The Phenomenon (Human Reaction to choices)

Since our group is curious to find out about the time taken for a person to choose. We decided to work in this area. During our research, we found out that Hick's Law proposed that the time taken for a human to choose something increases exponentially as the number of choices given to that person increases. Thus Logarithms is involved.

We decided to investigate in this area. And we found that the Hick's Law is the most suitable to calculate this phenomenon. To prove that we understand this topic on logarithms and to prove that Hick's Law is true, we did an experiment with several friends. (As this is all about human behaviour and human reaction)

We used the formula in Hick's Law, which is T=b x log2 (n+1).

To refresh your memories, T is the time taken to choose from a variety of choices. n is the number of choices given to the person. However, there is one more constant, which is the b. We concluded that b varies from person to person as every human is different. So, how did we execute this 'experiment'?

First we had to find the constant b. We formulated a question for our friend. (Anonymous to protect his identity -- Stated in the report) We gave decided to give him 4 choices of fruits to choose from. (Since we do not know the constant b, we had to work backwards).

Q: Choose with carefully consideration and do not be bias in this test. What would you choose? Orange, Apple, Pineapple or Honeydew?

A: (After 5 seconds) Apple.

And we worked backwards to find out his constant b.

T=b x log2 (4+1) --> 5s=b x log2 (5) --> 5/log2 (5) = b

And b is 2.15338279.

And we can continue with our experiment! Next, we formulated another question. But this time, we gave him 6 choices to choose from. Before we even asked him the question, we calculated and predicted the time he will take to choose from the 6 items.

T=b x log2 (6+1)

T=2.15338279 x log2 (7)

T=6.046559356

Round down to 6s.

Q: Choose among these drinks. Peach Tea, Green Tea, Ice Lemon Tea, H-Two-O, Bubble Tea or Apple Tea?

A: (After 7 seconds) Bubble Tea.

He took 7 seconds to answer this question. Although the answer and the predicted answer was not exactly the same, it was a difference of 1 second. Give and take 1~3 seconds for human reaction, he is in the range of our predicted answer and we have successfully completed our experiment!

However, we were not happy with just 2 questions. We formulated another questions. This time, we gave him 9 choices to choose from.

Again, the formula -->

T=b x log2 (9+1)

T=2.15338279 x log2 (10)

T=7.153382789

Predicted Time taken for him to choose from 9 choices is about 7s.

Q: Choose a brand of a car. Toyota, Honda, Ford, Fiat, Chevrolet, Volkswagon, Mitsubishi, Nissan, Proton.

And indeed, this time, he took 7 seconds to answer!

Isn't it amazing!?

pH Scale (Real-life Application) -- Continued

Here's another real-life application we did to confirm this phenomenon! Again, we experimented with a drink. But this time we experimented with orange juice.

So again, .

Hydromium Ions in orange juice is 3.2 x 10^-4

Thus the pH level of the Orange Juice is about 3.5. We have also crossed referenced our findings with other dietary websites and confirmed that the pH level of orange juice is indeed 3.5

http://www.indiacurry.com/chutney/foodphlevel.htm

http://answers.yahoo.com/question/index?qid=20110501122448AAmBXIS

http://www.onlinecancerinfo.com/docs/diet/pHbalance.htm

Real-Life Application of pH Scale

To prove that the pH Scale formula is correct and to relate what we have learnt about logarithms into reality, we have made this small little experiment.

For starters, we experimented on a drink that is very familiar to us -- Lemon Juice.

As we all know, Lemon is acidic. But how acidic is it? Let's find out!

By using the formula , we sub in some numbers:

Lemon has a Hydronium Ion number of 3.2 x 10^-3.

Thus

pH = -log(3.2 x 10^-3)

pH = 2.494

Round up to 2.49

Therefore the pH level of the Lemon Juice is 2.49. We cross-referenced it to several websites and confirmed that the acidity of Lemon juice is 2.49.

http://www.indiacurry.com/chutney/foodphlevel.htm

http://www.everydiet.org/diet/acid-alkaline-diets

http://answers.ask.com/Science/Chemistry/what_is_the_ph_level_of_lemon_juice

Hick's Law

Ah! This is interesting! Hick's Law describes the time it takes for a person to make a decision when he has choices to choose from.

