability = seasonal variability + inter-annual variability
See plots of all locations here.
..takes more than a minute to load, 120MB html downloading..
click on any circle to reveal station properties. Full-screen available in the top-left corner


11. 09. 2007
Handful of Sudoku fun: No pen or pencils needed
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Mathematics is important in everyday life and, with this is mind, the purpose of Mathematics at Great Waltham C of E Primary School is to develop an ability to solve problems, to reason, to think logically and to work systematically and accurately. Through our school’s ethos ‘Let Our Light Shine’ and our lifelong values ‘ASPIRE’, all children are challenged and encouraged to excel in Maths.
New mathematical concepts are introduced using a ‘Concrete, Pictorial and Abstract’ approach; enabling all children to experience hands-on learning when discovering new mathematical topics, and allows them to have clear models and images to aid their understanding. Arithmetic and basic math skills are practised daily to ensure key mathematical concepts are embedded and children can recall this information to see the links between topics in Maths. Learning is carefully sequenced, taking into account what has been taught before, and what knowledge and skills are needed for the next stage of our children’s mathematical development, ensuring it is accessible to all pupils from EYFS to Year 6 and develops them as mathematical thinkers.
Please find the progression maps below which are structured using the topic headings as they appear in the National Curriculum. Each document indicates what is taught in the different year groups.


Doosan and Bobcat Company have continued their support of STEM-related education and research by donating nine $1,000 scholarships to students enrolled at North Dakota and Minnesota colleges and technical schools.
Students receiving the scholarships are studying science, technology, engineering and math fields at Moorhead State Community and Technical College (M State), North Dakota State College of Science (NDSCS) and Bismarck State College (BSC).
Scholarship recipients are:
Support of STEM programs — Science, Technology, Engineering and Mathematics — is a key area of focus of Doosan and Bobcat community involvement initiatives. These educational and research areas help fuel innovation, which is the very foundation of Doosan and Bobcat, and they are key to the company’s continued growth and success. STEM areas also help generate community progress, which is very important to Doosan and Bobcat as well.
The company supports all educational levels — elementary, middle school, high school and college students — through grants, in-class participation, production facility tours/field trips and scholarship programs.
In January, Doosan and Bobcat Company announced a $3 million donation to North Dakota State University (NDSU) to establish a STEM scholarship endowment. The donation was matched with state funds, bringing the total gift to $4.5 million, which is the largest gift to establish a scholarship endowment in NDSU history.
“Our involvement in STEM programs continues to grow,” said Laura Ness Owens, director of communication for Bobcat Company and Doosan. “We want to do everything we can to help ensure there are more young people pursuing STEM-related professions. We know it won’t take long for all of us to see those investments pay off through business and community growth in the region.”


