# Normal distribution, sampling, function fitting & Regression

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## Introduction

In this chapter, you will use the mean, median, mode and standard deviation of a set of data to identify whether the data is normally distributed or whether it is skewed. You will learn more about populations and selecting different kinds of samples in order to avoid bias. You will work with lines of best fit, and learn how to find a regression equation and a correlation coefficient. You will analyse these measures in order to draw conclusions and make predictions.

## Investigation :

You are given a table of data below.

 75 67 70 71 71 73 74 75 80 75 77 78 78 78 78 79 91 81 82 82 83 86 86 87
1. Calculate the mean, median, mode and standard deviation of the data.
2. What percentage of the data is within one standard deviation of the mean?
3. Draw a histogram of the data using intervals $60\le x<64$ , $64\le x<68$ , etc.
4. Join the midpoints of the bars to form a frequency polygon.

If large numbers of data are collected from a population, the graph will often have a bell shape. If the data was, say, examination results, a few learners usually get very high marks, a few very low marks and most get a mark in the middle range. We say a distribution is normal if

• the mean, median and mode are equal.
• it is symmetric around the mean.
• $±68%$ of the sample lies within one standard deviation of the mean, $95%$ within two standard deviations and $99%$ within three standard deviations of the mean.

What happens if the test was very easy or very difficult? Then the distribution may not be symmetrical. If extremely high or extremely low scores are added to a distribution, then the mean tends to shift towards these scores and the curve becomes skewed.

If the test was very difficult, the mean score is shifted to the left. In this case, we say the distribution is positively skewed , or skewed right . If it was very easy, then many learners would get high scores, and the mean of the distribution would be shifted to the right. We say the distribution is negatively skewed , or skewed left .

## Normal distribution

1. Given the pairs of normal curves below, sketch the graphs on the same set of axes and show any relation between them. An important point to remember is that the area beneath the curve corresponds to 100%.
1. Mean = 8, standard deviation = 4 and Mean = 4, standard deviation = 8
2. Mean = 8, standard deviation = 4 and Mean = 16, standard deviation = 4
3. Mean = 8, standard deviation = 4 and Mean = 8, standard deviation = 8
2. After a class test, the following scores were recorded:
 Test Score Frequency 3 1 4 7 5 14 6 21 7 14 8 6 9 1 Total 64 Mean 6 Standard Deviation 1,2
1. Draw the histogram of the results.
2. Join the midpoints of each bar and draw a frequency polygon.
3. What mark must one obtain in order to be in the top 2% of the class?
4. Approximately 84% of the pupils passed the test. What was the pass mark?
5. Is the distribution normal or skewed?
3. In a road safety study, the speed of 175 cars was monitored along a specific stretch of highway in order to find out whether there existed any link between high speed and the large number of accidents along the route. A frequency table of the results is drawn up below.
 Speed (km.h ${}^{-1}$ ) Number of cars (Frequency) 50 19 60 28 70 23 80 56 90 20 100 16 110 8 120 5
The mean speed was determined to be around 82 km.h ${}^{-1}$ while the median speed was worked out to be around 84,5 km.h ${}^{-1}$ .
1. Draw a frequency polygon to visualise the data in the table above.
2. Is this distribution symmetrical or skewed left or right? Give a reason fro your answer.

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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