Definition of Hick's Law: The time M(n) required to make a choice from a menu of n items rises with the log to the base two of n.

It's relatively simple to understand, right?

So from now on, you can use the Hick's Law to find out the amount of time it takes for your friend to make a decision!

It is logarithmic as people subdivide the total collection of choices into categories and eliminate half of the remaining choices at each step. They DO NOT consider each and every choice one-by-one. (Linear Time)

The important fact that needs to be noted is the Rule of Large Menus. Guess why restaurants have big and large menus rather than small ones? It's because of the Rule of Large Menus! (One large menu is more time-efficient than several small submenus supporting the same choices, even if we ignore the time overhead of moving among submenus.)

http://www.catb.org/~esr/writings/taouu/html/ch04s03.html

Another important rule is the Rule of Target Size. The size of a button should be proportional to its expected frequency of use. This explains why our shirts have small buttons! It's because we don't use the buttons often! We only think of it when we put the shirt on!

http://www.catb.org/~esr/writings/taouu/html/ch04s03.html

The Rule of the Infinite Edge is also another rule derived from the Hick's Law. The easiest target rectangles on the screen are those adjacent to its edges.

http://www.catb.org/~esr/writings/taouu/html/ch04s03.html

Hick's Law is widely used. It is used in combat fights, sports (reaction time) and many other things in life! Some of Hick's Law's favourite quotes are:

"Lag time increases significantly with the greater number of techniques"

"It takes 58% more time to pick between two choices"

"It takes about a second to pick a tactic"

"Selection time gets compounded exponentially when a person has to select from several choices"

http://www.hockscqc.com/articles/hickslaw.htm

And how is the Hick's Law logarithmic? It's quite similar to the Fitt's Law. The more choices one is given, the time taken for the person to choose is exponentially proportional to the number of choices he is given. Again, the time taken increases by the factor of 10.

http://upload.wikimedia.org/math/f/c/f/fcfb68e3f4f84b245eb0074178873575.png

Here's the Hick's Law formula!

T -- Time

b -- Constant that can be determined empirically by fitting a line to measure data.

+1 -- Because there is uncertainty about whether to respond or not, as well as about which response to make

n -- The number of choices given

http://en.wikipedia.org/wiki/Hick's_law

The Weber-Fechner Law

The Weber-Fechner Law talks about the relationship between the physical magnitudes of stimuli and perceived intensity of the stimuli. The Weber-Fechner Law was created by 2 people, thus the name "Weber-Fechner". Weber made a law that expresses relationship between quantity an intensity of something and how much more needs to be added for people to be able to feel that something has been added. On the other hand, Fechner provides an explanation for Weber's Law, thus the name "Weber-Fechner".

http://www.cs171.org/lectures/M-Landy-Weber.pdf

Simply speaking, Weber experimented on the weight needed for a person to feel the difference when additional weight is added. Weber found out that the smallest noticeable difference in weight is proportional to the starting vale of the weight. This means that if you are holding a weight of 500g, an additional 500g is needed for you to feel a noticeable difference. If you are holding a weight of 2kg, you will need 2kg more worth of weight for you to feel a difference.

The relationship between the stimulus and the perception is logarithmic, just like pH scale and the Richter Scale. So have you found something interested about logarithms and the application of it in reality? Logarithms is used to calculate numbers that are very big. As the thing you are finding increases by 1 unit, the effect is increased by 10 times and then the following increases by a factor of 10.

Anyway, coming back to the Weber-Fechner Law, logarithms is used to calculate human perception of things. The Weber-Fechner Law does not only still true for the touch senses of a human, it is also valid for other stimuli and other sensory perceptions as well, such as our hearing and our vision.

Now, time for the formula that proves the this amazing phenomenon.

http://upload.wikimedia.org/math/7/8/0/780dc889d3ca8d5048baa2acf4806d3a.png

It is derived by a series of calculations, including perception and the constant of integration.