MEI A Level Mathematics Year 1 (AS) 4th Ed.
MEI A Level Mathematics Year 2 4th Ed.
MEI A Level Further Mathematics Year 1 (AS) 4th Ed.
MEI A Level Further Mathematics Year 2 4th Ed.
MEI A Level Further Mathematics Mechanics 4th Ed.
MEI A Level Further Mathematics Statistics 4th Ed.
MEI Further Maths: Numerical Methods 4th Ed.
MEI Further Maths: Modelling with Algorithms 4th Ed.
MEI Further Maths: Further Pure Maths with Technology
MEI Further Maths: Extra Pure Maths
OCR B (MEI) Year 1/AS Mathematics Exam Practice
OCR B (MEI) A Level Mathematics Exam Practice
My Revision Notes: OCR B (MEI) A Level Mathematics Year 1/AS (Pure)
My Revision Notes: OCR B (MEI) A Level Mathematics Year 1/AS (Applied)
My Revision Notes: OCR B (MEI) A Level Mathematics (Pure)
My Revision Notes: OCR B (MEI) A Level Mathematics (Applied)
OCR A Level Mathematics (AS) Year 1 576pp
OCR A Level Mathematics Year 2 608pp
OCR Core (AS) Year 1 216pp
OCR Core Year 2 352pp
OCR Statistics 352pp
OCR Mechanics 376pp
OCR Decision 168pp
Edexcel A Level Mathematics (AS) Year 1 576pp
Edexcel A Level Mathematics Year 2 600pp
Edexcel Core (AS) Year 1 256pp
Edexcel Core Year 2 328pp
Edexcel Statistics 448pp
Edexcel Mechanics 420pp
Edexcel Decision 232pp
AQA A Level Mathematics (AS) Year 1 576pp
AQA A Level Mathematics Year 2 600pp
AQA Core (AS) Year 1 312pp
AQA Core Year 2 472pp
AQA Statistics 304pp
AQA Mechanics 352pp
AQA Discrete 160pp
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2 edition of Mathematics in general found in the catalog.
Mathematics in general
A. E. F. Davis
Label mounted on t.p.: New York, Humanities Press.
|Statement||[by] A. E. F. Davis.|
|Series||Concept books, 4|
|LC Classifications||QA39 .D29|
|The Physical Object|
|Pagination||ix, 147 p.|
|Number of Pages||147|
|LC Control Number||71432814|
Spivak makes his writing on the main objective of the book — Stokes Theorem — painless and easy to grasp. Hence, students will be able to improve their marks through their good attempt of objective type paper. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Hence, Prepare your 10th class final exams or class test in a best way. Jones Review: Undergraduate math majors will find this book to be easily approachable but containing much depth. What drew you to the field of mathematics?
My passion is for education as a whole, rather than just mathematics. This hardcover volume is rich of illustrations and would be a nice coffee table book. The high point is Norman Kagan's utterly hilarious "The Mathenauts", in which only mathematicians can travel through space, because space is mathematical — and, conversely, anything mathematical can be reality. As the resident statistician here at Haese Mathematics, I get the pleasure of writing new statistics chapters and related material.
However, it slips in because it communicated to the world one of the very greatest ideas of all time: Nature has laws, and they can be expressed in the language of mathematics. The book consists of numerous selections from his columns, classified according to the mathematical area involved. It allows them to appreciate where mathematics came from, who created it, what it's good for, and where it's going, without getting tangled up in the technicalities. He clarifies deep concepts in a manner that boasts his mathematical capacity and skill while never leaving the reader behind.
Love in the Western World
Exploring Psychology 5e Paper & Scientific American Explores The Hidden Mind
The promoting effect of lanthanum in heterogeneous catalysts
introduction to the music of R. Vaughan Williams
In the best interest of the child
Conflict Analysis: The Formal Theory of Behavior
Andromeda Strain, The
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Caesar and contemporary Roman society.
Recreation and leisure time activities in the correctional setting
An Introduction Mathematics in general book Probability Theory and Its Applications by William Feller Review: In this first volume, William Feller paints a clear picture of probability theory and several of its interesting applications from the discrete viewpoint.
These days, Michael Mathematics in general book the editing and I do proofreading and audio. In the first chapter, Mendelson explains set theory in an unmitigated manner which primes readers for the following chapters.
The authors prompt readers to actually think about the ideas and methods mentioned rather than blindly swallow them down for later use. Condition: New. He also effectively begins each chapter with a short preamble that helps further instill understanding of the main concepts.
Self-study enthusiasts will fall in love with the clarity and simple progress of the writing. Dummit and Richard M. The material can be used as both refresher reading and as a primary study guide.
The enthusiasm he exhibits throughout will spread onto readers like wildfire. Horses, show jumping and course design, alpacas, badminton, running, art, history, faith, reading, hiking, photography On the borderline between popular science and an academic text, but eminently readable without specialist knowledge.
An isomorphomechanism is essential equipment. Tensors in general relativity[ edit ] Further information: Tensor One of the profound consequences of relativity theory was the abolition of privileged reference frames.
As Mathematics in general book bread, I used to do a good deal of consulting on defence problems, until I learned that writing books was in every way more rewarding, and since it did not involve handling classified materials, I could do it at home.
He contrives a Mathematics in general book and classical introduction to topology that targets continuity and linearity, the dominating themes. The Four Pillars of Geometry by John Stillwell Review: This is a beautifully written book that will help students connect the dots between four differing viewpoints in geometry.
For example, in the theory of manifolds, each point is contained in a by no means unique coordinate chartand this chart can be thought of as representing the 'local spacetime' around the observer Mathematics in general book by Mathematics in general book point.
They have done much to make science what some scientists have always hoped it would be: a part of the culture of our times.
Inhe was brought to work at Cambridge. As such, this title is the ideal coffee table book for math geeks. He encompasses all basic and advanced topics such as differentiation, Banach and Hilbert Spaces, Fourier analysis, etc.
Teukolsky, William T. An understanding of working with matrices is required. What interests you outside mathematics? When a new or revised syllabus is released for a curriculum that we write for, a lot of work goes into devising a structure for the book that addresses the syllabus.
By return mail, the magician names the selected card. The story of the student who believed nothing except his calculator is an object lesson for everyone who thinks mathematics is just 'sums'. The reasons why we do things, and the people behind them are often important things we forget to talk about.
During the course of reading this one, it will become blatantly clear to the reader that the author has created this work out of passion and a genuine love for the subject. I really love to travel. Strang has taught this material to thousands of students.Mathematics for the General Course in Engineering, Volume I covers the syllabus in mathematics for the G.1 year of the general course in engineering.
Provided in this text are 31 unworked examples, which form a comprehensive revision course that students are recommended to.
General Mathematics. Showing 1 - 14 of 14 Items. Narrow Your Results Sort by: View: Go to the first page Go to THE MATH BOOK. Edited by Karl Warsi. Hardbound. $ $ Add to Cart. THE MATHEMATICS OF SECRETS: Cryptography from Caesar Ciphers to Digital Encryption.
Joshua Holden. Dec 15, · 10th Class Mathematics Arts Group Book Solved Study Notes Board of Intermediate and Secondary Education (BISE) is responsible for conducting 9th and 10th class exams.
At the same time, all know that 9th class and 10th Class BISE hold annual exams normally before Intermediate exams. 9th class and 10th class board exams phase is considered as the.Examples: Decimals on the Pdf Line Example 5 a) Plot on the number line with a black dot.
b) Plot with a green dot. Solution: For we split the segment from 0 to 1 on the number line into ten equal pieces between 0 and 1 and then count.This book is download pdf to be used by children ages 5 to 6. Other age groups will also benefit from the book. Anyone can use this book globally, although the curriculum may differ slightly from one region to the other.
This is so because the core content of Mathematics is the same around the world.General Mathematics Secretary’s message This teacher ebook is to be used by teachers when implementing the Upper Secondary General Mathematics syllabus (Grades 11 and 12) throughout Papua New Guinea.
The General Mathematics syllabus identifies the learning outcomes and content of the subject as well as assessment requirements.


