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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.

The t distribution

In probability and statistics, the t-distribution or Student's distribution arises in the problem of estimating the mean of a normally distributed population when the sample size is small, as well as when (as in nearly all practical statistical work) the population standard deviation is unknown and has to be estimated from the data.

    Textbook problems treating the standard deviation as if it were known are of two kinds:

  • those in which the sample size is so large that one may treat a data-based estimate of the variance as if it were certain,
  • those that illustrate mathematical reasoning, in which the problem of estimating the standard deviation is temporarily ignored because that is not the point that the author or instructor is then explaining.

The t distribution

t Distribution
If Z is a random variable that is N ( 0,1 ) , if U is a random variable that is χ 2 ( r ) , and if Z and U are independent, then
T = Z U / r = X ¯ μ S / n
has a t distribution with r degrees of freedom.

Where μ is the population mean, x ¯ is the sample mean and s is the estimator for population standard deviation (i.e., the sample variance) defined by

s 2 = 1 N 1 i = 1 N ( x i x ¯ ) 2 .

If σ = s , t = z , the distribution becomes the normal distribution. As N increases, Student’s t distribution approaches the normal distribution . It can be derived by transforming student’s z -distribution using z x ¯ μ s and then defining t = z n 1 .

The resulting probability and cumulative distribution functions are:

f ( t ) = Γ [ ( r + 1 ) / 2 ] π r Γ ( r / 2 ) ( 1 + t 2 / r ) ( r + 1 ) / 2 ,
F ( t ) = 1 2 + 1 2 [ I ( 1 ; 1 2 r , 1 2 ) I ( r r + t 2 , 1 2 r , 1 2 ) ] sgn ( t ) = 1 2 i t B ( t 2 r ; 1 2 , 1 2 ( 1 r ) ) Γ ( 1 2 ( r + 1 ) ) 2 π | t | Γ ( 1 2 r )


  • r = n 1 is the number of degrees of freedom,
  • < t < ,
  • Γ ( z ) is the gamma function,
  • B ( a , b ) is the bets function,
  • I ( z ; a , b ) is the regularized beta function defined by I ( z ; a , b ) = B ( z ; a , b ) B ( a , b ) .

The effect of degree of freedom on the t distribution is illustrated in the four t distributions on the Figure 1 .

p.d.f. of the t distribution for degrees of freedom r =3, r =6, r = .

In general, it is difficult to evaluate the distribution function of T . Some values are usually given in the tables. Also observe that the graph of the p.d.f. of T is symmetrical with respect to the vertical axis t =0 and is very similar to the graph of the p.d.f. of the standard normal distribution N ( 0,1 ) . However the tails of the t distribution are heavier that those of a normal one; that is, there is more extreme probability in the t distribution than in the standardized normal one. Because of the symmetry of the t distribution about t =0, the mean (if it exists) must be equal to zero. That is, it can be shown that E ( T ) = 0 when r 2 . When r =1 the t distribution is the Cauchy distribution , and thus both the variance and mean do not exist.

Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Introduction to statistics. OpenStax CNX. Oct 09, 2007 Download for free at http://cnx.org/content/col10343/1.3
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