# 1.2 The mean, variance, and standard deviation

 Page 1 / 1
This course is a short series of lectures on Introductory Statistics. Topics covered are listed in the Table of Contents. The notes were prepared by EwaPaszek and Marek Kimmel. The development of this course has been supported by NSF 0203396 grant.

## Mean and variance

Certain mathematical expectations are so important that they have special names. In this section we consider two of them: the mean and the variance.

Mean Value

If X is a random variable with p.d.f. $f\left(x\right)$ of the discrete type and space R = $\left({b}_{1},{b}_{2},{b}_{3},...\right)$ , then $E\left(X\right)=\sum _{R}xf\left(x\right)={b}_{1}f\left({b}_{1}\right)+{b}_{2}f\left({b}_{2}\right)+{b}_{3}f\left({b}_{3}\right)+...$ is the weighted average of the numbers belonging to R , where the weights are given by the p.d.f. $f\left(x\right)$ .

We call $E\left(X\right)$ the mean of X (or the mean of the distribution ) and denote it by $\mu$ . That is, $\mu =E\left(X\right)$ .

In mechanics, the weighted average of the points ${b}_{1},{b}_{2},{b}_{3},...$ in one-dimensional space is called the centroid of the system. Those without the mechanics background can think of the centroid as being the point of balance for the system in which the weights $f\left({b}_{1}\right),f\left({b}_{2}\right),f\left({b}_{3}\right),...$ are places upon the points ${b}_{1},{b}_{2},{b}_{3},...$ .

Let X have the p.d.f.

$f\left(x\right)=\left\{\begin{array}{l}\frac{1}{8},x=0,3,\\ \frac{3}{8},x=1,2.\end{array}$

The mean of X is

$\mu =E\left[X=0\left(\frac{1}{8}\right)+1\left(\frac{3}{8}\right)+2\left(\frac{3}{8}\right)+3\left(\frac{1}{8}\right)=\frac{3}{2}.$

The example below shows that if the outcomes of X are equally likely (i.e., each of the outcomes has the same probability), then the mean of X is the arithmetic average of these outcomes.

Roll a fair die and let X denote the outcome. Thus X has the p.d.f. $f\left(x\right)=\frac{1}{6},x=1,2,3,4,5,6.$ Then,

$E\left(X\right)=\sum _{x=1}^{6}x\left(\frac{1}{6}\right)=\frac{1+2+3+4+5+6}{6}=\frac{7}{2},$

which is the arithmetic average of the first six positive integers.

Variance

It was denoted that the mean $\mu =E\left(X\right)$ is the centroid of a system of weights of measure of the central location of the probability distribution of X . A measure of the dispersion or spread of a distribution is defined as follows:

If $u\left(x\right)={\left(x-\mu \right)}^{2}$ and $E\left[{\left(X-\mu \right)}^{2}\right]$ exists, the variance , frequently denoted by ${\sigma }^{2}$ or $Var\left(X\right)$ , of a random variable X of the discrete type (or variance of the distribution) is defined by

${\sigma }^{2}=E\left[{\left(X-\mu \right)}^{2}\right]=\sum _{R}{\left(x-\mu \right)}^{2}f\left(x\right).$

The positive square root of the variance is called the standard deviation of X and is denoted by

$\sigma =\sqrt{Var\left(X\right)}=\sqrt{E\left[{\left(X-\mu \right)}^{2}\right]}.$

Let the p.d.f. of X by defined by $f\left(x\right)=\frac{x}{6},x=1,2,3.$

The mean of X is

$\mu =E\left(X\right)=1\left(\frac{1}{6}\right)+2\left(\frac{2}{6}\right)+3\left(\frac{3}{6}\right)=\frac{7}{3}.$

To find the variance and standard deviation of X we first find

$E\left({X}^{2}\right)={1}^{2}\left(\frac{1}{6}\right)+{2}^{2}\left(\frac{2}{6}\right)+{3}^{2}\left(\frac{3}{6}\right)=\frac{36}{6}=6.$

Thus the variance of X is ${\sigma }^{2}=E\left({X}^{2}\right)-{\mu }^{2}=6-{\left(\frac{7}{3}\right)}^{2}=\frac{5}{9},$

and the standard deviation of X is $\sigma =\sqrt{5}{9}}=0.745.$

Let X be a random variable with mean ${\mu }_{x}$ and variance ${\sigma }_{x}^{2}$ . Of course, $Y=aX+b$ , where a and b are constants, is a random variable, too. The mean of Y is