Casino Game Abilities
You can find currently three main categories of casino games available: cards, video poker games, and digital casino games. Video poker game titles are played by one person at a time and don’t require the participation of gambling establishment staff to play. These games are ever more popular among non-gamers because they are relatively easy to learn and play. They also have a tendency to provide a more competitive playing experience than almost every other games are.
Blackjack is just about the best-known card game on earth, and for good reason. Blackjack is often the initial casino game competitors are introduced to and is a good introduction to the more challenging games that the casino gives. While you are playing blackjack at an on the web casino, you will find that you have the very best odds of winning once you come to bet. That’s because the house always has blackjack cards, and the lowest possible house edge (or benefits) is two per card. The low your chances of getting an edge increase, the better off you’re. The minimum bet amount is always lower than the maximum bet amount, so the lower you are willing to go with an individual card, the better off you’re.
Another simple blackjack strategy is to use basic statistics to analyze the way the blackjack player could be behaving. One such statistic is named the binomial distribution and it compares the winning possibility of blackjack with the chances of picking any number in the set, up to a specified restriction. The binomial distribution plots the probability as a performance of the amount of successes and failures. For example, the standard deviation is used to plot the data points for each and every 100 hands played. The typical deviation uses a normal curve to plot the data points so that you could see the range of habit and how it differs with the amount of successful plays and the period of each successful have fun.
Blackjack players can make use of the typical deviation to determine the casino’s odds of winning more money in the popular games. Standard deviation is most useful in studying the random distribution of the results of gaming events, like spins on roulette wheels or spins on slot machines. It is also used to study the behavior of the chances in non-stop jackpots on slot machine games and video poker devices. One casino identifies these as “montee games,” because players can win a variety of prizes simultaneously over many palms.
The random distribution of the results of casino slot machines is comparable to the odds of striking the jackpot in slot machines. When players place wagers, they take the opportunity that the gambling house will hit a jackpot. If you are gambling with a 10-part slot machine, then the best odds which can be given to you’d be if you get yourself a hit every time with among your bets. But, because you don’t know what’s the best payout will be, you will not know whether you’re getting the very best odds on slot machines or not really.
To study the true odds at playing slot games, a player should multiply the amount wagered and also the casino’s expected payout for every spin. For instance, if a person is betting ten cents about the same spin, the casino’s expected payout per spin is $.10. 베스트카지노 With this particular number, the player can do a quick computation to determine the volume wagered, multiplied by the expected payout to find the casino’s true odds of winning. Then, he can multiply that number by the number of times he wants to bet and discover if he gets an absolute bet every time. This way, the player can calculate his likelihood of winning without having to depend on the numbers generated by the machine.
In roulette and blackjack, players have to follow a technique of matching the number of opponents which come up with a specific number. A good player should do this consistently to increase his chances of winning. This can be the same in playing other gambling house games such as craps and baccarat. It requires strategic thinking, which is portion of the real skill component.
On the whole, playing casino games online is way better because it has a lower house edge than slots games. Nevertheless, both roulette and slots game titles have the tendency to generate larger jackpots. There are also casino games online that work with random amount generators for drawing numbers. As compared with these, the home edge is smaller in gambling establishment games on the Internet.


MELISSA & DOUG- CHUNKY JUMBO NUMBERS PUZZLE
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Counting fun begins with this extra thick wooden puzzle that includes 20 easy-grasp, chunky pieces (1-20). A full-color, matching number appears under each piece, and counting pictures equal to each numbered piece are shown on the puzzle board. Number pieces stand up. Encourages hand-eye, fine motor and early math skills.


More than 2000 years ago, long before rockets were launched into orbit or explorers sailed around the globe, a Greek mathematician measured the size of the Earth using nothing more than a few facts about lines, angles, and circles. This course will start at the very beginnings of geometry, answering questions like "How big is an angle?" and "What are parallel lines?" and proceed up through advanced theorems and proofs about 2D and 3D shapes. Along the way, you'll learn a few different ways to find the area of a triangle, you'll discover a shortcut for counting the number of stones in the Great Pyramid of Giza, and you'll even come up with your own estimate for the size of the Earth. In this course, you'll be able to choose your own path within each lesson, and you can jump between lessons to quickly review earlier material. GeometryX covers a standard curriculum in high school geometry, and CCSS (common core) alignment is indicated where applicable. Learn more about our High School and AP* Exam Preparation Courses This course was funded in part by the Wertheimer Fund.