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If X is a normally distributed random variable and X ~ N(μ, σ) , then the z-score is:

z = x - μ σ

The z-score tells you how many standard deviations that the value x is above (to the right of) or below (to the left of) the mean, μ . Values of x that are larger than the mean have positive z-scores and values of x that are smaller than the mean have negative z-scores. If x equals the mean, then x has a z-score of 0 .

Suppose X ~ N(5, 6) . This says that X is a normally distributed random variable with mean μ = 5 and standard deviation σ = 6 . Suppose x = 17 . Then:

z = x - μ σ = 17 - 5 6 = 2

This means that x = 17 is 2 standard deviations (2σ) above or to the right of the mean μ = 5 . The standard deviation is σ = 6 .

Notice that:

5 + 2 6 = 17 (The pattern is μ + z σ = x . )

Now suppose x=1 . Then:

z = x - μ σ = 1 - 5 6 = - 0.67 (rounded to two decimal places)

This means that x = 1 is 0.67 standard deviations (- 0.67σ) below or to the left of the mean μ = 5 . Notice that:

5 + ( -0.67 ) ( 6 ) is approximately equal to 1 (This has the pattern μ + ( -0.67 ) σ = 1 )

Summarizing, when z is positive, x is above or to the right of μ and when z is negative, x is to the left of or below μ .

Some doctors believe that a person can lose 5 pounds, on the average, in a month by reducing his/her fat intake and by exercising consistently. Suppose weight loss has anormal distribution. Let X = the amount of weight lost (in pounds) by a person in a month. Use a standard deviation of 2 pounds. X ~ N(5, 2) . Fill in the blanks.

Suppose a person lost 10 pounds in a month. The z-score when x = 10 pounds is z = 2.5 (verify). This z-score tells you that x = 10 is ________ standard deviations to the ________ (right or left) of the mean _____ (What is the mean?).

This z-score tells you that x = 10 is 2.5 standard deviations to the right of the mean 5 .

Suppose a person gained 3 pounds (a negative weight loss). Then z = __________. This z-score tells you that x = -3 is ________ standard deviations to the __________ (right or left) of the mean.

z = -4 . This z-score tells you that x = -3 is 4 standard deviations to the left of the mean.

Suppose the random variables X and Y have the following normal distributions: X ~ N(5, 6) and Y ~ N(2, 1) . If x = 17 , then z  =  2 . (This was previously shown.) If y = 4 , what is z ?

z = y - μ σ = 4 - 2 1 = 2 where μ=2 and σ=1.

The z-score for y = 4 is z = 2 . This means that 4 is z = 2 standard deviations to the right of the mean. Therefore, x = 17 and y = 4 are both 2 (of their ) standard deviations to the right of their respective means.

The z-score allows us to compare data that are scaled differently. To understand the concept, suppose X ~ N(5, 6) represents weight gains for one group of people who are trying to gain weight in a 6 week period and Y ~ N(2, 1) measures the same weight gain for a second group of people. A negative weight gain would be a weight loss.Since x = 17 and y = 4 are each 2 standard deviations to the right of their means, they represent the same weight gain relative to their means .

The empirical rule

If X is a random variable and has a normal distribution with mean µ and standard deviation σ then the Empirical Rule says (See the figure below)
  • About 68.27% of the x values lie between -1 σ and +1 σ of the mean µ (within 1 standard deviation of the mean).
  • About 95.45% of the x values lie between -2 σ and +2 σ of the mean µ (within 2 standard deviations of the mean).
  • About 99.73% of the x values lie between -3 σ and +3 σ of the mean µ (within 3 standard deviations of the mean). Notice that almost all the x values lie within 3 standard deviations of the mean.
  • The z-scores for +1 σ and –1 σ are +1 and -1, respectively.
  • The z-scores for +2 σ and –2 σ are +2 and -2, respectively.
  • The z-scores for +3 σ and –3 σ are +3 and -3 respectively.
Empirical Rule
The Empirical Rule is also known as the 68-95-99.7 Rule.

Suppose X has a normal distribution with mean 50 and standard deviation 6.

  • About 68.27% of the x values lie between -1 σ = (-1)(6) = -6 and 1 σ = (1)(6) = 6 of the mean 50. The values 50 - 6 = 44 and 50 + 6 = 56 are within 1 standard deviation of the mean 50. The z-scores are -1 and +1 for 44 and 56, respectively.
  • About 95.45% of the x values lie between -2 σ = (-2)(6) = -12 and 2 σ = (2)(6) = 12 of the mean 50. The values 50 - 12 = 38 and 50 + 12 = 62 are within 2 standard deviations of the mean 50. The z-scores are -2 and 2 for 38 and 62, respectively.
  • About 99.73% of the x values lie between -3 σ = (-3)(6) = -18 and 3 σ = (3)(6) = 18 of the mean 50. The values 50 - 18 = 32 and 50 + 18 = 68 are within 3 standard deviations of the mean 50. The z-scores are -3 and +3 for 32 and 68, respectively.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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it is a goid question and i want to know the answer as well
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characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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
Do you know which machine is used to that process?
s.
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
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what is biological synthesis of nanoparticles
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Source:  OpenStax, Elementary statistics. OpenStax CNX. Dec 30, 2013 Download for free at http://cnx.org/content/col10966/1.4
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