Seeing formulas does not amaze us that much. However, when we apply it to reality, it will definitely WOW you! After we touch on the last law -- Hick's Law, we shall move on to the Real-life applications of all these formulas!!

http://en.wikipedia.org/wiki/Weber%E2%80%93Fechner_law

http://www.neuro.uu.se/fysiologi/gu/nbb/lectures/WebFech.html

Logarithms in Psychology (Fitt's Law)

Logarithms can be applied to Psychology too! In the introduction of logarithms, we introduced 3 laws in Logarithms in Psychology. They are

1) Fitt's Law

2) Hick's Law

3) Weber-Fechner Law

Basically, these are laws used to calculate human reaction time. Fitt's Law is model the act of pointing, either physically touching an object with a hand or finger, or virtually, by pointing to an object on a computer monitor using a pointing device.

http://en.wikipedia.org/wiki/Fitt%27s_law

The person who invented this Law, Fitt's, discovered a formal relationship that models speed or accuracy tradeoffs in rapid, aimed movements. According to Fitt's Law, the time to move and point to a target of width W at a distance A is a logarithmic function of the spatial relative error.

http://www.cs.umd.edu/class/fall2002/cmsc838s/tichi/fitts.html

There are a few variations to this magical Fitt's Law. (the traditional way and the improved method) However, they all use the concept of logarithms. The variations are the use of different symbols. However, if you closely compare the different formulas, they use the same theory and are the same.

Here's the formula that we find is the easiest to understand:

http://upload.wikimedia.org/math/e/7/e/e7e6cee6e7664d150f8db606c7f6fc02.png

Definitions:

T -- Average time taken to complete the movement

a -- Represents the start and stop time of the device (intercept)

b -- Represents the inherent speed of the device (slope)

*Note that a and b can be determined by fitting a straight line to the measured data

D -- The distance from the starting point to the center of the target

W -- The width of the target measured along the axis of motion.

http://en.wikipedia.org/wiki/Fitt%27s_law

However, traditionally, researchers used different symbols for this formula.

T can be replaced by MT, which is the movement time

1 can be replaced c, which is a constant of 0, 0.5 or 1. (We use 1 as c)

D can be replaced by A, which is the amplitude

And W corresponds to "accuracy"

http://www.cs.umd.edu/class/fall2002/cmsc838s/tichi/fitts.html

By using this formula, we can predict speed of the action of pointing to or tapping an object.

http://sixrevisions.com/usabilityaccessibility/improving-usability-with-fitts-law/

http://cdn.sixrevisions.com/0128-03_diagram.jpg

As you can see from this diagram, the object's distance and width does play a very important role in calculating the speed and accuracy of the action.

http://sixrevisions.com/usabilityaccessibility/improving-usability-with-fitts-law/

The pH Scale

Is Logarithms only usable in geography only? (Richter Scale)

The answer is definitely NO!

Logarithms is used everywhere around us. It is so common that we don't even know it exist in our lives! Another example of applying logarithms is in Chemistry. Remember those strips of paper you use in the Chemistry Laboratory to test for acidity and alkali? It makes use of the pH level right? Guess what? The pH Scale also makes use of Logarithms!

How the pH Scale works is almost the same as the Richter Scale. The pH Scale ranges from 0 to 14 when pH level of 7 is neutral. pH less than 7 is acidic and increases it's acidity as it goes down the scale. A pH greater than 7 is alkali and it increases alkali when it goes up the scale. As mentioned just now, the pH scale is logarithmic. How much more acidic is pH 4 to pH 3? How much more alkali is pH 11 to pH 12? It applies the same concept as the Richter Scale -- Magnitude. For every unit, the acidity/alkali increases/decreases by factors of 10. For example, pH 4 is 10 times more acidic than pH 5. pH 3 is 100 times more acidic than pH 5. This is very similar to the magnitude of the earthquake.

So what is the magic formula for pH Scale? Here it is!

http://www.okc.cc.ok.us/maustin/Eval_Logs/Image827.gif

is the hydronium ion concentration in moles per litre.

Here's an example: (We are going to calculate the pH of milk)

(pH Formula)

(Substitution)

pH = - (-6.4)

pH = 6.4

And thus the pH of milk is about 6.4.