Moreover, the variance of Y is

${\sigma }_{Y}^{2}=E\left[{\left(Y-{\mu }_{Y}\right)}^{2}\right]=E\left[{\left(aX+b-a{\mu }_{X}-b\right)}^{2}\right]=E\left[{a}^{2}{\left(X-{\mu }_{X}\right)}^{2}\right]={a}^{2}{\sigma }_{X}^{2}.$

Moments of the distribution

Let r be a positive integer. If $E\left({X}^{r}\right)=\sum _{R}{x}^{r}f\left(x\right)$ exists, it is called the r th moment of the distribution about the origin. The expression moment has its origin in the study of mechanics.

In addition, the expectation $E\left[{\left(X-b\right)}^{r}\right]=\sum _{R}{x}^{r}f\left(x\right)$ is called the r th moment of the distribution about b . For a given positive integer r.

$E\left[{\left(X\right)}_{r}\right]=E\left[X\left(X-1\right)\left(X-2\right)\cdot \cdot \cdot \left(X-r+1\right)\right]$ is called the r th factorial moment .

The second factorial moment is equal to the difference of the second and first moments: $E\left[X\left(X-1\right)\right]=E\left({X}^{2}\right)-E\left(X\right).$

There is another formula that can be used for computing the variance that uses the second factorial moment and sometimes simplifies the calculations.

First find the values of $E\left(X\right)$ and $E\left[X\left(X-1\right)\right]$ . Then ${\sigma }^{2}=E\left[X\left(X-1\right)\right]+E\left(X\right)-{\left[E\left(X\right)\right]}^{2},$ since using the distributive property of E , this becomes ${\sigma }^{2}=E\left({X}^{2}\right)-E\left(X\right)+E\left(X\right)-{\left[E\left(X\right)\right]}^{2}=E\left({X}^{2}\right)-{\mu }^{2}.$

Let continue with example 4 , it can be find that

$E\left[X\left(X-1\right)\right]=1\left(0\right)\left(\frac{1}{6}\right)+2\left(1\right)\left(\frac{2}{6}\right)+3\left(2\right)\left(\frac{3}{6}\right)=\frac{22}{6}.$

Thus ${\sigma }^{2}=E\left[X\left(X-1\right)\right]+E\left(X\right)-{\left[E\left(X\right)\right]}^{2}=\frac{22}{6}+\frac{7}{3}-{\left(\frac{7}{3}\right)}^{2}=\frac{5}{9}.$

Recall the empirical distribution is defined by placing the weight (probability) of 1/ n on each of n observations ${x}_{1},{x}_{2},...,{x}_{n}$ . Then the mean of this empirical distribution is $\sum _{i=1}^{n}{x}_{i}\frac{1}{n}=\frac{\sum _{i=1}^{n}{x}_{i}}{n}=\overline{x}.$

The symbol $\overline{x}$ represents the mean of the empirical distribution . It is seen that $\overline{x}$ is usually close in value to $\mu =E\left(X\right)$ ; thus, when $\mu$ is unknown, $\overline{x}$ will be used to estimate $\mu$ .

Similarly, the variance of the empirical distribution can be computed. Let v denote this variance so that it is equal to

$v={\sum _{i=1}^{n}\left({x}_{i}-\overline{x}\right)}^{2}\frac{1}{n}=\sum _{i=1}^{n}{x}_{i}^{2}\frac{1}{n}-{\overline{x}}^{2}=\frac{1}{n}\sum _{i=1}^{n}{x}_{i}^{2}-{\overline{x}}^{2}.$

This last statement is true because, in general, ${\sigma }^{2}=E\left({X}^{2}\right)-{\mu }^{2}.$

There is a relationship between the sample variance ${s}^{2}$ and variance v of the empirical distribution, namely ${s}^{2}=ns/\left(n-1\right)$ . Of course, with large n , the difference between ${s}^{2}$ and v is very small. Usually, we use ${s}^{2}$ to estimate ${\sigma }^{2}$ when ${\sigma }^{2}$ is unknown.
BERNOULLI TRIALS and BINOMIAL DISTRIBUTION

#### 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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Introduction to statistics. OpenStax CNX. Oct 09, 2007 Download for free at http://cnx.org/content/col10343/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Introduction to statistics' conversation and receive update notifications? By Qqq Qqq  By   By Anindyo Mukhopadhyay    By