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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

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Source:  OpenStax, Principles of business statistics. OpenStax CNX. Aug 05, 2009 Download for free at http://cnx.org/content/col10874/1.5
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