Most Promising Batsman Cricket Has Ever Seen - Find Out Mathematically

When comparing different batsmen, the statistic that is invariably brought out is the batting average. It is a fair enough indicator of a batsman's ability too, for it suggests the number of runs he scores per dismissal - Brian Lara makes 53 runs per dismissal to Ramnaresh Sarwan's 40, hence Lara is clearly a superior batsman.

While the efficacy of averages is inarguable, it has its limitations. For instance, it doesn't tell us the consistency levels of a player: a batsman who scores 0, 200, 25 has exactly the same average - 75 - as one who makes 70, 80, 75, though it's obvious which one of the two has been more consistent.

Enter a statistical tool called the standard deviation. As the name suggests, this method indicates how much a sequence of numbers deviates from its average. 

The problem with average is that if one leg of yours is in the oven, and the other in a freezer, on an average you are comfortable. This is a big learning in statistics – the word average makes no sense if the standard deviation is very high. Thats why if a batsmen scores 200 in one inning and goes for duck in next 3, he would still have an excellent average of 50, but player not consistence. that means standard deviation is very high.

 

You'd obviously want greater consistency from a batsman, but check this sequence out: 16, 15, 17, 20, 22, 14, 18. Mr X is obviously extremely consistent - the standard deviation is only 2.61 - but at an average of 17.43, he isn't doing much to help the cause of his team. 

In the two run-sequences given earlier, for example, the second one has a standard deviation of just 4.08, while for the first, it's a whopping 88.98

A meaningful stat, then, is one which combines batting averages - for that is an indication of the sheer volume of runs he scores each time he bats - with a consistency index which measures how much he deviates from his average score. For the purpose of this exercise, the batting average has been divided by the standard deviation to arrive at an index. Batting index is exactly inverse to another stastical term called  coefficient of variation (CV) which is defined as the ration of standard deviation to mean.

The table below lists the ones with the most favourable batting index for players with at least 5000 Test runs, and it's interesting to see the ones who make the cut. On top of the ranking is Jacques Kallis, the batting machine from South Africa

Batsman	Runs	Average	SD	Batting index (Average/ SD)
Jacques Kallis	7940	56.31	44.54	1.26
Allan Border	11,174	50.56	40.49	1.25
Ken Barrington	6806	58.67	47.36	1.24
Jack Hobbs	5410	56.95	46.68	1.22
Arjuna Ranatunga	5105	35.70	29.44	1.21

Here are some other star player who had not made it to top 10 either, Ricky Ponting (1.13), Rahul Dravid (1.12), Adam Gilchrist and Sourav Ganguly (both 1.10). Inzamam-ul-Haq manages an index of 1.07, while Sachin Tendulkar has 1.03, both slightly better than two stalwarts from the 1980s, Sunil Gavaskar and Viv Richards (both 1.02, rounded off to the second decimal).

Let's now lower the bar to 3000 runs and look for consistency alone. How many would have guessed that Shaun Pollock would have had the lowest standard deviation among this group? In fact, the top six are all lower middle order batsmen who have consistently bailed their teams out in crises. Their averages aren't so impressive, but the standard deviations indicate just how consistently they have performed.

The six most consistent ones ( 3000 Test runs)

Batsman	Runs	Average	SD
Shaun Pollock	3406	31.25	23.44
Rodney Marsh	3633	26.52	25.91
Richard Hadlee	3124	27.17	26.31
Mark Boucher	3357	29.97	26.65
Ian Healy	4356	27.40	26.69
Jeff Dujon	3322	31.94	29.01

 [All the stats from test cricket only]

Quick Fact - Don Bradman, had a staggering average of 99.94 everyone knows, but at the same time a standard deviation of nearly 87 is also highest (most inconsistent) among all batsmen with at least 3000 run.

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Samsung or Mr. Modi ? "Mode" is Most Popular

Study of statistics is all about collecting, arranging and interpretation of data. Data could be scientific observations, survey, or public opinion as we see in case of elections in democratic system.

