The largest prime

The largest prime number ever seen by humankind was found on January 7, 2016.  It weighs in at  22,338,618 digits in length. It is what is known as a Mersenne prime (2^74,207,281-1).  Mersenne primes are a group of prime numbers of the form 2ⁿ -1 where n is  is assumed to be prime. This is a fascinating find. Prime numbers are the "building blocks" of all other numbers.  Their unique nature has led many number theorists and math enthusiasts to study them in great detail. Although they are used heavily in cryptography, the latest find is far too large to be used in any encryption schemes to date. But as computation theory evolves and computer architectures become more sophisticated these extremely large primes may find their place.

The "common core check guy"

In the news recently, we see the story of Doug Herrmann. Doug is a father from Ohio who out of frustration with his child's math homework, wrote a check using common core methods. He was frustrated because he was unable to understand what the school was teaching his son, and therefore was unable to help his son with his math homework.

The ten card system that his “viral” check picture was intended to highlight is not a terrible method of mental math. One exception is that for some it abstracts the idea so far that it seems to become separate from the actual “math”. Utilizing grouping techniques is in no way a “bad” technique to understand mental math. However, it becomes a problem if too much emphasis is placed on a single rigid method.

Using the ten card system is one of many useful techniques for understanding grouping. We only run into problems when the standardized testing systems force children to “believe” there is only one right way. The testing should only test for the ability to solve problems and not for the strict adherence to a single “correct” way of getting there. The practice tests I have looked at appear as though they are actually structured that way. The second grade tests do emphasize place value, but do not seem to indicate a single method. Now as far as the rigidity of the teachers, I have not seen how this comes into play. I do know that in many instances students fall behind due to teachers being too strict in the techniques that you use.

Please understand that I am well aware that there has been a rising trend in math illiteracy. I am not completely blasting common core. I , along with other parents, have some questions about the implementation and flexibility of these standards.

Some questions I have include:

• How flexible are the methods used ?
  • Are students being taught that the ten card system is the “only correct” method ?
  • Will students be penalized for utilizing other techniques to arrive at their answers ?

• How are conflicts between common core, and how the parents teach their children handled ?
  • Do teachers penalize their students for the way their parents teach them to solve problems ?
• Is there an awareness of the different ways people understand concepts ?
  • Is there an accommodation in place for students who understand place value through other means than just grouping visualizations? ( e.g. positional notation instead of ten cards.)

These questions go beyond just common core. The rigid adherence to single techniques has long been the culprit behind the fall in mathematics education.

On a personal note, I have been a victim to the rigidity of teachers. When I was in high school(many years before common core) I failed a math class simply due to my teacher not approving of my mental math techniques. To be fair, I would like to emphasize the math teacher was not actually a math teacher, she was a soccer coach moonlighting as a math teacher due to a poor student teacher ratio.

The conflict arose when I was forced to show my work(and by show my work I mean that my teacher wanted to see the remedial addition, subtraction, multiplication, and division) This was an algebra class, where showing your work meant to show the steps taken to simplify the expressions. However, this instructor insisted we not just show that we had to multiply she wanted to see the actual steps we took to multiply. I thought this was odd in an algebra class, because learning to multiply was second grade. However, I did comply and attempted to be verbose with “showing my work”. However, my method of “long hand” multiplication was different from what she was taught. Most people are taught the “only” way to multiply is to start in the one column and carry and borrow and all these other concepts. I used a more “short hand” method which lends itself well to mental math. I would start with the highest power of ten column and work left to right instead of right to left. For example if I were to multiply two digit numbers I would start in the tens column and multiply no need for carrying and borrowing. You are just simply adding up zeros. This method isn't understood by all, but it was how I understood numbers and place value. Unfortunately, my math teacher was strict and rigid in how she wanted it done and failed me on all of my quizzes. I understood the algebra part and understood the basic calculations part, and arrived at the correct answer. I just applied a different method for the basic calculations.

