# 1.3 Levels of measurement

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The data can be put in order from lowest to highest: 20, 68, 80, 92.

The differences between the data have meaning. The score 92 is more than the score 68 by 24 points. Ratios can be calculated. The smallest score is 0. So 80 is four times 20. The score of 80 is four times better than the score of 20.

## Frequency

Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows:

• 5
• 6
• 3
• 3
• 2
• 4
• 7
• 5
• 2
• 3
• 5
• 6
• 5
• 4
• 4
• 3
• 5
• 2
• 5
• 3
.

[link] lists the different data values in ascending order and their frequencies.

Frequency table of student work hours
DATA VALUE FREQUENCY
2 3
3 5
4 3
5 6
6 2
7 1

A frequency is the number of times a value of the data occurs. According to [link] , there are three students who work two hours, five students who work three hours, and so on. The sum of the values in the frequency column, 20, represents the total number of students included in the sample.

A relative frequency is the ratio (fraction or proportion) of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes. To find the relative frequencies, divide each frequency by the total number of students in the sample–in this case, 20. Relative frequencies can be written as fractions, percents, or decimals.

Frequency table of student work hours with relative frequencies
DATA VALUE FREQUENCY RELATIVE FREQUENCY
2 3 $\frac{3}{20}$ or 0.15
3 5 $\frac{5}{20}$ or 0.25
4 3 $\frac{3}{20}$ or 0.15
5 6 $\frac{6}{20}$ or 0.30
6 2 $\frac{2}{20}$ or 0.10
7 1 $\frac{1}{20}$ or 0.05

The sum of the values in the relative frequency column of [link] is $\frac{20}{20}$ , or 1.

Cumulative relative frequency is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies tothe relative frequency for the current row, as shown in [link] .

Frequency table of student work hours with relative and cumulative relative frequencies
DATA VALUE FREQUENCY RELATIVE
FREQUENCY
CUMULATIVE RELATIVE
FREQUENCY
2 3 $\frac{3}{20}$ or 0.15 0.15
3 5 $\frac{5}{20}$ or 0.25 0.15 + 0.25 = 0.40
4 3 $\frac{3}{20}$ or 0.15 0.40 + 0.15 = 0.55
5 6 $\frac{6}{20}$ or 0.30 0.55 + 0.30 = 0.85
6 2 $\frac{2}{20}$ or 0.10 0.85 + 0.10 = 0.95
7 1 $\frac{1}{20}$ or 0.05 0.95 + 0.05 = 1.00

The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated.

## Note

Because of rounding, the relative frequency column may not always sum to one, and the last entry in the cumulative relative frequency column may not be one. However, they each should be close to one.

[link] represents the heights, in inches, of a sample of 100 male semiprofessional soccer players.

Frequency table of soccer player height
HEIGHTS
(INCHES)
FREQUENCY RELATIVE
FREQUENCY
CUMULATIVE
RELATIVE
FREQUENCY
59.95–61.95 5 $\frac{5}{100}$ = 0.05 0.05
61.95–63.95 3 $\frac{3}{100}$ = 0.03 0.05 + 0.03 = 0.08
63.95–65.95 15 $\frac{15}{100}$ = 0.15 0.08 + 0.15 = 0.23
65.95–67.95 40 $\frac{40}{100}$ = 0.40 0.23 + 0.40 = 0.63
67.95–69.95 17 $\frac{17}{100}$ = 0.17 0.63 + 0.17 = 0.80
69.95–71.95 12 $\frac{12}{100}$ = 0.12 0.80 + 0.12 = 0.92
71.95–73.95 7 $\frac{7}{100}$ = 0.07 0.92 + 0.07 = 0.99
73.95–75.95 1 $\frac{1}{100}$ = 0.01 0.99 + 0.01 = 1.00
Total = 100 Total = 1.00

The data in this table have been grouped into the following intervals:

• 59.95 to 61.95 inches
• 61.95 to 63.95 inches
• 63.95 to 65.95 inches
• 65.95 to 67.95 inches
• 67.95 to 69.95 inches
• 69.95 to 71.95 inches
• 71.95 to 73.95 inches
• 73.95 to 75.95 inches

#### Questions & Answers

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
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Damian Reply
absolutely yes
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how to know photocatalytic properties of tio2 nanoparticles...what to do now
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characteristics of micro business
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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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Introductory statistics. OpenStax CNX. Aug 09, 2016 Download for free at http://legacy.cnx.org/content/col11776/1.26
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