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Distribution of data

Symmetric and skewed data

The shape of a data set is important to know.

Shape of a data set

This describes how the data is distributed relative to the mean and median.

  • Symmetrical data sets are balanced on either side of the median.
  • Skewed data is spread out on one side more than on the other. It can be skewed right or skewed left.

Relationship of the mean, median, and mode

The relationship of the mean, median, and mode to each other can provide some information about the relative shape of the data distribution. If the mean, median, and mode are approximately equal to each other, the distribution can be assumed to be approximately symmetrical. With both the mean and median known, the following can be concluded:

  • (mean - median) 0 then the data is symmetrical
  • (mean - median) > 0 then the data is positively skewed (skewed to the right). This means that the median is close to the start of the data set.
  • (mean - median) < 0 then the data is negatively skewed (skewed to the left). This means that the median is close to the end of the data set.

Distribution of data

  1. Three sets of 12 pupils each had test score recorded. The test was out of 50. Use the given data to answer the following questions.
    Cumulative Frequencies for Data Set 2.
    Set 1 Set 2 Set 3
    25 32 43
    47 34 47
    15 35 16
    17 32 43
    16 25 38
    26 16 44
    24 38 42
    27 47 50
    22 43 50
    24 29 44
    12 18 43
    31 25 42
    1. For each of the sets calculate the mean and the five number summary.
    2. For each of the classes find the difference between the mean and the median. Make box and whisker plots on the same set of axes.
    3. State which of the three are skewed (either right or left).
    4. Is set A skewed or symmetrical?
    5. Is set C symmetrical? Why or why not?
  2. Two data sets have the same range and interquartile range, but one is skewed right and the other is skewed left. Sketch the box and whisker plots and then invent data (6 points in each set) that meets the requirements.

Scatter plots

A scatter-plot is a graph that shows the relationship between two variables. We say this is bivariate data and we plot the data from two different sets using ordered pairs. For example, we could have mass on the horizontal axis (first variable) and height on the second axis (second variable), or we could have current on the horizontal axis and voltage on the vertical axis.

Ohm's Law is an important relationship in physics. Ohm's law describes the relationship between current and voltage in a conductor, like a piece of wire. When we measure the voltage (dependent variable) that results from a certain current (independent variable) in a wire, we get the data points as shown in [link] .

Values of current and voltage measured in a wire.
Current Voltage Current Voltage
0 0,4 2,4 1,4
0,2 0,3 2,6 1,6
0,4 0,6 2,8 1,9
0,6 0,6 3 1,9
0,8 0,4 3,2 2
1 1 3,4 1,9
1,2 0,9 3,6 2,1
1,4 0,7 3,8 2,1
1,6 1 4 2,4
1,8 1,1 4,2 2,4
2 1,3 4,4 2,5
2,2 1,1 4,6 2,5

When we plot this data as points, we get the scatter plot shown in [link] .

If we are to come up with a function that best describes the data, we would have to say that a straight line best describes this data.

Ohm's law

Ohm's Law describes the relationship between current and voltage in a conductor. The gradient of the graph of voltage vs. current is known as the resistance of the conductor.

Research project : scatter plot

The function that best describes a set of data can take any form. We will restrict ourselves to the forms already studied, that is, linear, quadratic or exponential. Plot the following sets of data as scatter plots and deduce the type of function that best describes the data. The type of function can either be quadratic or exponential.

  1. x y x y x y x y
    -5 9,8 0 14,2 -2,5 11,9 2,5 49,3
    -4,5 4,4 0,5 22,5 -2 6,9 3 68,9
    -4 7,6 1 21,5 -1,5 8,2 3,5 88,4
    -3,5 7,9 1,5 27,5 -1 7,8 4 117,2
    -3 7,5 2 41,9 -0,5 14,4 4,5 151,4
  2. x y x y x y x y
    -5 75 0 5 -2,5 27,5 2,5 7,5
    -4,5 63,5 0,5 3,5 -2 21 3 11
    -4 53 1 3 -1,5 15,5 3,5 15,5
    -3,5 43,5 1,5 3,5 -1 11 4 21
    -3 35 2 5 -0,5 7,5 4,5 27,5
  3. Height (cm) 147 150 152 155 157 160 163 165
    168 170 173 175 178 180 183
    Weight (kg) 52 53 54 56 57 59 60 61
    63 64 66 68 70 72 74
outlier

A point on a scatter plot which is widely separated from the other points or a result differing greatly from others in the same sample is called an outlier.

The following simulation allows you to plot scatter plots and fit a curve to the plot. Ignore the error bars (blue lines) on the points.

Phet simulation for scatter plots

Scatter plots

  1. A class's results for a test were recorded along with the amount of time spent studying for it. The results are given below.
    Score (percent) Time spent studying (minutes)
    67 100
    55 85
    70 150
    90 180
    45 70
    75 160
    50 80
    60 90
    84 110
    30 60
    66 96
    96 200
    1. Draw a diagram labelling horizontal and vertical axes.
    2. State with reasons, the cause or independent variable and the effect or dependent variable.
    3. Plot the data pairs
    4. What do you observe about the plot?
    5. Is there any pattern emerging?
  2. The rankings of eight tennis players is given along with the time they spend practising.
    Practice time (min) Ranking
    154 5
    390 1
    130 6
    70 8
    240 3
    280 2
    175 4
    103 7
    1. Construct a scatter plot and explain how you chose the dependent (cause) and independent (effect) variables.
    2. What pattern or trend do you observe?
  3. Eight childrens sweet consumption and sleep habits were recorded. The data is given in the following table.
    Number of sweets (per week) Average sleeping time (per day)
    15 4
    12 4,5
    5 8
    3 8,5
    18 3
    23 2
    11 5
    4 8
    1. What is the dependent (cause) variable?
    2. What is the independent (effect) variable?
    3. Construct a scatter plot of the data.
    4. What trend do you observe?

Questions & Answers

what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
Other chapter Q/A we can ask
Moahammedashifali Reply

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Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
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