Now my story is sadly similar to many other students. For most, these situations reinforce their frustrations with mathematics. It can quickly lead students to make snap judgments about the efficacy of learning and understanding math. They soon start to see math as a foolish endeavor with rigid methodologies not worth their time. Or they could simply just give up believing they will never understand and struggle to merely pass.

 I don't exclusively place the blame on teachers for the problem of rigidity. The parents also hold a share of the blame. As with the “common core check guy” Doug Hermann and some of the comments on his posting, they also find themselves rigid in their understanding. Simply not understanding a technique of solving a problem does not condemn it. Rather, it simply means you should probably learn more about it before condemnation.  

Propositional calculus: an introduction

Propositional calculus is the branch of mathematics that deals with the rules of logic and evaluation of statements or propositions. It is sometimes referred to as sentential calculus for its use of the sentential conjunctions. It is primarily concerned generating laws for evaluating conjugated statements. Propositional calculus is part of a broader science of logic and proof.

Some of the symbols used in propositional calculus:

Symbol	Meaning
¬	Not or negation
^	and
v	or
→	implies
↔	If and only if
P	Sentential Statement
Q	Sentential Statement
T	True
F	False

In propositional calculus the above symbols are used in constructing truth tables for evaluating compound or conjugated sentences. The truth tables provide a short hand tool for deriving new laws from the simple compound sentential statements. Truth tables like the one below shows some of the simple rules and the new rules derived from those.

P	Q	P→Q	P^Q	PvQ	(P^Q)→P	P ↔ Q
T	T	T	T	T	T	T
F	T	T	F	T	T	F
T	F	F	F	T	T	F
F	F	T	F	F	T	T

This is just a simple introduction to propositional calculus, it is part of a much broader branch of propositional logic . It is useful in the formulation of logic rules, and methods of proof.

The "Reality" of math and logic

Logic has been used as a methodology even before its formalization. In order to survive, our progenitors had to be able to deduce whether a particular place or event was dangerous. These more earlier versions of ourselves, needed to determine what was safe to eat and where food was located. In this sense they were using logic intuitively.

As societies evolved, we became aware of a need for a system of accounting. We developed symbols to serve as abstract models for real world objects. Empirically , we can demonstrate the need for a modeling system. We can also demonstrate that these models of real world objects work. The models functionality means they can be used for future discoveries, and aids in civilization building.

Enumeration permeates our entire civilization and history on multiple levels. It observably exists separate from language. From the simplest counting systems, patterns began to emerge. The laws of mathematics became more complex, and a method of proof was needed that was empirically verifiable. The earliest proofs were purely empirical as used by the Pythagoreans and Thales of Miletus. Over time, mathematical proofs slowly became less heuristic and a formalized system of logic was becoming more prominent. Logic was largely an emergent aspect of these methods of empirical verification. The methods of logical proof slowly grew over the next two and half centuries from Thales around 550 BCE to Euclid around 300 BCE. Euclid is credited with formulation of the axiomatic method of proof.

The Euclidean style of proof through axioms, appears assumptive, or only justifiable a priori.

The axioms served as descriptions or definitions of geometric objects. The definitions could be demonstrated empirically in general, but they did rely on assumptions. For example, take the statement 2 points determine a line. This is a postulate proposed as self-evident, and deduced from it is many other theorems. If you were to take this as meaning all statements derived rely on a non-empirical premise you would be incorrect. The statement can be demonstrated as empirically true. You would simply have to draw a line through only one point. The line drawn, would be a point and not a line. To construct a line you would have no choice but to have it pass through multiple points. So, we can see the statement is both self-evident and empirically demonstrable.

The evolution of mathematical systems of proof allowed for a more formalized system of logic. All of which finds its origins in our innate pattern recognition. The ability to recognize patterns and natural desire to seek them out led to the mathematical modeling of real world objects. The modeling techniques made a complete system of logic and mathematics more easily constructed and shared. We formulated logical and mathematically complete laws and theorems from some of the simplest models.

