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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

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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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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