Devry MATH225 Week 2 Assignment Frequency Tables and Histograms Latest 2019 JULY Question # 00603503 Subject: Mathematics Due on: 08/08/2019 Posted On: 08/08/2019 08:30 AM Tutorials: 1 Rating: 4.5/5

MATH225 Statistical Reasoning for the Health Sciences

Week 2 Assignment Frequency Tables and Histograms

Question The
histogram below represents the prices of digital SLR camera models at a store.
Describe the shape of the distribution.

A histogram has a horizontal axis labeled
Camera Prices in dollars from 0 to 2000 in increments of 500 and a vertical
axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains
vertical bars of width 250 starting at the horizontal axis value 0. The heights
of the bars are as follows, where the left horizontal axis label is listed
first and the frequency is listed second: 0, 5; 250, 7; 500, 4; 750, 3; 1000,
3; 1250, 2; 1500, 1; 1750, 1.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question Describe
the shape of the given histogram.

A histogram has a horizontal axis from 0 to 16
in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in
increments of 2. The histogram contains vertical bars of width 1 starting at
the horizontal axis value 0. The heights of the bars are as follows, where the
horizontal axis label is listed first and the frequency is listed second: 0, 0;
1, 0; 2, 0; 3, 0; 4, 1; 5, 2; 6, 2; 7, 4; 8, 6; 9, 7; 10, 6; 11, 5; 12, 3; 13,
2; 14, 1; 15, 0.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question A
restaurant is open for both lunch and dinner. One day, the owner kept track of
the number of occupied tables in the dining area and created a histogram
showing the results for each half hour of the day. What shape does the
distribution have?

A histogram has a horizontal axis labeled Time
of Day from 9 to 21 in increments of 3 and a vertical axis labeled Frequency
from 0 to 20 in increments of 5. The histogram contains vertical bars of width
0.5 starting at the horizontal axis value 9. The heights of the bars are as
follows, where the left horizontal axis label is listed first and the frequency
is listed second: 9, 1; 9.5, 1; 10, 2; 10, 5; 11, 9; 11.5, 10; 12, 11; 12.5,
10; 13, 7; 13.5, 3; 14, 3; 14.5, 4; 15, 4; 15.5, 6; 16, 4; 16.5, 6; 17, 7;
17.5, 10; 18, 12; 18.5, 15; 19, 12; 19.5, 10; 20, 7; 20.5, 6. All vertical
coordinates are approximate.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question A
bookstore took an inventory of the prices of its books and created a histogram
to show the results. What shape does the distribution have?

A histogram has a horizontal axis labeled Book
Prices in dollars from 0 to 200 in increments of 50 and a vertical axis labeled
Frequency from 0 to 30 in increments of 10. The histogram contains vertical
bars of width 10 starting at the horizontal axis value 0. The heights of the
bars are as follows, where the left horizontal axis label is listed first and
the frequency is listed second: 0, 5; 10, 20; 20, 25; 30, 21; 40, 15; 50, 13;
60, 14; 70, 16; 80, 16; 90, 21; 100, 24; 110, 26; 120, 22; 130, 21; 140, 20;
150, 15; 160, 9; 170, 7; 180, 5; 190, 3. All vertical coordinates are
approximate.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question Describe
the shape of the given histogram.

A histogram has a horizontal axis from 0 to 16
in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in
increments of 2. The histogram contains vertical bars of width 1 starting at
the horizontal axis value 0. The heights of the bars are as follows, where the
horizontal axis label is listed first and the frequency is listed second: 0, 0;
1, 0; 2, 0; 3, 0; 4, 1; 5, 1; 6, 2; 7, 2; 8, 3; 9, 4; 10, 6; 11, 7; 12, 8; 13,
8; 14, 5; 15, 0.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question Describe
the shape of the given histogram.

A histogram has a horizontal axis from 0 to 16
in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in
increments of 2. The histogram contains vertical bars of width 1 starting at
the horizontal axis value 0. The heights of the bars are as follows, where the
horizontal axis label is listed first and the frequency is listed second: 0, 0;
1, 1; 2, 1; 3, 2; 4, 3; 5, 5; 6, 6; 7, 6; 8, 5; 9, 4; 10, 2; 11, 1; 12, 1; 13,
0; 14, 0; 15, 0.

uniform

unimodal and symmetric

unimodal and left skewed

unimodal and right skewed

bimodal

Identify
and Labels Shapes of Histograms

Histogram
Shapes

A histogram
is a graph that helps show the distribution of values in a set of data. An
example of a histogram is shown below. The horizontal axis is labeled with data
values. It is divided into several sections that all have the same width. Then
a bar is drawn above each section, and the height of the bar is related to how
many of the data values are within the corresponding range on the horizontal
axis. The vertical axis (and the height of the bars) can be either counts of
data values (called a frequency) or the fraction of the data set in the range
(called the relative frequency).

A histogram has a horizontal axis labeled from
0 to 10 in increments of 1 and a vertical axis labeled Frequency from 0 to 10
in increments of 2. The histogram contains vertical bars of width 1 starting at
the horizontal axis value 0. The heights of the bars are as follows, where the
left horizontal axis label is listed first and the frequency is listed second:
0, 1; 1, 2; 2, 4; 3, 6; 4, 7; 5, 7; 6, 6; 7, 4; 8, 2; 9, 1.