We see that these models are intuitive,empirical, and justified in their use.

Positional Notation

Positional Notation

Our number system is base ten, as many learned in elementary school. Which means that there is only ten symbols to represent numbers. 0,1,2,3,4,5,6,7,8,9 . To represent ten we use 1 and 0 (10). Which when we learned place value, we recognized this as 1 in the tens place and 0 in the ones place. What this means is that the position of the symbol will determine the value of the number. For example when we see the symbol 34 we recognize it as 3 tens and 4 ones. We would often see something like:

Tens	ones
3	4

 We are mostly taught this and just accept it as is and move on. To dig a little deeper into to place value I will introduce a new way of looking at this (Positional Notation) . To rewrite our example ,34, into positional notation we get this:

3 x 10¹ + 4 x 10º

The reason I write it this way is that is represents a number as powers of the base it is in. In this case it is base 10. Positional notation gives us a way of looking at numbers relevant to their base. The reason this is useful to learn, is that it gives us a quick way to convert numbers from other bases to base ten. For example, if you need to convert the number 101 in binary to base 10 you can rewrite 101 in positional notation and convert it easily. Since 101 is in binary we will use powers of 2 instead of 10 and we get

1 x 2² + 0 x 2¹ + 1 x  2º

1 x 4 + 0 x 2 + 1 x 1

4 + 0 + 1

5

As you can see from the table above just by rewriting the number into positional notation relevant to the base it is in you add it all up and you get the number in base 10. So when we convert 101 from binary to base ten we get 5.


 This is just a neat little trick I use to quickly convert numbers from other bases to base 10. With a little practice, you can convert numbers from most bases back to base ten almost effortlessly. There is other ways of converting number to other bases, I just really like this one.    

Python and Prime number Lists

Prime numbers, although simple in definition, provide several complexities in the field of mathematics, as well as in computer science. For reference, prime numbers are natural numbers which are only divisible by one and themselves. On the surface, this definition seems straightforward and not an important area for research. However, there are a great many applications of prime numbers.  Prime numbers are essentially the building blocks of all other numbers.  For centuries, this unique set of numbers has mesmerized and even baffled some of the greatest mathematicians.  The distribution of prime numbers is another big area of mathematics research. It would be great if we could find a formula for the nth prime. Yet, it seems as though it would be impossible. We can however use computers to do this for us up to the limits of the processing power available.  No real efficient algorithms exist to find primes other than methods of iterating through every number up to the square root of a given number to determine if it is prime.  Finding the patterns with some high level mathematics, is being used but no solid proofs exist yet to revolutionize this area of research.  With this in mind I would like to continue with discussing python in mathematics. 

I previously showed a fairly simple program for Slope-intercept using python. The program served to output a formula in slope-intercept form given two points.   In another post on computer programming in mathematics, I included some Java code for testing if a number is prime. Below I have a Python code to ask for 2 numbers to serve as the starting point and ending point for  listing primes within that range.  I have not included any error checking or input checking,  so it is important to ensure this code works you must enter an integer greater than 1. If you do not do this you will not get a meaningful output or you will get errors.   This was written and tested in Idle 3.2 so it may not work as expected in other versions.  This code also utilizes the standard sieving algorithm to perform its prime checking so it is by no means efficient for larger inputs and can be very processor intensive

.   CAUTION!!!  only use this code to list primes in small ranges CAUTION!!!   The caution is here because I have locked up my computer trying to list primes from 2 – 10,000,000 I would recommend running it on ranges in the 100's It was originally written for educational and research purposes only.   I generally use it to write primes to text files to print out as educational resource materials.

