Saturday, October 30, 2010

Median, Average and Standard Deviation

Mathematicians often use the terms "median," "average" and "standard deviation." They are important for data analysis. I will provide examples to illustrate how these can be applied.

Imagine that seven students write an exam and obtain the following scores: 60%, 65%, 65%, 75%, 80%, 90% and 95%. The total number of exams is seven. The median is the number which separates the higher half from the lower half. This is known as the central value. In this case, it is the fourth highest/lowest score. There are three lower scores and three higher scores from the central value. The median is precisely in the middle. In this case, the median is 75%.

With an odd number it is easy to find the median. With an even number, however, the method is a little different. Imagine that instead of seven exams we have eight. The scores are: 60%, 65%, 65%, 75%, 80%, 90%, 95% and 100%. To calculate the median it is necessary to determine the average of the two scores in the middle. In this case we have two scores below the top three and two scores above the top three. They are 75% and 80%. To calculate the average of these two scores, we add them together and then divide by two. 75+80=155. 155/2=77.5. 77.5 is the median of the eight exam scores.

The average in a data set is the value obtained by adding up all the values in a set and dividing them by the number of values. For example, if we have the scores 60%, 65%, 65%, 75%, 80%, 90%, 95% and 100%, the total is 630. We now divide this number by eight, the total number of scores. 630/8=78.75. Thus, 78.75 is the average of the eight exam scores.

Standard deviation measures average variation from the average value of a data set. Low standard deviation indicates that values tend to be close to the average and high standard deviation indicates that they are spread over a wide range. To calculate standard deviation, we first determine the average value of our data. Next we compute the difference of each value from the average and square the result of each. Then we divide the this number by the total number of values minus one and calculate the square root.

Consider the following scores on an quiz out of ten: 5, 6, 7, 8, 9, 9, 10, 10. To find the average, we need the sum of the eight scores. This is 64. We now divide 64 by 8. 64/8=8. The average value is 8.

Next we compute the difference of each value from the average. Here are the results:


Negative values are not a problem for calculating standard deviation. Now we square these values. -3x-3=9, -2x-2=4, -1x-1 =1, 0x0=0, 1x1=1, 1x1=1, 2x2=4 and 2x2=4.

The next step is to calculate the sum of these values. 9+4+1+0+1+1+4+4=24. The sum is 24.

We must now divide by the number of values minus one. We have eight values so we divide by seven. 24/7=3.42.

The final step is to determine the square root of 3.42. The square root of 3.42 is approximately 1.85. This means that the average variation from the value of 8 is 1.85. In other words, most exam scores deviate 1.85 points from the average value of 8 which is from 6.15 to 9.85. Of the eight exam scores, five fall within this range.

Mathematicians often calculate the median, average and standard deviation of a data set. They are very useful in the interpretation of data. For this reason, it is a good idea to become familiar with their many applications.

Tuesday, October 26, 2010

Grammatical Case

All languages have grammatical case. The number of cases varies from one language to another but all have nominative case. This is the subject form of a noun/pronoun. Grammatical case refers to the change in form of a noun/pronoun in a sentence.

English has two cases for nouns and three for pronouns. The cases for nouns are nominative and genitive. For example, the noun "friend" is "friend" in nominative and "friend's" in genitive. Nominative case is for subjects and genitive case is for possession.

Of course English can express many other cases but they are not inflected. Accusative case is used for direct objects but English nouns are the same in nominative and accusative. Dative case is used for indirect objects but this is also indistinguishable from nominative in English. This is illustrated by the following examples:

I want to meet your friend. (accusative)
I gave the ticket to my friend. (dative)

English pronouns have three cases. They are nominative, accusative and genitive. The following sentences illustrate the three cases:

I can play the piano. (nominative)
He doesn't know me. (accusative)
This is my car. (genitive)

The first person singular pronoun has three forms: I, me and my. Another form is "mine," a possessive pronoun, but this can also be analyzed as genitive.

Many languages have more inflected cases than English. German pronouns have four (nominative, accusative, dative, genitive) and Russian has six (nominative, accusative, dative, genitive, instrumental, locative).

Instrumental case indicates the manner in which an action is performed. For example, in the sentence "He kicked the ball with his left foot," the noun "foot" is in instrumental case.