One must know there are number of properties in study of statistics to interpreted data,like mean ,median, mode, variance and standard deviation. So what kind of property require to elect someone during elections? By making a choice among mean ,median and mode one can easily come to result that public opinion we get in terms of election are purely one choice based, as a voter can not say he is 60% with candidate A and 40% with candidate B. A voter has choose one the candidate between candidate A and B. You can not take mean of candidate A and B,thus Mode fits best in this case.

Even median also can not tell you about frequency of data. In this way Mode comes in story which is all about most repeated value from the group of data or we can say most popular value from a system of data.

Suppose there are 30 Students in a class and and a class representative has to be elected from candidates A, B and C. following pattern of voter slip is found inside ballot box (you don't need to count)
A  A  A  C  C B  B  A  B  A  C  A  B  C  A  B  A  C  C  C  C  C  C  C  A  C  C  C  B  B

As we see C is the most popular candidate among all them, appearing 14 times our mode(winner) is C.  'Mode' is Mr. Narendra Modi for Indian General Election 2014, 'Mode' is Barak Obama for US Presidential Election 2012.

It was a simple examples of statistical term Mode, but not that simple since election is National or International level phenomenon usually occurring in 4 or 5 years. 

Here is another application of Mode we use everyday without even knowing it. A practical example might go like this. If you want to buy a new phone, you ask your friends on what they are using. Say you asked ten of them. You came to know that 5 of them are using the same phone(let Samsung) and 3 of them are using some other type(iPhone?) and rest 2 of them are using some other phones. Assuming all of them are happy with their phones. You might think buying the one which is used by more number of people. In this way Mode proves to be a daily life phenomenon.

Calculation of Mode in case of bar graphs become more easier as the most frequent appearing value is the tallest bar, for example

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Indian Cricket Team - ft. Mr. Mean, Mr. Median & Mr. Mode

Indian National Cricket team has to check its batting line-up strength ahead of an International tournament. BCCI has appointed Mr. Panda for this task and asked three veteran bears to assist Mr. Panda

So, First there is Mr.Panda

Then there are Mr. Mean, Mr. Median, and Mr. Mode. They are our average bears:

Mr. Panda wanted to know how the players of Indian national team were doing. He called for Mr. Mean, Mr. Median, and Mr. Mode and said:
Mr. Panda: Go to the stadium and find out how the average player is scoring.

[Averages look at groups of things (or individuals) and point out what’s common, normal, or ordinary]

Mr. Mean went to Stadium first. The Cricketers had differing number of runs

Mr. Mean is mean. He took all their runs and counted them. There were two hundred and seventy runs in total by nine batsmen. He divided all the runs into nine piles. Each player got 30 runs:

This made some of the players happy and some mad!


Mr. Mean went back to Mr. Panda and said:
Mr. Mean: The players are great. The average player has 30 runs.
But after Mr. Mean left, the players who lost runs took them back. So all the players had the same number of runs as before.

Mr. Median went to Stadium second. Mr. Median loves the middle. He found the player with the middle amount of runs and reported this number to Mr. Panda.

Mr. Median: The players are okay. The average player has 21 runs.
Finally Mr. Mode went to see the squirrels. Mr. Mode loves whatever happens most often.

He came back and said…
Mr. Mode: The batsmen are terrible!!! The average player only has thirteen run.
Whom should Mr. Panda believe? He sent the Three Average Bears to Cricket Stadium and they all came back with different answers. What do you think is the best average?

Significance of Mean, Median and Mode for Engineers:

"As an engineer, I often faced situations with more than one “right” answer. A good engineer can usually make the numbers say almost anything. However, a great engineer uses her judgement to find the best and most accurate way to present the data. Some clients pressure you to give them the answers they want. However, this never pays off in the long run. Someone could get hurt. And even if no one gets hurt, someone else is bound to look at the data later. If they find data splicing, your client could get sued and lose more money than if you had told them the truth. Also, you could lose your licence for bad engineering practice. Engineering is more than number crunching; it’s about being honest and using good judgement" - By Hannah Holt
Hannah Holt is A Children's Author, the concept of bears is taken from her blog lightbulbbooks.com

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Mean, Median, Mode - Mathematical Description

One thing that it always hard for new statistics students is the idea that the mean, median, and mode each have two different (but related) meanings.  Each can refer to a property of some specific set of data that has been collected or to the property of an entire distribution.  

