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    For any data set, no matter what the distribution of the data is:

  • At least 75% of the data is within two standard deviations of the mean.
  • At least 89% of the data is within three standard deviations of the mean.
  • At least 95% of the data is within 4.5 standard deviations of the mean.
  • This is known as Chebyshev's Rule.

    For data having a distribution that is bell-shaped and symmetric:

  • Approximately 68% of the data is within one standard deviation of the mean.
  • Approximately 95% of the data is within two standard deviations of the mean.
  • More than 99% of the data is within three standard deviations of the mean.
  • This is known as the Empirical Rule.
  • It is important to note that this rule only applies when the shape of the distribution of the data is bell-shaped and symmetric. We will learn more about this when studying the "Normal" or "Gaussian" probability distribution in later chapters.

References

Data from Microsoft Bookshelf.

King, Bill.“Graphically Speaking.” Institutional Research, Lake Tahoe Community College. Available online at http://www.ltcc.edu/web/about/institutional-research (accessed April 3, 2013).

Chapter review

The standard deviation can help you calculate the spread of data. There are different equations to use if are calculating the standard deviation of a sample or of a population.

  • The Standard Deviation allows us to compare individual data or classes to the data set mean numerically.
  • s = ( x x ¯ ) 2 n 1 or s = f ( x x ¯ ) 2 n 1 is the formula for calculating the standard deviation of a sample. To calculate the standard deviation of a population, we would use the population mean, μ , and the formula σ = ( x μ ) 2 N or σ = f ( x μ ) 2 N .

Formula review

s x = f m 2 n x ¯ 2 where s x =  sample standard deviation x ¯  = sample mean

Practice

Use the following information to answer the next two exercises : The following data are the distances between 20 retail stores and a large distribution center. The distances are in miles.
29; 37; 38; 40; 58; 67; 68; 69; 76; 86; 87; 95; 96; 96; 99; 106; 112; 127; 145; 150

Use a graphing calculator or computer to find the standard deviation and round to the nearest tenth.

s = 34.5

Find the value that is one standard deviation below the mean.

Two baseball players, Fredo and Karl, on different teams wanted to find out who had the higher batting average when compared to his team. Which baseball player had the higher batting average when compared to his team?

Baseball Player Batting Average Team Batting Average Team Standard Deviation
Fredo 0.158 0.166 0.012
Karl 0.177 0.189 0.015

For Fredo: z = 0.158  –  0.166 0.012 = –0.67

For Karl: z = 0.177  –  0.189 0.015 = –0.8

Fredo’s z -score of –0.67 is higher than Karl’s z -score of –0.8. For batting average, higher values are better, so Fredo has a better batting average compared to his team.

Use [link] to find the value that is three standard deviations:

  • above the mean
  • below the mean


Find the standard deviation for the following frequency tables using the formula. Check the calculations with the TI 83/84 .

Find the standard deviation for the following frequency tables using the formula. Check the calculations with the TI 83/84.

  1. Grade Frequency
    49.5–59.5 2
    59.5–69.5 3
    69.5–79.5 8
    79.5–89.5 12
    89.5–99.5 5
  2. Daily Low Temperature Frequency
    49.5–59.5 53
    59.5–69.5 32
    69.5–79.5 15
    79.5–89.5 1
    89.5–99.5 0
  3. Points per Game Frequency
    49.5–59.5 14
    59.5–69.5 32
    69.5–79.5 15
    79.5–89.5 23
    89.5–99.5 2
  1. s x = f m 2 n x ¯ 2 = 193157.45 30 79.5 2 = 10.88
  2. s x = f m 2 n x ¯ 2 = 380945.3 101 60.94 2 = 7.62
  3. s x = f m 2 n x ¯ 2 = 440051.5 86 70.66 2 = 11.14

Questions & Answers

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
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
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
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Statistics i - math1020 - red river college - version 2015 revision a - draft 2015-10-24. OpenStax CNX. Oct 24, 2015 Download for free at http://legacy.cnx.org/content/col11891/1.8
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