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The number line may help you understand standard deviation. If we were to put 5 and 7 on a number line, 7 is to the right of 5. We say, then, that 7 is one standard deviation to the right of 5 because
5 + (1)(2) = 7 .

If 1 were also part of the data set, then 1 is two standard deviations to the left of 5 because
5 + (-2)(2) = 1 .

A number line labeled from 0 to 7.

  • In general, a value = mean + (#ofSTDEV)(standard deviation)
  • where #ofSTDEVs = the number of standard deviations
  • 7 is one standard deviation more than the mean of 5 because: 7=5+ (1) (2)
  • 1 is two standard deviations less than the mean of 5 because: 1=5+ (−2) (2)

The equation value = mean + (#ofSTDEVs)(standard deviation) can be expressed for a sample and for a population:

  • sample: x = x + (#ofSTDEV)(s)
  • Population: x = μ + (#ofSTDEV)(σ)
The lower case letter s represents the sample standard deviation and the Greek letter σ (sigma, lower case) represents the population standard deviation.

The symbol x is the sample mean and the Greek symbol μ is the population mean.

Calculating the standard deviation

If x is a number, then the difference " x - mean" is called its deviation . In a data set, there are as many deviations as there are items in the data set. The deviations are used to calculate the standard deviation. If the numbers belong to a population, in symbols a deviation is x μ . For sample data, in symbols a deviation is x - x .

The procedure to calculate the standard deviation depends on whether the numbers are the entire population or are data from a sample. The calculations are similar, but not identical. Therefore the symbol used to represent the standard deviation depends on whether it is calculated from a population or a sample. The lower case letter s represents the sample standard deviation and the Greek letter σ (sigma, lower case) represents the population standard deviation. If the sample has the same characteristics as the population, then s should be a good estimate of σ .

To calculate the standard deviation, we need to calculate the variance first. The variance is an average of the squares of the deviations (the x - x values for a sample, or the x μ values for a population). The symbol σ 2 represents the population variance; the population standard deviation σ is the square root of the population variance. The symbol s 2 represents the sample variance; the sample standard deviation s is the square root of the sample variance. You can think of the standard deviation as a special average of the deviations.

If the numbers come from a census of the entire population and not a sample, when we calculate the average of the squared deviations to find the variance, we divide by N , the number of items in the population. If the data are from a sample rather than a population, when we calculate the average of the squared deviations, we divide by n-1 , one less than the number of items in the sample. You can see that in the formulas below.

Formulas for the sample standard deviation

  • s = size 12{s={}} {} Σ ( x x ¯ ) 2 n 1 or s = size 12{s={}} {} Σ f · ( x x ¯ ) 2 n 1
  • For the sample standard deviation, the denominator is n-1 , that is the sample size MINUS 1.

    Formulas for the population standard deviation

  • σ = size 12{σ={}} {} Σ ( x μ ¯ ) 2 N or σ = size 12{σ={}} {} Σ f · ( x μ ¯ ) 2 N
  • For the population standard deviation, the denominator is N , the number of items in the population.

Questions & Answers

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
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Collaborative statistics for mt230. OpenStax CNX. Aug 18, 2011 Download for free at http://legacy.cnx.org/content/col11345/1.2
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