Mode:  This one is pretty straightforward.  If you have a set of numbers (i.e. data points), then the mode is the number (or numbers) appearing most frequently.  It doesn't matter if the most frequent number is the smallest in the list, the largest, or anywhere in the middle.  If it shows up on the list the largest number of times, it is the mode.

Median:  This one is also pretty easy to understand.  You take your list of numbers, put them in order from smallest to largest, and then pick the one that is exactly in the middle.  That's the median.  There are some details left to nail down that aren't very interesting (and I'll discuss them at the end of this part), but the big idea is that the median has the property that half of the data points are below it and half are above it. 

Now notice that it could be the case that the small numbers on the list are really "bunched up" while the larger number are really spread out (or vice versa) -- something like:  1, 2, 3, 4, 10, 100, 1000.  The median is 4 because 1,2,3 are below it and 10, 100, 1000 are above it.  So the median can be really close to the smallest numbers and really far from the largest numbers (or vice versa).

One uninteresting detail has to do with the number of data points.  If the number of data points is odd, then there is always a data point that is the median (like in the example above).  However, if the number of data points is even -- for example: 1, 4, 5, 11 -- what do you do?  In this case, people typically find the two numbers in the middle of the list (4 and 5 from my example) and average them to get the median.  So it would be 4.5 for this example.  Notice that 4.5 has the desired property that half the data points are less than 4.5 and half a greater.  So the median need not be one of the data points. 

The other uninteresting detail has to do with repeated data points.  For example: 1, 2, 3, 3, 3, 3, 4, 4, 5.  In this sorted list, the third 3 is the middle data point so it is the median.  If we include two of the threes as "smaller than the median" and one of the threes as "larger than the median" then exactly half are larger and half are smaller.  The more natural idea of only looking at data points strictly larger than three and strictly smaller fails since 2 data points are actually smaller and 3 are actually larger.  There does not exist a number that has the property that exactly half the data are strictly larger.  Nevertheless, the median is three.

Mean:  It's a little harder to understand the "physical significance" of the mean.  With the median, we had a notion of the "middle" of some data that had the property that it didn't matter how much a data point was above (or below) the "middle" only that it was above (or below) the "middle."  In the 1, 2, 3, 4, 10, 100, 1000 example, that notion makes the "middle" of this set of numbers equal to 4.  That "middle" wouldn't change if the 1000 data point were changed to 100,000.  In some ways, that is a nice property to have, but in other ways, it seems rather odd.  I mean, 4 doesn't really seem to capture the idea that the numbers are getting really spread out on the top end. 

The mean is a way to describe the "middle" so that numbers far from the middle have a bigger influence than numbers close to it.  The best physical analog is the idea of the center of mass but if you haven't thought carefully about physics, this analog won't help much.  Instead, let me ask this question.  If you are generally an "A student" in a class, and you happen to fail an exam, would you rather fail it with a 50% or with a 0%?  Why?  In either case, that failing grade will be your lowest grade so why should it matter whether it was bad or REALLY bad.  The reason is that the 0% score brings down the average more than the 50% score does.  (They have the same impact on the median.)  The mean is an idea of the "middle" that is sensitive to how "stretched out" the data is.

Finally We Have:

Three Ways to Find an Average:

1. Mean: Add up all the parts and divide by the number of pieces. (40+12+8+7+1+1+1)/7 This is the most commonly used average.

2. Median: Arrange all the numbers from most to least. Pick the middle number. If you have an even number of data, take the mean of the two middle numbers.

3. Mode: Look through all the numbers and count how often each number happens. Pick the number that happens most often.

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How Chanakya Deals With Snakebites? ft. Dr. Edward Jenner

Chanakya was the royal advisor of the first Mauryan emperor Chandragupta who played an important role in the establishment of the largest empire yet seen in Indian history, the Maurya Empire and Edward Jenner is know as the father of modern vaccination which ruled out numbers of invincible diseases from the history of mankind. How Chanakya, a philospher comes into story of a doctor and his discovery?