#primecheck(x) checks if x is prime returns boolean#

def primecheck(x):

    i=2

    for i in range(2,x):

        if x%i == 0:

            return False

    return True

#primelist(start,stop) lists all prime numbers between start and stop#

def primelist():

    start = int(input('Enter starting point must be an integer greater than 1:  '))   

    stop = int(input('Enter ending point must be an integer greater than 1:   '))  

    i=2

    primeList = []

    for i in range(start,stop):

        if primecheck(i) == True:

            primeList.append(i)

    return primeList

print (primelist())

Python Slope-Intercept

Python  is a wonderful programming language for quickly coding solutions to many problems. You can download python by going to python.org/download and selecting the version that’s best for your operating system.  Python can run from command line or though Idle that comes with most python downloads.   I have been working with python for almost a year now and it has become my “go to “for  

a lot of work in mathematics. It is not necessarily the “best” language for processor intensive mathematical algorithms, but it can offer efficient and quick solutions that can be easily applied to larger projects.  Recently, I have been doing some review of linear algebra and slope-intercept formulas. This is a fairly simple subject in mathematics but I thought it might be interesting to use python to solve some of these problems.   My solution might not be elegant but it is fairly simple.  I designed the program to take in two points (x1, y1), (x2, y2)  and output an equation of the line in slope-intercept form.   For those who may not be familiar slope-intercept form it is  y = mx + b where m is the slope and b is the y- intercept. The slope is calculated from the two given points by the formula (y1-y2)/(x1-x2) and the y -intercept can be found by 

b = y – mx.

Slope-intercept is useful in linear depreciation models.  For example:  if you bought a car at $81,292 and after 10 years it is only worth $33,121 to find out how much it will be worth after 13 years you can use slope-intercept to get a depreciation formula. The two points are at 0 years it is 81,292 (0, 81,292) and at 10 years it is 33,121 (10 , 33,121) substituting these numbers into our formulas you first find the slope which will represent the rate of depreciation.  You use the slope formula (y1-y2)/(x1-x2) 

(33,121 – 81,292) / (10-0) = -48,171 / 10 = -4817.1   and the y intercept is b = y – mx. 81292- 0(4817.1) = b= 81292 and putting it together you get y = (-4817.1) x + 81292 using this formula we can then determine the value of the car at 13 years by substituting 13 in for x

y = (-4817.1)13 + 81292 and we get 18,669.70 so after 13 years our car is worth $18,669.70

The Python code to find our linear depreciation formula is fairly straight forward


def slope():

    result = ''

    x = float(input("enter x one"))

    y = float(input("enter y one"))

    a = float(input("enter x two"))

    b = float(input("enter y two"))

    slope = (y-b)/(x-a)

    yint = y - (slope*x)
    if yint < 0:

        sign='-'

    else:

        sign='+'

    result = str("y="),str(slope),str("x"),sign,str(yint)

    return result

print(slope())

I use the input() method to take in the two points (x,y), (a,b)

then create a variable named “slope” and then immediately assign it to the formula to find the slope given 2 points (y-b)/(x-a). I then create a variable yint and assign it to the formula for the y-intercept. The if-else statement I use to aid with formatting because generally the + (plus sign) isn't displayed when you get a positive. The final line print(slope()) is there only for when you are running the code in console or idle if you were to convert this to a python module it won't be needed.  

I am sure there are more elegant and efficient ways of doing this as well as built-in python methods to make it display much better but that’s the beauty of python it’s easy to modify and play around with to get desired output style. 

Standardized thinking.......

    Despite efforts from Washington, math education is still in trouble in the U.S.  We are now experiencing the so called “math wars”.  Educators are competing with policy makers, and mathematicians are challenging both.  The argument is that educators are focused on the mission and purity of education as a whole.  Education is the basis for lifelong learning and free thought.  When mathematics is concerned there is a wedge between mechanical learning and reasoning.  University mathematicians believe that teaching reasoning before standard algorithms is crippling student‘s abilities to learn the more advanced concepts in college level mathematics.  It is argued that before learning mathematical and numerical reasoning students must understand the standards of math.

