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Data Frequency Relative Frequency Cumulative Relative Frequency
33 1 0.032 0.032
42 1 0.032 0.064
49 2 0.065 0.129
53 1 0.032 0.161
55 2 0.065 0.226
61 1 0.032 0.258
63 1 0.032 0.29
67 1 0.032 0.322
68 2 0.065 0.387
69 2 0.065 0.452
72 1 0.032 0.484
73 1 0.032 0.516
74 1 0.032 0.548
78 1 0.032 0.580
80 1 0.032 0.612
83 1 0.032 0.644
88 3 0.097 0.741
90 1 0.032 0.773
92 1 0.032 0.805
94 4 0.129 0.934
96 1 0.032 0.966
100 1 0.032 0.998 (Why isn't this value 1? ANSWER: Rounding)

Standard deviation of grouped frequency tables

Recall that for grouped data we do not know individual data values, so we cannot describe the typical value of the data with precision. In other words, we cannot find the exact mean, median, or mode. We can, however, determine the best estimate of the measures of center by finding the mean of the grouped data with the formula: M e a n   o f   F r e q u e n c y   T a b l e = f m f
where f = interval frequencies and m = interval midpoints.

Just as we could not find the exact mean, neither can we find the exact standard deviation. Remember that standard deviation describes numerically the expected deviation a data value has from the mean. In simple English, the standard deviation allows us to compare how “unusual” individual data is compared to the mean.

Find the standard deviation for the data in [link] .

Class Frequency, f Midpoint, m f * m f ( m - x ) 2
0–2 1 1 1 * 1 = 1 1 ( 1 - 7.58 ) 2 = 43.26
3–5 6 4 6 * 4 = 24 6 ( 4 - 7.58 ) 2 = 76.77
6-8 10 7 10 * 7 = 70 10 ( 7 - 7.58 ) 2 = 3.33
9-11 7 10 7 * 10 = 70 7 ( 10 - 7.58 ) 2 = 41.10
12-14 0 13 0 * 13 = 0 0 ( 13 - 7.58 ) 2 = 0
26=n x = 197 26 = 7.58 s 2 = 306.35 26 - 1 = 12.25

For this data set, we have the mean, x = 7.58 and the standard deviation, s x = 3.5. This means that a randomly selected data value would be expected to be 3.5 units from the mean. If we look at the first class, we see that the class midpoint is equal to one. This is almost two full standard deviations from the mean since 7.58 – 3.5 – 3.5 = 0.58. While the formula for calculating the standard deviation is not complicated, s x = Σ ( m x ) 2 f n 1 where
s x = sample standard deviation, x = sample mean, the calculations are tedious. It is usually best to use technology when performing the calculations.

Comparing values from different data sets

The standard deviation is useful when comparing data values that come from different data sets. If the data sets have different means and standard deviations, then comparing the data values directly can be misleading.

  • For each data value, calculate how many standard deviations away from its mean the value is.
  • Use the formula: value = mean + (#ofSTDEVs)(standard deviation); solve for #ofSTDEVs.
  • # o f S T D E V s = value – mean standard deviation
  • Compare the results of this calculation.

#ofSTDEVs is often called a " z -score"; we can use the symbol z . In symbols, the formulas become:

Sample x = x + zs z = x     x s
Population x = μ + z = x     μ σ

Two students, John and Ali, from different high schools, wanted to find out who had the highest GPA when compared to his school. Which student had the highest GPA when compared to his school?

Student GPA School Mean GPA School Standard Deviation
John 2.85 3.0 0.7
Ali 77 80 10

For each student, determine how many standard deviations (#ofSTDEVs) his GPA is away from the average, for his school. Pay careful attention to signs when comparing and interpreting the answer.

z = # of STDEVs = value  mean standard deviation = x - μ σ

For John, z = # o f S T D E V s = 2.85 3.0 0.7 = 0.21

For Ali, z = # o f S T D E V s = 77 80 10 = - 0.3

John has the better GPA when compared to his school because his GPA is 0.21 standard deviations below his school's mean while Ali's GPA is 0.3 standard deviations below his school's mean.

John's z -score of –0.21 is higher than Ali's z -score of –0.3. For GPA, higher values are better, so we conclude that John has the better GPA when compared to his school.

Questions & Answers

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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Business statistics -- bsta 200 -- humber college -- version 2016reva -- draft 2016-04-04. OpenStax CNX. Apr 05, 2016 Download for free at http://legacy.cnx.org/content/col11969/1.5
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