# 7.8 Central limit theorem: summary of formulas

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Formula

## Central limit theorem for sample means

$\overline{X}$ ~ $N\left({\mu }_{X},\frac{{\sigma }_{X}}{\sqrt{n}}\right)\phantom{\rule{35pt}{0ex}}$ The Mean $\left(\overline{X}\right)$ : $\phantom{\rule{10pt}{0ex}}{\mu }_{X}$

Formula

## Central limit theorem for sample means z-score and standard error of the mean

$z=\frac{\overline{x}-{\mu }_{X}}{\left(\frac{{\sigma }_{X}}{\sqrt{n}}\right)}\phantom{\rule{25pt}{0ex}}$ Standard Error of the Mean (Standard Deviation $\left(\overline{X}\right)$ ): $\phantom{\rule{10pt}{0ex}}\frac{{\sigma }_{X}}{\sqrt{n}}$

Formula

## Central limit theorem for sums

$\mathrm{\Sigma X}$ ~ $N\left[\left(n\right)\cdot {\mu }_{X},\sqrt{n}\cdot {\sigma }_{X}\right]\phantom{\rule{10pt}{0ex}}$ Mean for Sums $\left(\mathrm{\Sigma X}\right)$ : $\phantom{\rule{10pt}{0ex}}n\cdot {\mu }_{X}$

Formula

## Central limit theorem for sums z-score and standard deviation for sums

$z=\frac{\mathrm{\Sigma x}-n\cdot {\mu }_{X}}{\sqrt{n}\cdot {\sigma }_{X}}\phantom{\rule{25pt}{0ex}}$ Standard Deviation for Sums $\left(\mathrm{\Sigma X}\right)$ : $\phantom{\rule{25pt}{0ex}}\sqrt{n}\cdot {\sigma }_{X}$

## Average

• A number that describes the central tendency of the data. There are a number of specialized averages, including the arithmetic mean, weighted mean, median, mode, and geometric mean.

## Central limit theorem

• Given a random variable (RV) with known mean μ and known standard deviation σ. We are sampling with size n and we are interested in two new RVs - the sample mean, $\overline{x}$ , and the sample sum, ΣX.If the size n of the sample is sufficiently large, then $\overline{X}$ ~ $N\left({\mu }_{X},\frac{{\sigma }_{X}}{\sqrt{n}}\right)$ and $\mathrm{\Sigma X}$ ~ $N\left(n\cdot {\mu }_{X},\sqrt{n}\cdot {\sigma }_{X}\right)$ . If the size n of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The mean of the sample means will equal the population mean and the mean of the sample sums will equal n times the population mean. The standard deviation of the distribution of the sample means,, is called the standard error of the mean

## Mean

• A number that measures the central tendency. A common name for mean is 'average.' The term 'mean' is a shortened form of 'arithmetic mean.' By definition, the mean for a sample (denoted by $\overline{x}$ ) is $\overline{x}$ (the sum of all values in the sample divided by the number of values in the sample), and the mean for a population (denoted byμ) is μ (the sum of all the values in the population divided by the number of values in the population).

## Standard error of the mean

• The standard deviation of the distribution of the sample means, $\frac{{\sigma }_{}}{\sqrt{n}}$

#### Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
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
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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 using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
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