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This module introduces the history of the normal curve.

In the chapter on probability, we saw that the binomial distribution could be used to solve problems such as "If afair coin is flipped 100 times, what is the probability of getting 60 or more heads?" The probability of exactly x heads out of N flips is computed using the formula: P x N x N x x 1 N x where x is the number of heads (60), N is the number of flips (100), and is the probability of a head (0.5). Therefore, to solve this problem, you computethe probability of 60 heads, then the probability of 61 heads, 62 heads, etc, and add up all these probabilities. Imagine how longit must have taken to compute binomial probabilities before the advent of calculators and computers.

Abraham de Moivre, an 18th century statistician and consultant to gamblers was often called upon to make these lengthycomputations. de Moivre noted that when the number of events (coin flips) increased, the shape of the binomial distributionapproached a very smooth curve. Binomial distributions for 2, 4, and 12 flips are shown in .

Examples of binomial distributions. The heights of the blue bars represent the probabilities.
de Moivre reasoned that if he could find a mathematical expression for this curve, he would be able to solve problemssuch as finding the probability of 60 or more heads out of 100 coin flips much more easily. This is exactly what he did, andthe curve he discovered is now called the normal curve .

The normal approximation to the binomial distribution for 12 coin flips. The smooth curve is the normal distribution. Notehow well it approximates the binomial probabilities represented by the heights of the blue lines.

The importance of the normal curve stems primarily from the fact that the distribution of many natural phenomena are at leastapproximately normally distributed. One of the first applications of the normal distribution was to the analysis oferrors of measurement made in astronomical observations, errors that occurred because of imperfect instruments and imperfectobservers. Galileo in the 17th century noted that these errors were symmetric and that small errors occurred more frequentlythan large errors. This led to several hypothesized distributions of errors, but it was not until the early 19thcentury that it was discovered that these errors followed a normal distribution. Independently the mathematicians Adrian in1808 and Gauss in 1809 developed the formula for the normal distribution and showed that errors were fit well by thisdistribution.

This same distribution had been discovered by Laplace in 1778 when he derived the extremely important central limit theorem , the topic of a later section of this chapter. Laplace showed that even if a distribution is not normally distributed, the means of repeatedsamples from the distribution would be very nearly normal, and that the the larger the sample size, the closer the distributionwould be to a normal distribution. Most statistical procedures for testing differences between means assume normaldistributions. Because the distribution of means is very close to normal, these tests work well even if the distribution itselfis only roughly normal.

Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Professors help document. OpenStax CNX. Aug 27, 2010 Download for free at http://cnx.org/content/col11223/1.1
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