Lets try to dig it out, starting with some terrifying data about 3rd member of this article,

QuickFact  Snakebites, most of them are caused by non-venomous snakes. Of the roughly 3,000 known species of snake found worldwide, only 15% are considered dangerous to humans. According to The American Society of Tropical Medicine and Hygiene more than 2.5 lakh cases of snake bites and 46000 death due to it are reported every year in India making it the most heavily affected country in the world.

Antivenom is a biological product used in the treatment of venomous bites or stings. Antivenom is created by milking venom from a relevant snake, spider, insect, or fish to initiate an immune system response to external agent (i.e. venom). The resulting neutralizer enzyme or antibodies are then harvested from the animal's blood and stored to give to the victim of snakebite or sting.

Antivenoms bind to and neutralize the venom, halting further damage, but do not reverse damage already done. Thus, they should be administered as soon as possible after the venom has been injected, but are of some benefit as long as venom is present in the body. Since the advent of antivenoms, some bites which were previously invariably fatal have become only rarely fatal provided that the antivenom is administered soon enough.

The principle of antivenom is based on that of vaccines, developed by Edward Jenner; however, instead of inducing immunity in the patient directly, it is induced in a host animal(horse or sheep) and the hyperimmunized serum is transfused into the patient.

The credit for discovery of modern vaccination is given to an English doctor named Edward Jenner in 1796.

Jenner introduced fluid containing the cowpox virus to healthy people body. Cowpox is an infection that usually infects cows (as the name suggests), but it can also infect humans. It’s very similar to smallpox, but is a much less dangerous disease.

Jenner found that by injecting people with the cowpox virus, they were protected from infection by the smallpox virus. He also discovered that this immunity could be passed from one person to another. This process became known as vaccination.

But The earliest documented examples of vaccination are from India and China in the 17th century, where vaccination with powdered scabs from people infected with smallpox was used to protect against the disease. Smallpox used to be a common disease throughout the world and 20 to 30% of infected persons died from the disease. Smallpox was responsible for 8 to 20% of all deaths in several European countries in the 18th century.

Another story found about history of vaccination related with Mouraya emperor Chandragupt and his royal advisor Chanakya which lead us further back to 300 BCE.

The emperor Bindusara was the son of the first Mauryan emperor Chandragupta Maurya and his queen Durdhara. According to the Rajavalikatha, a Jain work, the original name of this emperor was Simhasena. According to a legend mentioned in the Jain texts, Chandragupta's Guru and advisor Chanakya used to feed the emperor with small doses of poison to build his immunity against possible poisoning attempts by the enemies. One day, Chandragupta, not knowing about the poison, shared his food with his pregnant wife, Queen Durdhara, who was 7 days away from delivery. The queen not immune to the poison collapsed and died within a few minutes. Chanakya entered the room the very time she collapsed, and in order to save the child in the womb, he immediately cut open the dead queen's belly and took the baby out, by that time a drop of poison had already reached the baby and touched its head due to which child got a permanent blueish spot (a "bindu") on his forehead. Thus, the newborn was named "Bindusara".

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Visha Kyanya - Truth Behind Fiction

The Visha Kanya (English: Poison girl), refers most often as fiction character with historic origin. They are young women used as assasins, often against powerful enemies and  mostly found in Indian mythology and literatures.

Visha Kanya were used by kings to destroy enemies during the times of the ancient India. The story goes that young girls were raised on a carefully crafted diet of poison and antidote from a very young age. Although many would not survive, those that did were immune to other poisons and their body fluids would be poisonous to others; sexual contact would thus be lethal to other humans.

To justify the authenticity of being a Visha Kanya we have to understand the concept of poison or venom.