Locative case is used for location. In the sentence "They live in Vancouver", the city "Vancouver" is in locative case.

Another case found in a number of languages is vocative. This is used to address people. In the question "Julius, how are you?" the name "Julius" is in vocative case.

Case is an important feature of grammar. Found in every language, it is thus universal. However, the number of cases can vary significantly from one language to another. English nouns are inflected for two cases, Russian for six, Finnish for fifteen and Hungarian for eighteen.

Tuesday, October 19, 2010

Pig Latin

Pig Latin is the name of a pseudo-language used by English speakers. It is often spoken by children and adults who do not wish to be understood by others. It can thus be characterized as a secret language.

The rules of Pig Latin are simple. To change an English word to Pig Latin the onset of the first syllable is transferred to the end and "ay" is added. For example, "Pig Latin" becomes "Igpay Atinlay and "speak" becomes "eakspay." If the word begins with a vowel, the onset is unchanged but "way" is added. Other variations of Pig Latin add "ay" and "hay".

Here are eight questions with their Pig Latin equivalents:

How are you? Owhay areway ouyay?
What's your name? Atswhay ouryay amenay?
Can you help me? Ancay ouyay elphay emay?
Do you speak English? Oday ouyay eakspay Englishway?
What did you eat for breakfast? Atwhay idday ouyay eatway orfay eakfastbray?
Where do you live? Erewhay oday ouyay ivelay?
How old are you? Owhay oldway areway ouyay?
Do you like French cuisine? Oday ouyay ikelay Enchfray isinecuay?

Pig Latin is a fun game which many enjoy. The rules of Pig Latin are simple but it can be a challenge to have an entire conversation in this language. Even if one is familiar with the rules of Pig Latin, it takes practice to become good at speaking and understanding it fluently.

Sunday, October 17, 2010

Victory with a Material Deficit

Most chess games are won with more material or the same amount as that of one's opponent. It is not so common to win with less material. In my case, however, I managed to win with two fewer pieces than my opponent. Though he had an extra rook and bishop, I was victorious. My opponent was Rootsrocker of England who played black. Here are the moves and commentary of this game of speed chess:

1. e4 e5
2. Nf3 Nc6
3. Bc4 Nf6

Black employs the Two Knights Defence. A popular fourth move for white here is Ng5 which targets f7. I choose to play a different move.

4. 0-0 Nxe4

I do not mind the loss of my e-pawn because black has not completed his development and the knight on e4 is unprotected.

5. d4 d6
6. dxe dxe

Material is almost even but the black king is exposed.

7. Bxf7+ Kxf7

I sacrifice my bishop to prevent the black king from castling.

8. Nxe5+ Ke6

Black cannot capture the knight because then I can capture the black queen.

9. Qg4+ Kxe5

The black king is exposed but black has seven pieces and I have only five. I must also be careful because my queen is attacked by black's light-squared bishop.

10. Bf4+ Kd5

My bishop joins the attack. Black's king moves to a vulnerable square. Kf6 is a better move for black.

11. Rd1+ Nd4

Black must block the check to prevent the loss of the queen.

12. Nc3+ Nxc3
13. Rxd4+ Kxd4
14. bxc3+ Kc4

Black does not capture the pawn on c3. He fears that Kxc3 exposes his king to too much danger by bringing it too close to enemy territory. Another option on move 14 is Bxc7+. The queen cannot capture the bishop because of the check from my queen. With this move I can capture the black queen on move 15. I miss this move and thus fail to play it.

15. Qe2+ Kxc3

Black now changes his mind and captures the pawn on c3. Once again Bxc7+ is an option but my move is effective.

16. Be5+ Kb4
17. Rb1+ Kc5

The checks are relentless.

18. Qb5#

Though I have less material than black, I am victorious. The keys to victory are my ability to develop quickly, protect my king and exploit black's exposed king. I manage to check the black king eleven times in a row. This is critical because it allows me to seize the initiative.

Thursday, October 14, 2010

Currencies of the World

Despite the introduction of the Euro as a common currency and a shared name for many currencies, a number of countries have unique names. With the adoption of the Euro, currencies such as the Dutch guilder, German mark, Portuguese escudo, Greek drachma, French franc, Spanish peseta and Italian lira ceased to exist. Other countries share the dollar, shilling, pound, peso and dinar. Nevertheless, many currencies have very different names.

