# 2.6 Measures of the spread of the data  (Page 2/7)

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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
$\mathrm{5 + \left(1\right)\left(2\right) = 7}$ .

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

• 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: $x=\overline{x}+\mathrm{\left(#ofSTDEV\right)\left(s\right)}$
• Population: $x=\mu +\mathrm{\left(#ofSTDEV\right)\left(\sigma \right)}$
The lower case letter $s$ represents the sample standard deviation and the Greek letter $\sigma$ (sigma, lower case) represents the population standard deviation.

The symbol $\overline{x}$ is the sample mean and the Greek symbol $\mu$ 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 $x-\mu$ . For sample data, in symbols a deviation is $x-$ $\overline{x}$ .

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 $\sigma$ (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 $\sigma$ .

To calculate the standard deviation, we need to calculate the variance first. The variance is an average of the squares of the deviations (the $x-$ $\overline{x}$ values for a sample, or the $x-\mu$ values for a population). The symbol ${\sigma }^{2}$ represents the population variance; the population standard deviation $\sigma$ 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

• $s=$ $\sqrt{\frac{\Sigma {\left(x-\overline{x}\right)}^{2}}{n-1}}$ or $s=$ $\sqrt{\frac{\Sigma {f·\left(x-\overline{x}\right)}^{2}}{n-1}}$
• For the sample standard deviation, the denominator is n-1 , that is the sample size MINUS 1.

## Formulas for the population standard deviation

• $\sigma =$ $\sqrt{\frac{\Sigma {\left(x-\overline{\mu }\right)}^{2}}{N}}$ or $\sigma =$ $\sqrt{\frac{\Sigma {f·\left(x-\overline{\mu }\right)}^{2}}{N}}$
• For the population standard deviation, the denominator is N , the number of items in the population.

#### Questions & Answers

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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