Numbers of venomous creatures are found around us, snake is one of the most notorious among them. If snake bites someone its venom may cause one of two effects to our body

•  Venom may cut the nervous system, hence isolating the control of brain over various organs most importantly to Heart.
•  Venom mixed into blood may further dissolve body tissues to blood thus affecting natural body function.
   

To prevent this, an antidote of venom is prepared by injecting dilute solution of it into a horse (or ship), thus horse's immune system produces its own antidote or antibody against the venom which is further derived to give in form of injection to a human being bitten by snake.

In this way our body can develop a self antidote for  a particular or a group of venom if it is given in dilute amount, but problem is that to fight against the high concentrated venom injected by snake(or any venomous insect or reptile) needs a high concentration of antidote. Human body can not store such a high concentrated antidote over a long period, that why its sored in form anti-venom and given externally to the victim.

If someone started taking dilute amount of venom every month to keep his immune system ready to fight against any external poison or venom given to him we will have a profound amount of anti-venom stored in his body and note that it will be the anti-venom (antidote) which will be stored by taking small quantity of poison or venom over a long period.

If the theory behind origin of Visha Kanya is treating them with poison over childhood , they would grow into a living antidote, and their bite or somehow contact to another person will result as sharing of a useful antidote not the deadly poison. And yes, it is way similar to vaccination.

This way, a person sharing blood, saliva or sexual contact with Visha Kanya will go vaccinated against any future snakebite. Thus Visha Kanya would have been much useful(and may be pretty as well) than dangerous.

QuickFact When taken orally it is called poison, when injected by some other means like bite or syringe it is called venom, otherwise both are same thing and have same hazardous effect to anatomy.

Laboratories use extracted snake venom to produce antivenom. Photo: Wikipedia

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Guessing a MCQ Answer - Negative Marking Principle

Most of the competitive exams being held now days offers multiple choice questions (abbreviated MCQs), so even a person who don't have any idea about the solution may answer a guess from available option, unlike in case of conventional exam where one go for a solution only if he has to write something fact-full with respect to question asked.

If number of possible options associated with question is 4(option A, B, C, D ), and one makes a guess for answer he get the right answer in one of the each four trials. Since there are four options for each question and correct answer for a question is distributed over four options A, B,C and D. There would be equal chances for all of the options to be correct(i.e. 1/4 or 25% each). If one choose to mark 4 consecutive questions with option C, he will be correct with at least (as average) one guess out of four.

This is known as probability in mathematical studies, probability stands for chances. If one has to randomly choose one of the available option out of the four option provided the chance of being it correct would be 1/4 or 0.25 or 25%.

With only two option given for a question and making guess for one out of two question will result in probability of making 1/2 or 0.50 or 50 % of it correct.

In another case if five options are given with reference to a question and one has to guess an answer he can guess on out of five, that means 1/5. Since number of options are 5 in this case at least 1 out his 5 guesses would be correct and we can say that chances or probability of getting a right answer is 1/5 or 0.20 or 20% in this case.  As we see number of options increases chances (probability) of guessing a correct answer is decreasing.

Its good from the perspective of aspirant giving a 4 option MCQs exam, he may have a chance to get 25% easily without knowing anything useful and by just making random guesses.

But the teacher, administrative or anyone conducting exam does not want any one to score  based on random guesesses, the purpose of exam is to screen only reward marks based of knowledge of the individual. So how to deal with guessers, here Negative marking comes into picture but how much?

Let, Fill is math teacher he conducts a MCQ based test for his students with 4 options provided with every question on the test sheet, also each question carries equal makes and that is equal to 1. He knows that if one goes with guessing 1 out of 4 questions will be correct, but at the same time 3 will be incorrect. He wants to reward something for incorrect options so that this guessing pattern of 3 incorrect and 1 correct(luckily) out of 4 question will result in zero marks. He  assumes to assign 'N' marks for incorrect answer
so

Number of incorrect guess * N + marks of correct guess * marks for correct answer = 0

 3*N + 1* number for correct answer = 0

By making this way overall effect of guess results in zero, if N= -1/3 of marks for correct answer.