Here is a list of currencies used by countries around the world. This list includes countries whose currencies are relatively unfamiliar.

Costa Rica-colon
Czech Republic-koruna
Papua New Guinea-kina
South Africa-rand

The reason many of the currencies on this list are relatively unfamiliar is that they are not likely to be listed in a foreign exchange rate table and are not in high demand. In many countries with weak currencies, the US dollar and Euro are very popular and can be freely used. Nevertheless, the use of a separate currency serves as a mark of sovereignty, history and pride.

Monday, October 11, 2010

Winning the Queen with a Sacrifice

It seldom happens that I win my opponent's queen with a sacrifice. However, it happened in a recent game of speed chess that I played at My opponent was Gattacci of Italy who played black. Here are the moves of the game along with my commentary:

1. e4 e5
2. Nf3 d6
3. Bc4 h6

Black wants to prevent my bishop from going to g5.

4. 0-0 Nf6
5. Re1 Nc6
6. c3 a6
7. d4 exd
8. cxd Be7

Black can now castle.

9. e5 dxe
10. Nxe5 0-0
11. Nxc6 bxc6
12. Nc3 Bd6
13. Be3 Re8
14. Qc2 Be6
15. Bd3 Nd5
16. Nxd5 Bxd5
17. Qd2 Qf6
18. a3 g6

Black makes a bad move. Qh4 is a better move because it forces me to weaken the defence of my king.

19. Bxh6 Kh7
20. Bg5 Qxd4

Black is now equal in material but he fails to see my next move.

21. Bxg6+ Kxg6
22. Qxd4 Rh8

I sacrifice my bishop to win the black queen. With Rh8 black threatens to capture my pawn on h2.

23. Bf4 Bxf4
24. Qxf4 f5
25. Qg3+ Kf6
26. h4 Rag8
27. Qc3+ Kg6
28. g3 Kf7
29. Qe5 Rxh4

Black makes a move that takes me by surprise. I cannot take his rook because this puts me in check.

30. Qxf5+ Kg7
31. Re7+ Kh8

The black king is now trapped in the corner.

32. Qf6+ Rg7
33. Qxg7#

Black threatens mate with Rh1 but I do not give him the opportunity to play it. My bishop sacrifice which wins the black queen is the turning point of the game. It gives me a decisive material advantage which I convert to victory.

Monday, October 4, 2010

Position of Adverbs in English Sentences

The position of English adverbs often varies in active and passive sentences. It tends to be more restricted in active voice than in passive. In certain cases, however, it is equally distributed in passive and active.

Consider the following sentences:

a) He detonated the bomb carefully.
b) He carefully detonated the bomb.

Though sentence a is more common than b, both are acceptable.

c) The bomb was detonated carefully.
d) The bomb was carefully detonated.

In sentences a-d the adverb can be placed before or after the verb. Here the adverb position is equally distributed in active and passive.

Now consider the following sentences:

e) They treated her unfairly.
f) *They unfairly treated her.

g) She was treated unfairly.
h) She was unfairly treated.

In active voice the adverb must be placed after the verb. In passive voice, however, the adverb can also be placed before the verb.

Here are more examples:

i) Laura sent the invitations electronically.
j) *Laura electronically sent the invitations.

k) The invitations were sent electronically.
l) *The invitations were electronically sent.

In sentences i-k the adverb must be placed after the verb in both active and passive.

Now consider the following sentences:

m) She sang beautifully.
n) *She beautifully sang.

o) It was sung beautifully.
p) It was beautifully sung.

In sentences m-p the adverb position is not equally distributed. The adverb must come after the verb in active but is not restricted in passive.

The rules for adverb position in English can vary in the active and passive voices. In certain cases the adverb can precede or follow the verb in both voices. Sometimes it is equally restricted and must be placed after the verb. Other times the adverb can precede or follow the verb in passive but must follow the verb in active. Though the adverb in English sentences can be equally distributed in active and passive, the adverb position tends to be more restricted in active than in passive.

Featured Post

Finding the Proto-Form

Related languages have a number of words which are similar to one another. In the branch of linguistics known as historical linguistics, the...