The dangers of sticking to a mechanical standard algorithm approach are flexibility, intellectual growth, and individual ability. Flexibility refers to having a standard such as right to left based calculation. This concept alone can prove very confusing especially in younger students who are still honing their reading skills. In most languages, specifically English, students are learning to read from left to right. Every time they are exposed to any form of writing they are taught that it should be understood from left to right. When children begin learning math they are taught to read math in contradiction to all the other reading that they learn. In essence this can cause confusion in both subjects for most students.  Not only is the left to right approach quicker and easier, it also flows logically with the way we are taught to read.

Another issue is that when teachers are grounded with this single algorithm method, students who understand it differently are often punished with bad grades.  Some teachers are so set in the standard right to left and carry method, that if a student chooses to use a different method they are given a bad grade.  The stigma put on outside the norm computation will stunt the students drive and understanding of math. It places in them the “why am I wrong when I got the same answer?” argument.     I had a personal experience with this receiving a failing grade on a test because on the page where you “show your work” I did all multiplication from left to right. The teacher of the class I was in had no clue how that method worked and therefore failed me on my exam “for cheating”. The reason given was that the only way to multiply is the standard right to left so the work shown was an attempt to cover up cheating.  Now this is a specific individual example that does not reflect all teachers, but there are still some that are so set in the standard that “out of box” thinking is “out the window”.

Aside from flexibility, we have the issue of intellectual growth.  Intellectual growth is arguably the central idea of education.  The ability to think and reason is devastatingly uncommon among many adults and children.  Most educators wish to remedy this situation, by trying to step outside of the memorize and regurgitate on the test paradigm.  Standardized testing is standardizing thinking and learning which is destroying innovation and lifelong learning.  While most agree that some concepts should be tested and retained to be successful in college, the ways by which we test and use these concepts need to be modified. 

The last issue is individual ability. Going back to the personal example of being labeled a cheater for utilizing a method that was easiest to me was a stab at my individual abilities. That failure caused me to have to take a much lower level math class then I should have been in twice.  Luckily, the next time around I had a better informed teacher that removed me from that class put me into pre-calculus instead where I aced the exams.  The point is that no matter how you try to standardize people, we are all individuals with individual ability to understand the world around us. 

The Beal Prize

Andrew Beal a self-made billionaire has announced the prize of one million dollars to the person who can prove the Beal conjecture.  The conjecture is similar to the famous last theorem of Pierre de Fermat.  Fermat’s last theorem, originally written in 1636 remained unsolved until 1995.    The theorem states that no three positive integers a, b, and c can satisfy the equation (aⁿ + bⁿ = cⁿ ), for any integer value of n>2.                  The Beal conjecture begins with the equation A ^X + B ^Y = C ^Z and states that if A, B, C, X, Y, and Z are all positive integers > 2 then A, B, and C must share a common factor.  The novice to mathematics might ask the question, what does all of this mean or more importantly why is this conjecture important?  This conjecture along with its predecessor Fermat’s last theorem is closely related to Diophantine equations.  Diophantine equations are a special set of equations named for Greek mathematician Diophantus of Alexandria.

The Diophantine equations all dealt with whole numbers and involved multiple unknown quantities. The most famous of these  equations is the Pythagorean Theorem, which is a² +b²= c ² where a and b are the two legs and c is the hypotenuse of a right triangle.  The Pythagorean equation shows the relationship of the lengths of all of the sides.  The real world applications of the Pythagorean Theorem are in engineering, architecture, cartography, and many other areas.  Diophantine equations all find their way into many separate real world applications.   The Beal conjecture is no exception the solution could lead to more applications, than just a simple intellectual curiosity.  Hopefully, this one doesn’t take 3 centuries to solve as did its predecessor.  

The proof of Fermat’s last Theorem was over 100 pages long and took 7 years of isolation for Sir Andrew John Wiles of Britain to complete. Will the Beal conjecture proof take as long? Now that the Fermat proof, which is closely related, is floating around, I propose that the solution to the Beal conjecture is within the Fermat proof or perhaps even an expansion or generalization of that proof. 