(if 1 mark is decided for every correct answer, -1/3 will be for incorrect answer)

If number of options provided for each question is 2. Due to random guess 1 out  each 2 answers will be correct(luckily) and 1 will be incorrect(mathematically). in this case

1*marks for incorrect answer guess(N) + 1* Marks for correct answer = 0
which provides, marks for incorrect guess equal to minus of marks for correct answer.

In case of questions with five options negative marking is equal to minus of one fourth of marks for correct answer by following similar calculation.

QuickFact Writer has got 36 % (>25% yeahh!) of maximum marks by guessing all the answer with option B in a 4 option based MCQ exam with zero negative marking. This could be happen because correct answer is randomly distributed over A, B, C and D. We give equal weight (25% each in this case) to every possible choice in study of Probability.

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Better Representative of Indian Economy - Median

Problem:	The Panda family has 5 children, aged 9, 12, 7, 16 and 13. What is the age of the middle child?
Solution:	Ordering the children's ages from least to greatest, we get:
	7,  9, 12,  13,  16
Answer:	The age of the middle child is the middlemost number in the data set, which is 12

But since our title does not stand for Pandas, better we move to Indian Economy,

Indian GDP was 1,708,46 billion US dollars in 2010 and it is expected to touch the target of 3,315,36 billion US dollars by 2020, What would be the value of GDP when we can say India has reached its half of the target during this growth period? As obvious it would be the GDP of middlemost year, i.e. 2015.

The "middle" value in the list of numbers is known as "median". The median is also the number that is halfway into the set. To find the median, the data should be arranged in order from least to greatest. If there is an even number of items in the data set, then the median is found by taking the mean (average) of the two middlemost numbers.

Median is more used for illustration purposes when the mean is misleading. For example, according to Forbes with 111 billionaires, India is 3rd on rich list at the same time World bank marked about 276 million people of India living below $1.25 per day. In 2012, the Indian government stated 21.9% of its population is below its official poverty limit. by looking at the median household income in countries. Here the median, unlike the mean provide better representation of data, doesn't let the super rich make the normal man look not-so-poor.

Another example for median may go like this, if you participated in a competitive exam and got some score and some rank according to your score. You can assess how much difference a mark can make. That is, if you find the median of the score, which is the score of the person who got exactly the middle rank. And compare it with the highest and lowest, you'll get a better idea of how tough the competition was. 

Median holds a strong scientific representation for half life theory. Radioactive material (like Uranium) is often summarized by its half-life, which is also a median. This decides what we want to use the material for. If low, it might be used as a radioactive contrast. If high, maybe atomic clock.

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Golden Ration - Beauty of Math

The golden ratio (symbol is the Greek letter "phi" shown at left) is a special number approximately equal to 1.618 It appears many times in geometry, art, architecture and other areas.

The Idea Behind It

We find the golden ratio when we divide a line into two parts so that: the whole length divided by the long part is also equal to the long part divided by the short part

Beauty

This rectangle has been made using the Golden Ratio, Looks like a typical frame for a painting, doesn't it?

Some artists and architects believe the Golden Ratio makes the most pleasing and beautiful shape.

Do you think it is the "most pleasing rectangle"?

Maybe you do or don't, that is up to you!





Dividing a Golden ratio rectangle continuously into new golden ratio rectangles, we get following pattern

Many buildings and artworks have the Golden Ratio in them, such as the Parthenon in Greece, but it is not really known if it was designed that way.

The Actual Value

The Golden Ratio is equal to:

1.61803398874989484820... (etc.)

The digits just keep on going, with no pattern. In fact the Golden Ratio is known to be an irrational number and I will tell you more about it later.

Calculating It

You can calculate it yourself by starting with any number and following these steps:

• A) divide 1 by your number (=1/number)
• B) add 1
• C) that is your new number, start again at A

With a calculator, just keep pressing "1/x", "+", "1", "=", around and around. I started with 2 and got this:

Number	1/Number	Add 1
2	1/2=0.5	0.5+1=1.5
1.5	1/1.5 = 0.666...	0.666... + 1 = 1.666...
1.666...	1/1.666... = 0.6	0.6 + 1 = 1.6
1.6	1/1.6 = 0.625	0.625 + 1 = 1.625
1.625	1/1.625 = 0.6154...	0.6154... + 1 = 1.6154...
1.6154...