Extracted from http://www.okc.cc.ok.us/maustin/Eval_Logs/Evaluating%20Logarithms.htm

We follow the hydroxide ions by this chart

[H3O+]	pH	[OH-]	Example
1 X 100	0	1 X 10-14	HCl (4%)
1 X 10-1	1	1 X 10-13	Stomach acid
1 X 10-2	2	1 X 10-12	Lemon juice
1 X 10-3	3	1 X 10-11	Vinegar
1 X 10-4	4	1 X 10-10	Soda
1 X 10-5	5	1 X 10-9	Rainwater (unpolluted)
1 X 10-6	6	1 X 10-8	Milk
1 X 10-7	7	1 X 10-7	Pure water
1 X 10-8	8	1 X 10-6	Egg whites
1 X 10-9	9	1 X 10-5	Baking Soda
1 X 10-10	10	1 X 10-4	Ammonia
1 X 10-11	11	1 X 10-3
1 X 10-12	12	1 X 10-2	Drano®
1 X 10-13	13	1 X 10-1	NaOH (4%)
1 X 10-14	14	1 X 100

http://web.jjay.cuny.edu/~acarpi/NSC/7-ph.htm

Isn't Logarithms interesting!?

It is easy to understand too! The same goes for decibels too!

http://www.elmhurst.edu/~chm/vchembook/184ph.html

The Magic Formulas (Richter Scale)

In the previous post, we introduced a formula for calculating the magnitude of earthquakes. The formula is

http://upload.wikimedia.org/math/1/3/e/13e5e0759d863fbb88d35d2ca1421b7b.png

WOW! You may be shocked and at a loss because this is one large chunk of logarithms and funny symbols! Do not worry. We will try to explain this formula so that it's easier for everyone to understand.

Before we start, let me introduce you to another formula very similar to the first one:

M = log₁₀ (A / A_zero)

A = A_zero 10^M

Don't understand? We will start with this formula first because it's simpler to understand and because you must be able to understand this formula before moving on to the original formula. (The original formula is derived from a few equations, including the ones we just introduced)

M = log₁₀ (A / A_zero)

Definitions:

M is the magnitude

A is the maximum excursion of the seismograph

A₀ is the empirical function depending on the epicenter distance

So as the for every 1 unit the magnitude increases, the amplitude increases by a factor of 10. The same 1 unit of increase in magnitude would give you an approximate of 32 times the earthquake's energy! Scary, isn't it?

So, by inserting the variables A and A_{0, you will get the magnitude of the earthquake and then you can find out the energy of the earthquake.}

_{Now you may ask, "How do we find the epicenter distance?"}

_{During an earthquake, seismic waves are spread spherically out from the hypo-center. You can find the epicenter distance by measuring the time difference on the seismograph and the distance on a travel-time graph at which the P-wave and S-wave have the same separation.}

_{A picture worth a thousand words. So let's let the animation tell you something about the P-wave and the S-wave.}

http://upload.wikimedia.org/wikipedia/commons/6/62/Onde_compression_impulsion_1d_30_petit.gif

http://upload.wikimedia.org/wikipedia/commons/6/6d/Onde_cisaillement_impulsion_1d_30_petit.gif

Left to right: P-wave (Primary Wave), S-wave (Secondary Wave)

And you on to the magic formula!

It is the same logic as the previous formulas we have just mentioned. While M is the magnitude and A is the amplitude. There's something new -- δ.δ the epicenter distance of the station, which is where the earthquake takes place. This formula is derived from the previous formulas.

Note: Magnitude greater than 4.6 are strong enough to be recorded b a seismograph. Any figures below 4.6 are not considered strong enough to be considered an earthquake.

And TADAAAA. You have your Richter Scale!

http://en.wikipedia.org/wiki/Epicentral_distance#Epicentral_distance

_{http://en.wikipedia.org/wiki/P-wave}

_{http://en.wikipedia.org/wiki/S-wave}

_{http://www.vibrationdata.com/earthquakes/magnitude.html}

The Richter Scale Chart

Before we start, here's something that you may be interested to know:

Magnitude	Description	Earthquake effects	Frequency of occurrence
Less than 2.0	Micro	Micro earthquakes, not felt.[12]	About 8,000 per day
2.0–2.9	Minor	Generally not felt, but recorded.	About 1,000 per day
3.0–3.9		Often felt, but rarely causes damage.	49,000 per year (est.)
4.0–4.9	Light	Noticeable shaking of indoor items, rattling noises. Significant damage unlikely.	6,200 per year (est.)
5.0–5.9	Moderate	Can cause major damage to poorly constructed buildings over small regions. At most slight damage to well-designed buildings.	800 per year
6.0–6.9	Strong	Can be destructive in areas up to about 160 kilometres (100 mi) across in populated areas.	120 per year
7.0–7.9	Major	Can cause serious damage over larger areas.	18 per year
8.0–8.9	Great	Can cause serious damage in areas several hundred kilometres across.	1 per year
9.0–9.9		Devastating in areas several thousand kilometres across.	1 per 20 years
10.0+	Massive	Never recorded, widespread devastation across very large areas; see below for equivalent seismic energy yield.	Extremely rare (Unknown)

(Based on U.S. Geological Survey documents

http://en.wikipedia.org/wiki/Richter_magnitude_scale

The Richter Scale

Now we are on to the use of logarithms! Logarithms isn't hard, right?

The Richter Scale is a logarithm scale that is used to calculate the magnitude of earthquakes.

The Richter Scale refers the measure of the amount of energy contained in an earthquake.

Cool Fact:

There is often a misconception that an earthquake that measure 5.0 on the Richter scale is just 1.0 larger than that of an earthquake of 4.0. This is WRONG.

An earthquake that measure 5.0 has an amplitude of 10 times larger than an earthquake that measures 4.0.

And 6.0 releases 100 times the energy of 4.0.

So, a magnitude of 9.0 is SCARY, isn't it?

And the secret formula is...

http://upload.wikimedia.org/math/1/3/e/13e5e0759d863fbb88d35d2ca1421b7b.png

We will go into more detail in the later postings...

Why is Logarithm needed?

Earthquake magnitudes and calculations are of a very large range. Thus Logarithm comes to the rescue! Logarithms are used to deal with numbers that span a very large range by using exponents of ten.

Here's the relation between Magnitude and Ground Amplitude before we end today's post.

Magnitude	Ground amplitude (mm) at a distance of 100 km
0	0.00000048
1	0.0000048
2	0.000048
3	0.00048
4	0.0048
5	0.048
6	.48
7	4.8
8	48
9	480

http://en.wikipedia.org/wiki/Richter_magnitude_scale

http://www.annegloag.com/AlgebraII/BuildingConnections/Earthquakes.pdf

http://zhidao.baidu.com/question/236687313.html

http://tremor.nmt.edu/faq/how.html

The Laws of Logarithms

Logarithms also have its own set of laws! The 5 main laws are:

1) Log_aa = 1

2) Log_a1 = 0

3) Log_aX + Log_aY = Log_aXY

4) Log_aX - Log_aY = Log_aX/Y

5) Log_aB^x = X Log_aB

http://www.themathpage.com/aprecalc/logarithms.htm#laws

And there you have it! The 5 logarithm laws!

Recap of Logarithms

Let's recap what we learnt about logarithms before we start rattling on the use of Logarithms.

Logarithms is almost parallel to Indices. So what is the relationship between these two? Let's take a look at this animation.

http://www.purplemath.com/modules/logs/relatani.gif

The animation is pretty much self-explanatory. Hope this post made you remember all you know about logarithms!

Introduction to Logarithms

Logarithms are inverses of exponential. Before actually researching into this topic--logarithms, my group and I used to think that logarithms was redundant in our daily lives. However, Logarithms is not as redundant is we thought it was! It is so commonly used in our daily lives! What is the use of Logarithms? The use of Logarithm scales is to reduce wide-ranging quantities to smaller scopes.

Some examples would be:

1) Richter Scale (Used to calculate the magnitude of earthquakes)

2) pH Scale (Used to calculate the acidity level or alkali level of a solution)

3) Decibels (Unit used to quantify sound pressure and voltage ratio)

Some Mathematical Laws make use of Logarithms too. For example, in Psychology:

1) The Hick's Law

2) The Fitt's Law

3) The Weber-Fechner Law

Generally, these 3 laws uses the idea of logarithms to calculate Human Reaction.

Hope you are starting to get interested in Logarithms after reading this!

Unfortunately, we will only be touching on the Ritcher Scale, the laws in Psychology and a little bit on pH scale...

http://en.wikipedia.org/wiki/Logarithms