Better learning through abstraction..…

There are many barriers to learning and teaching mathematics, some of which include:  anxiety, comprehension, and focus.  How do we get past these?  First, let’s look at anxiety. Why does the mere mention of mathematics strike fear and confusion in students?   It seems there are so many different causes of this to look at, that we are unable to pin point a single method to fix the problem.  In fact to find the solution, we need use the solution.   You will say   “well that sounds impossible”.   If we don’t know the solution, how can we use the solution to find it? There is a simple answer, abstraction.

I touched upon the idea of abstraction in Algebra: the true heart of mathematics.  So rather than just focusing on a solution to all of the different barriers to learning mathematics, we instead use abstraction to find a general solution.  We can then begin teaching abstraction rather than just mathematics.   If we find ways to take the classic “math” out of the picture we avoid the instant fear that many students face.  The statement “Today, class we are going to invent a new dictionary” is much less daunting than “Today, class we are going to learn algebra”.   The first statement will immediately spark a curiosity. The students will begin asking questions, and wanting to know more. 

What does inventing a new dictionary have to do with algebra?  To answer this, we first need to think what is algebra?  Algebra is basically a “dictionary” of formulas, theorems, and proofs which apply to a wide range of problems. 

How do these relate to each other?  Let’s look at the commutative property of addition. In algebra this is “defined” as   (a + b = b + a) .    To look at this in words rather than algebraic symbols, you get 

 Commutative property of addition: When adding two numbers together the order of the numbers does not change the result of the addition.

Where do we go from here?  This seems no different than how algebra is introduced normally.  What I am proposing is taking the math out of it in the beginning, and then slowly adding the math in after the concept of abstraction is understood.  We ask the students to look at some common problems they may face in their daily lives.  We will then list out some of the top suggestions. The students will then be asked to find the commonalities between the problems to group them together. Then the students are asked to make a name for each of the problem groups.  They can then begin defining the general solution to those problem types. Keep it simple, only spend about a week introducing the concept of abstraction with simple problems.   After the idea of abstraction seems clear then you could start adding the Math back into the process. You will begin with asking the students about math related problems. You could start with something simple like what is an even number. Many of the students by this point are familiar with the idea of what an even number is.  However, this time you can approach the same way you approached your dictionary idea. You ask the students to create their own definition of an even number. Allowing them to create their own mathematical rules to use empowers them with the ability to not only recognize the steps of problem solving, but to understand it rather than just memorize it.

               Teaching abstraction leaves a wide open doorway to many different possibilities. The hardest part of learning math for many students is the daunting amount of formulas and rules to remember. A student who knows how and why these rules work, and can look at a problem and devise their own algorithms for solving it, will be way ahead of the curve when they are faced with solving any problem they encounter. When you think about it you use this concept every day. As adults we face many different types of problems, and we solve them by first identifying the problem then abstracting a set of procedures for solving them. We sometimes have similar problems that we have solved before, and we encounter new problems that need new solutions.  However, being successful in life is dependent on the solidarity and efficiency of our ability to solve problems. When I say problems I don’t mean just mathematical, it is the seemingly mundane parts of life for some and the more perplexing or dire situations we encounter that I call problems.         Abstraction, logic, deductive and inductive reasoning are so important to a successful life that part of the curriculum of each grade level should be dedicated to them.

Analysis of the MegaMIllions lottery

I compiled some data in Excel using the winning numbers data from 5-17-2002 – 3-23-2012.A few of the charts created using the winning numbers for the Mega Millions lottery are found below.

This first chart describes the number of times each number is drawn for each pick. From the data we see that the number 2 is more likely to be the first number drawn because it shows up as the first pick more often. The number of times 2 has appeared first is 75.4% of the total number of times a 2 has been in the winning numbers.

