<< Chapter < Page Chapter >> Page >

Consider the following data set.
4; 5; 6; 6; 6; 7; 7; 7; 7; 7; 7; 8; 8; 8; 9; 10

This data set can be represented by following histogram. Each interval has width one, and each value is located in the middle of an interval.

This histogram matches the supplied data. It consists of 7 adjacent bars with the x-axis split into intervals of 1 from 4 to 10. The heighs of the bars peak in the middle and taper symmetrically to the right and left.

The histogram displays a symmetrical distribution of data. A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other. The mean, the median, and the mode are each seven for these data. In a perfectly symmetrical distribution, the mean and the median are the same. This example has one mode (unimodal), and the mode is the same as the mean and median. In a symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median.

The histogram for the data:

  • 4
  • 5
  • 6
  • 6
  • 6
  • 7
  • 7
  • 7
  • 7
  • 8
is not symmetrical. The right-hand side seems "chopped off" compared to the left side. A distribution of this type is called skewed to the left because it is pulled out to the left.

This histogram matches the supplied data. It consists of 5 adjacent bars with the x-axis split into intervals of 1 from 4 to 8. The peak is to the right, and the heights of the bars taper down to the left.

The mean is 6.3, the median is 6.5, and the mode is seven. Notice that the mean is less than the median, and they are both less than the mode. The mean and the median both reflect the skewing, but the mean reflects it more so.

The histogram for the data:

  • 6
  • 7
  • 7
  • 7
  • 7
  • 8
  • 8
  • 8
  • 9
  • 10
, is also not symmetrical. It is skewed to the right .

This histogram matches the supplied data. It consists of 5 adjacent bars with the x-axis split into intervals of 1 from 6 to 10. The peak is to the left, and the heights of the bars taper down to the right.

The mean is 7.7, the median is 7.5, and the mode is seven. Of the three statistics, the mean is the largest, while the mode is the smallest . Again, the mean reflects the skewing the most.

To summarize, generally if the distribution of data is skewed to the left, the mean is less than the median, which is often less than the mode. If the distribution of data is skewed to the right, the mode is often less than the median, which is less than the mean.

Skewness and symmetry become important when we discuss probability distributions in later chapters.

Statistics are used to compare and sometimes identify authors. The following lists shows a simple random sample that compares the letter counts for three authors.

Terry: 7; 9; 3; 3; 3; 4; 1; 3; 2; 2

Davis: 3; 3; 3; 4; 1; 4; 3; 2; 3; 1

Maris: 2; 3; 4; 4; 4; 6; 6; 6; 8; 3

  1. Make a dot plot for the three authors and compare the shapes.
  2. Calculate the mean for each.
  3. Calculate the median for each.
  4. Describe any pattern you notice between the shape and the measures of center.
  1. This dot plot matches the supplied data for Terry. The plot uses a number line from 1 to 10. It shows one  x over 1, two x's over 2, four x's over 3, one  x over 4, one x over 7, and one x over 9. There are no x's over the numbers 5, 6, 8, and 10.
    Terry’s distribution has a right (positive) skew.
    This dot plot matches the supplied data for Davi. The plot uses a number line from 1 to 10. It shows two  x's over 1, one x over 2, five x's over 3, and two x's over 4. There are no x's over the numbers 5, 6, 7, 8, 9, and 10.
    Davis’ distribution has a left (negative) skew
    This dot plot matches the supplied data for Mari. The plot uses a number line from 1 to 10. It shows one x over 2, two x's over 3, three x's over 4, three x's over 6, and one  x over 8. There are no x's over the numbers 1, 5, 7, 9, and 10.
    Maris’ distribution is symmetrically shaped.
  2. Terry’s mean is 3.7, Davis’ mean is 2.7, Maris’ mean is 4.6.
  3. Terry’s median is three, Davis’ median is three. Maris’ median is four.
  4. It appears that the median is always closest to the high point (the mode), while the mean tends to be farther out on the tail. In a symmetrical distribution, the mean and the median are both centrally located close to the high point of the distribution.

Try it

Discuss the mean, median, and mode for each of the following problems. Is there a pattern between the shape and measure of the center?

a.

This dot plot matches the supplied data. The plot uses a number line from 0 to 14. It shows two  x's over 0, four x's over 1, three x's over 2, one x over 3, two x's over the number 4, 5, 6, and 9, and 1 x each over 10 and 14. There are no x's over the numbers 7, 8, 11, 12, and 13.

b.

The Ages Former U.S Presidents Died
4 6 9
5 3 6 7 7 7 8
6 0 0 3 3 4 4 5 6 7 7 7 8
7 0 1 1 2 3 4 7 8 8 9
8 0 1 3 5 8
9 0 0 3 3
Key: 8|0 means 80.

c.

This is a histogram titled Hours Spent Playing Video Games on Weekends. The x-axis shows the number  of hours spent playing video games with bars showing values at intervals of 5. The y-axis shows the number of students. The first bar for 0 - 4.99 hours has a height of 2. The second bar from 5 - 9.99 has a height of 3. The third bar from 10 - 14.99 has a height of 4. The fourth bar from 15 - 19.99 has a height of 7. The fifth bar from 20 - 24.99 has a height of 9.

