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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.

A normal distribution

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 x < 64 , 64 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.

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
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
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
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 did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
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