How many times each numbers is drawn
	first pick	second pick	third pick	fourth pick	fifth pick	mega ball
1s	82	0	0	0	0	19
2's	92	9	0	0	0	21
3s	78	7	1	0	0	27
4's	70	23	1	0	0	24
5's	68	22	3	1	0	18
6's	61	21	3	0	0	22
7's	63	24	6	1	0	28
8's	53	27	4	0	0	22
9's	52	36	6	0	0	27
10's	50	46	13	0	0	27
11's	39	32	10	3	0	17
12's	35	44	16	2	0	21
13's	27	49	15	2	0	27
14's	46	40	19	2	1	16
15's	24	39	19	4	0	23
16's	32	42	16	10	0	15
17's	17	50	30	3	0	17
18's	23	40	27	10	1	21
19's	12	38	30	11	0	22
20's	13	40	38	7	1	24
21's	10	33	35	13	2	30
22's	14	34	32	13	4	25
23's	9	27	30	12	2	18
24's	14	32	36	10	3	23
25's	10	33	29	19	4	20
26's	5	25	39	19	2	22
27's	9	25	42	16	7	20
28's	5	20	35	34	5	9
29's	4	26	33	30	7	25
30's	2	23	27	29	8	21
31's	2	25	40	29	12	21
32's	1	19	34	35	10	15
33's	1	11	25	29	13	23
34's	0	8	31	29	11	26
35's	1	12	30	29	16	28
36's	1	10	25	41	21	32
37's	1	5	33	37	7	17
38's	1	6	25	40	22	25
39's	1	7	30	46	23	18
40's	0	5	19	45	23	20
41's	0	3	20	28	26	19
42's	0	3	18	43	29	23
43's	0	4	22	27	31	20
44's	0	1	18	32	42	21
45's	0	1	19	38	37	19
46's	0	1	11	44	47	17
47's	0	0	7	35	40	7
48's	0	0	12	41	55	4
49's	0	0	3	33	54	7
50's	0	0	2	25	74	4
51's	0	0	4	26	81	1
52's	0	0	2	17	89	10
53's	0	0	2	16	56	0
54's	0	0	1	8	50	0
55's	0	0	0	4	49	0
56's	0	0	0	0	63	0

In this next chart I have laid out the average, mode, median, harmonic mean and other analysis of the number distributions across all of the different picks. 

	first pick	second pick	third pick	fourth pick	fifth pick	mega ball
median all	7.5	18	28	38	49	24
Average all	9.278210117	18.6624514	28.3015564	37.42607	46.4542802	23.990272
mode all	2	17	27	39	52	36
standard deviation	7.137233978	9.10326996	10.2470703	9.59011029	7.6711884	13.842004
Geometric mean	6.565968069	16.139646	26.1471469	35.9559721	45.6978254	18.277583
Harmonic mean	4.207752217	13.1495089	23.5028125	34.0860285	44.7726832	10.698143
Mean - stddev	2.140976139	9.5591814	18.0544862	27.8359598	38.7830918	10.148268
Mean + stddev	14.63723398	27.10327	38.2470703	47.5901103	56.6711884	37.842004
avg mean+-stddev	8.389105058	18.3312257	28.1507782	37.713035	47.7271401	23.995136

This final chart is all of the data above rounded to the nearest whole number.

first pick	second pick	third pick	fourth pick	fifth pick	mega ball
8	18	28	38	49	24
9	19	28	37	46	24
2	17	27	39	52	36
7	9	10	10	8	14
7	16	26	36	46	18
4	13	24	34	45	11
2	10	18	28	39	10
15	27	38	48	57	38
8	18	28	38	48	24

It is important to note, that all these numbers represent the various distribution statistics of the numbers that were drawn at each of the televised drawings, and do not represent any regressive or other types of predictive models. These numbers are just simply a statistical analysis of the given data for educational purposes only. All data represents the real numbers published at http://www.megamillions.com/numbers/pastdrawings.asp The website offers a tool at the top of the page to select your data range and get all of the drawings