It is getting closer and closer!

But it takes a long time to get even close, but there are better ways and it can be calculated to thousands of decimal places quite quickly.

Drawing It

Here is one way to draw a rectangle with the Golden Ratio:

• Draw a square (of size "1")
• Place a dot half way along one side
• Draw a line from that point to an opposite corner (it will be √5/2 in length)
• Turn that line so that it runs along the square's side

Then you can extend the square to be a rectangle with the Golden Ratio.

The Formula

That rectangle above shows us a simple formula for the Golden Ratio.

When one side is 1, the other side is:

The square root of 5 is approximately 2.236068, so the Golden Ratio is approximately (1+2.236068)/2 = 3.236068/2 = 1.618034. This is an easy way to calculate it when you need it.

Interesting fact: the Golden Ratio is also equal to 2 × sin(54°), get your calculator and check!

Fibonacci Sequence

There is a special relationship between the Golden Ratio and the Fibonacci Sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

(The next number is found by adding up the two numbers before it.)

And here is a surprise: when we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio.

In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few:

A	B		B/A
2	3		1.5
3	5		1.666666666...
5	8		1.6
8	13		1.625
...	...		...
144	233		1.618055556...
233	377		1.618025751...
...	...		...

We don't even have to start with 2 and 3, here I chose 192 and 16 (and got the sequence 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ...):

A	B		B / A
192	16		0.08333333...
16	208		13
208	224		1.07692308...
224	432		1.92857143...
...	...		...
7408	11984		1.61771058...
11984	19392		1.61815754...
...	...		...

The Most Irrational ...

I believe the Golden Ratio is the most irrational number Here is why ...

One of the special properties of the Golden Ratio is that it can be defined in terms of itself, like this:
	(In numbers: 1.61803... = 1 + 1/1.61803...)
That can be expanded into this fraction that goes on for ever (called a "continued fraction"):

So, it neatly slips in between simple fractions.

But many other irrational numbers are reasonably close to rational numbers (for example Pi = 3.141592654... is pretty close to 22/7 = 3.1428571...)

Pentagram

No, not witchcraft! The pentagram is more famous as a magical or holy symbol. And it has the Golden Ratio in it:

• a/b = 1.618...
• b/c = 1.618...
• c/d = 1.618...

Other Names

The Golden Ratio is also sometimes called the golden section, golden mean, golden number, divine proportion, divine section and golden proportion.

 

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Irrational Numbers

An Irrational Number is a real number that cannot be written as a simple fraction.

Irrational means not Rational

Examples:

Rational Numbers

OK. A Rational Number can be written as a Ratio of two integers (ie a simple fraction).

Example: 1.5 is rational, because it can be written as the ratio 3/2

Example: 7 is rational, because it can be written as the ratio 7/1

Example 0.333... (3 repeating) is also rational, because it can be written as the ratio 1/3

Irrational Numbers

But some numbers cannot be written as a ratio of two integers ...

...they are called Irrational Numbers.

It is irrational because it cannot be written as a ratio (or fraction), not because it is crazy!

Example: π (Pi) is a famous irrational number.

π = 3.1415926535897932384626433832795 (and more...) You cannot write down a simple fraction that equals Pi.

The popular approximation of ²²/₇ = 3.1428571428571... is close but not accurate.

Another clue is that the decimal goes on forever without repeating.

Rational vs Irrational

So you can tell if it is Rational or Irrational by trying to write the number as a simple fraction.

Example: can be written as a simple fraction like this:

9.5 = ¹⁹/₂

So it is a rational number (and so is not irrational)

Here are some more examples:

Number	As a Fraction	Rational or Irrational?
1.75	7/4	Rational
.001	1/1000	Rational
√2 (square root of 2)	?	Irrational !

Square Root of 2

Let's look at the square root of 2 more closely.

