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It is important to keep in mind that every statistic, not just the mean has a sampling distribution. For example, shows all possible outcomes for the range of two numbers (larger number minus the smaller number). shows the frequencies for each of the possible ranges and shows the sampling distribution of the range.

All possible outcomes when two balls are sampled.
Outcome Ball 1 Ball 2 Range
1 1 1 0
2 1 2 1
3 1 3 2
4 2 1 1
5 2 2 0
6 2 3 1
7 3 1 2
8 3 2 1
9 3 3 0
Frequencies of ranges for n = 2.
Range Frequency Relative Frequency
0 3 0.333
1 4 0.444
2 2 0.222
Distribution of ranges for N 2 .

It is also important to keep in mind that there is a sampling distribution for various sample sizes. For simplicity, we havebeen using N 2 . The sampling distribution of the range for N 3 is shown in .

Distribution of ranges for N 3 .

Continuous distributions

In the previous section, the population consisted of three pool balls. Now we will consider sampling distributions whenthe population distribution is continuous. What if we had a thousand pool balls with numbers ranging from 0.001 to 1.000in equal steps. (Although this distribution is not really continuous, it is close enough to be considered continuous forpractical purposes.) As before, we are interested in the distribution of means we would get if we sampled two balls andcomputed the mean of these two. In the previous example, we started by computing the mean for each of the nine possibleoutcomes. This would get a bit tedious for this problem since there are 1,000,000 possible outcomes (1,000 for the firstball x 1,000 for the second.) Therefore, it is more convenient to use our second conceptualization of sampling distributionswhich conceives of sampling distributions in terms of relative frequency distributions. Specifically, the relative frequencydistribution that would occur if samples of two balls were repeatedly taken and the mean of each sample computed.

When we have a truly continuous distribution, it is not only impractical but actually impossible to enumerate all possible outcomes. Moreover, in continuous distributions, theprobability of obtaining any single value is zero. Therefore, as discussed in our introduction to Distributions , these values are called probability densities rather than probabilities.

Sampling distributions and inferential statistics

As we stated in the beginning of this chapter, sampling distributions are important for inferential statistics. In theexamples given so far, a population was specified and the sampling distribution of the mean and the range weredetermined. In practice, the process proceeds the other way: you collect sample data and, from these data, you estimateparameters of the sampling distribution. This knowledge of the sampling distribution can be very useful. For example, knowingthe degree to which means from different samples would differ from each other and from the population mean would give you asense of how close your particular sample mean is likely to be to the population mean. Fortunately, this information isdirectly available from a sampling distribution: The most common measure of how much sample means differ from each otheris the standard deviation of the sampling distribution of the mean. This standard deviation is called the standard error of the mean . If all the sample means were very close to the population mean, then the standard error of themean would be small. On the other hand, if the sample means varied considerably, then the standard error of the mean wouldbe large.

To be specific, assume your sample mean were 125 and you estimated that the standard error of the mean were 5 (using a methodshown in a later section). If you had a normal distribution, then it would be likely that your sample mean would be within10 units of the population mean since most of a normal distribution is within two standard deviations of the mean.

Keep in mind that all statistics have sampling distributions, not just the mean. In later sections we will be discussing the sampling distribution of the variance , the sampling distribution of the difference between means , and the sampling distribution of Pearson's correlation , among others.

Questions & Answers

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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
How can I make nanorobot?
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
how can I make nanorobot?
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
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 (custom online version modified by t. short). OpenStax CNX. Jul 15, 2013 Download for free at http://cnx.org/content/col11476/1.5
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