  1. mean = 4.25, median = 3.5, mode = 1; The mean>median>mode which indicates skewness to the right. (data are 0, 1, 2, 3, 4, 5, 6, 9, 10, 14 and respective frequencies are 2, 4, 3, 1, 2, 2, 2, 2, 1, 1)
  2. mean = 70.1 , median = 68, mode = 57, 67 bimodal; the mean and median are close but there is a little skewness to the right which is influenced by the data being bimodal. (data are 46, 49, 53, 56, 57, 57, 57, 58, 60, 60, 63, 63, 64, 64, 65, 66, 67, 67, 67, 68, 70, 71, 71, 72, 73, 74, 77, 78, 78, 79, 80, 81, 83, 85, 88, 90, 90 93, 93).
  3. These are estimates: mean =16.095, median = 17.495, mode = 22.495 (there may be no mode); The mean<median<mode which indicates skewness to the left. (data are the midponts of the intervals: 2.495, 7.495, 12.495, 17.495, 22.495 and respective frequencies are 2, 3, 4, 7, 9).

Chapter review

Looking at the distribution of data can reveal a lot about the relationship between the mean, the median, and the mode. There are three types of distributions. A right (or positive) skewed distribution has a shape like [link] . A left (or negative) skewed distribution has a shape like [link] . A symmetrical distrubtion looks like [link] .

Practice

Use the following information to answer the next three exercises: State whether the data are symmetrical, skewed to the left, or skewed to the right.

  • 1
  • 1
  • 1
  • 2
  • 2
  • 2
  • 2
  • 3
  • 3
  • 3
  • 3
  • 3
  • 3
  • 3
  • 3
  • 4
  • 4
  • 4
  • 5
  • 5

The data are symmetrical. The median is 3 and the mean is 2.85. They are close, and the mode lies close to the middle of the data, so the data are symmetrical.

  • 16
  • 17
  • 19
  • 22
  • 22
  • 22
  • 22
  • 22
  • 23

  • 87
  • 87
  • 87
  • 87
  • 87
  • 88
  • 89
  • 89
  • 90
  • 91

The data are skewed right. The median is 87.5 and the mean is 88.2. Even though they are close, the mode lies to the left of the middle of the data, and there are many more instances of 87 than any other number, so the data are skewed right.

When the data are skewed left, what is the typical relationship between the mean and median?

When the data are symmetrical, what is the typical relationship between the mean and median?

When the data are symmetrical, the mean and median are close or the same.

What word describes a distribution that has two modes?

Describe the shape of this distribution.

This is a historgram which consists of 5 adjacent bars with the x-axis split into intervals of 1 from 3 to 7. The bar heights peak at the first bar and taper lower to the right.

The distribution is skewed right because it looks pulled out to the right.

Describe the relationship between the mode and the median of this distribution.

This is a histogram which consists of 5 adjacent bars with the x-axis split into intervals of 1 from 3 to 7. The bar heights peak at the first bar and taper lower to the right. The bar ehighs from left to right are: 8, 4, 2, 2, 1.

Describe the relationship between the mean and the median of this distribution.

This is a histogram which  consists of 5 adjacent bars with the x-axis split into intervals of 1 from 3 to 7. The bar heights peak at the first bar and taper lower to the right. The bar heights from left to right are: 8, 4, 2, 2, 1.

The mean is 4.1 and is slightly greater than the median, which is four.

Describe the shape of this distribution.

This is a histogram which consists of 5 adjacent bars with the x-axis split into intervals of 1 from 3 to 7. The bar heights peak in the middle and taper down to the right and left.

Describe the relationship between the mode and the median of this distribution.

This is a histogram which consists of 5 adjacent bars with the x-axis split intervals of 1 from 3 to 7. The bar heights peak in the middle and taper down to the right and left.

The mode and the median are the same. In this case, they are both five.

Are the mean and the median the exact same in this distribution? Why or why not?

This is a histogram which consists of 5 adjacent bars with the x-axis split into intervals of 1 from 3 to 7. The bar heights from left to right are: 2, 4, 8, 5, 2.

Describe the shape of this distribution.

This is a histogram which consists of 5 adjacent bars over an x-axis split into intervals of 1 from 3 to 7. The bar heights from left to right are: 1, 1, 2, 4, 7.

The distribution is skewed left because it looks pulled out to the left.

Describe the relationship between the mode and the median of this distribution.

This is a histogram which consists of 5 adjacent bars over an x-axis split into intervals of 1 from 3 to 7. The bar heights from left to right are: 1, 1, 2, 4, 7.

Describe the relationship between the mean and the median of this distribution.

This is a histogram which consists of 5 adjacent bars over an x-axis split into intervals of 1 from 3 to 7. The bar heights from left to right are: 1, 1, 2, 4, 7.

The mean and the median are both six.

The mean and median for the data are the same.

  • 3
  • 4
  • 5
  • 5
  • 6
  • 6
  • 6
  • 6
  • 7
  • 7
  • 7
  • 7
  • 7
  • 7
  • 7

Is the data perfectly symmetrical? Why or why not?

Which is the greatest, the mean, the mode, or the median of the data set?

  • 11
  • 11
  • 12
  • 12
  • 12
  • 12
  • 13
  • 15
  • 17
  • 22
  • 22
  • 22

The mode is 12, the median is 13.5, and the mean is 15.1. The mean is the largest.

Which is the least, the mean, the mode, and the median of the data set?

  • 56
  • 56
  • 56
  • 58
  • 59
  • 60
  • 62
  • 64
  • 64
  • 65
  • 67

Of the three measures, which tends to reflect skewing the most, the mean, the mode, or the median? Why?

The mean tends to reflect skewing the most because it is affected the most by outliers.

In a perfectly symmetrical distribution, when would the mode be different from the mean and median?

Questions & Answers

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 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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Statistics i - math1020 - red river college - version 2015 revision a - draft 2015-10-24. OpenStax CNX. Oct 24, 2015 Download for free at http://legacy.cnx.org/content/col11891/1.8
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Statistics i - math1020 - red river college - version 2015 revision a - draft 2015-10-24' conversation and receive update notifications?

Ask