If you draw a square of size "1", what is the distance across the diagonal?

The answer is the square root of 2, which is 1.4142135623730950...(etc)
But it is not a number like 3, or five-thirds, or anything like that ...

Famous Irrational Numbers

	Pi is a famous irrational number. People have calculated Pi to over a quadrillion decimal places and still there is no pattern. The first few digits look like this: 3.1415926535897932384626433832795 (and more ...)
	The number e (Euler's Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. The first few digits look like this: 2.7182818284590452353602874713527 (and more ...)
	The Golden Ratio is an irrational number. The first few digits look like this: 1.61803398874989484820... (and more ...)
	Many square roots, cube roots, etc are also irrational numbers. Examples: √3 1.7320508075688772935274463415059 (etc) √99 9.9498743710661995473447982100121 (etc)
But √4 = 2 (rational), and √9 = 3 (rational) ... ... so not all roots are irrational.

Note on Multiplying Irrational Numbers

Have a look at this:

• π × π = π² is irrational
• But √2 × √2 = 2 is rational

So be careful ... multiplying irrational numbers might result in a rational number!

QuickFact : Apparently Hippasus (one of Pythagoras' students) discovered irrational numbers when trying to write the square root of 2 as a fraction (using geometry, it is thought). Instead he proved you couldn't write the square root of 2 as a fraction and so it was irrational.

But followers of Pythagoras could not accept the existence of irrational numbers, and it is said that Hippasus was drowned at sea as a punishment from the gods!

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Normal Distribution, Standard Deviation And Graduate Aptitude Test in Engineering

Let a Professor is making a test
Here are the students results (out of 60 points):
20, 15, 26, 32, 18, 28, 35, 14, 26, 22, 17
Most students didn't even get 30 out of 60, and most will fail. (marks below 20 conventionally)
The test must have been really hard, so the Prof decides to Standardize all the scores and only fail people 1 standard deviation below the mean.
The Mean is 23 (μ), and the Standard Deviation(σ) is 6.6 by

Standard Deviation":

And and standardized score z;

 

and these are the Standard Scores, which can be treated as relative scores with respect to hardness of the test:
-0.45, -1.21, 0.45, 1.36, -0.76, 0.76, 1.82, -1.36, 0.45, -0.15, -0.91
Only 2 students will fail (the ones who scored 15 and 14 on the test)


The Standard Deviation is a measure of how spread out numbers are and Normal Distribution is the distribution of data around the mean, which is the hill point of the curve. Distribution of standard deviation around mean is shown in image.




look at the cumulative percentage in orange, it indicates position of a particular data value in entire collection of data, for example if student has got 26 marks in our example, which has Z score 0.45 (let 0.5 ) he stands above 69.1% (50mean+19.1) of students of the class, and 31.9% has higher marks than him.

We have seen an exam could be set very hard in response to increasing competition.One can not fix conventional 33% as passing limit in this case.

This is what exactly happens in case of Graduate Aptitude Test in Engineering, known as GATE, conducted by one of 7 IITs or IISc Banglore every year to provide scale technical and logical abilities of engineering graduates seeking for higher education or job opportunities.

To decide cut-off marks for a particular stream of engineering they choose to go with mean + 1 standard deviation (you can find in GATE information brochure), that means z score +1 is set as cut-off.

As we have seen mean + 1 standard deviation (z score 1)covers 84.1 % (50% for mean and 34 % for +1Standard deviation) of the total distribution , Z score 1 is set as cut-off score, which means cut-off score covers 84% candidates below cut-off marks and select rest 15.9% those are considered to be eligible (passed the exam).

So if cut-off marks for a particular exam is 26 (out of 100), it means only 16 % of candidates appeared in exam got marks more than 26, and rest 84.1% stands below.





An interesting thing to mention here, clueless candidates who are giving exam without preparation will get marks in lower limit of distribution thus pulling down the mean but increases the standard deviation(spread marks), what would be the overall effect on quantity MEAN + STANDARD DEVIATION (i.e. CUT-OFF) due to clueless candidates? Test your math.

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