The general
shape of a histogram can be described as uniform, unimodal, bimodal, or
multimodal. An example of each of these is given below. A uniform histogram has
bars that are all close to the same height. A unimodal histogram has a single
peak, and a multimodal histogram has more one peak. Sometimes the case with two
peaks is also called bimodal.

A histogram has a horizontal axis from 0 to 10
in increments of 1 and a vertical axis labeled Frequency from 0 to 10 in
increments of 2. The histogram contains vertical bars of width 1 starting at
the horizontal axis value 0. The heights of the bars are as follows, where the
left horizontal axis label is listed first and the frequency is listed second:
0, 6; 1, 6; 2, 6; 3, 6; 4, 6; 5, 6; 6, 6; 7, 6; 8, 6; 9, 6.

Uniform

A histogram
has a horizontal axis labeled from 0 to 10 in increments of 1 and a vertical
axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains
vertical bars of width 1 starting at the horizontal axis value 0. The heights
of the bars are as follows, where the left horizontal axis label is listed
first and the frequency is listed second: 0, 1; 1, 2; 2, 4; 3, 6; 4, 7; 5, 7;
6, 6; 7, 4; 8, 2; 9, 1.

Unimodal

A histogram has a horizontal axis from 0 to 16
in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in
increments of 2. The histogram contains vertical bars of width 1 starting at
the horizontal axis value 0. The heights of the bars are as follows, where the
left horizontal axis label is listed first and the frequency is listed second:
0, 1; 1, 2; 2, 4; 3, 6; 4, 7; 5, 7; 6, 6; 7, 4; 8, 4; 9, 6; 10, 7; 11, 7; 12,
6; 13, 4; 14, 2; 15, 1.

Bimodal

A histogram
has a horizontal axis from 0 to 20 in increments of 4 and a vertical axis
labeled Frequency from 0 to 10 in increments of 2. The histogram contains
vertical bars of width 1 starting at the horizontal axis value 0. The heights
of the bars are as follows, where the left horizontal axis label is listed
first and the frequency is listed second: 0, 1; 1, 2; 2, 4; 3, 5; 4, 4; 5, 3;
6, 4; 7, 6; 8, 7; 9, 8; 10, 7; 11, 6; 12, 5; 13, 4; 14, 4; 15, 5; 16, 6; 17, 4;
18, 2; 19, 1.

Multimodal

A histogram can also be described by it
symmetry or skewness. A symmetric histogram has two halves that are
approximately mirror images of each other. The uniform, unimodal, and bimodal
histograms shown above are all symmetric. A histogram is skewed left when the
tail of bars extending towards smaller data values is longer than the tail
extending towards larger data values. A histogram is skewed right when the tail
of bars extending towards larger data values is longer than the tail extending
towards smaller data values. Examples of skewed histograms are shown below.

A histogram has a horizontal axis from 0 to 16
in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in
increments of 2. The histogram contains vertical bars of width 1 starting at
the horizontal axis value 0. The heights of the bars are as follows, where the
left horizontal axis label is listed first and the frequency is listed second:
0, 0; 1, 1; 2, 1; 3, 1; 4, 2; 5, 2; 6, 3; 7, 3; 8, 4; 9, 5; 10, 6; 11, 7; 12,
7; 13, 5; 14, 3; 15, 1.

Skewed Left

A histogram
has a horizontal axis from 0 to 16 in increments of 2 and a vertical axis
labeled Frequency from 0 to 10 in increments of 2. The histogram contains
vertical bars of width 1 starting at the horizontal axis value 0. The heights
of the bars are as follows, where the left horizontal axis label is listed
first and the frequency is listed second: 0, 1; 1, 3; 2, 5; 3, 7; 4, 7; 5, 6;
6, 5; 7, 4; 8, 3; 9, 3; 10, 2; 11, 2; 12, 1; 13, 1; 14, 1; 15, 0.

Skewed
Right

Real world
distributions tend to be right skewed when there is a lower bound for the
values, most values are clustered in a range, but it is not impossible for very
large values to occur. A classic example of this is income distribution is some
countries. The lower bound is zero, most of the population have incomes within
a few standard deviations of the mean, but there are people with much larger
incomes.

Similarly,
left skewed distributions tend to occur for quantities that have a natural
upper bound when most of the population tend to be near that bound. For
example, student scores on a easy test will tend to all be fairly high and
create a peak near 100%. However, it is possible for a few students to create a
tail that extends down to much lower scores.

Uniform
distributions are less common. One simple example is rolling a fair dice. Every
number from 1to 6 has the same probability of appearing. Bimodal distributions
occur when there is a reason for two different peaks. For example, the
distribution of the people attending Disney Land at different timings could be
bimodal. It could have a peak at 11 am and another peak at 2 pm.

The
unimodal symmetric distribution, also called bell shaped is a very common
distribution. For example, the distribution of the mean wages of many random
samples of 30 people will be a unimodal symmetric distribution.

Example

Question Describe
the shape of the histogram shown below.

A histogram has a horizontal axis from 0 to 16
in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in
increments of 2. The histogram contains vertical bars of width 1 starting at
the horizontal axis value 0. The heights of the bars are as follows, where the
left horizontal axis label is listed first and the frequency is listed second:
0, 2; 1, 7; 2, 8; 3, 7; 4, 6; 5, 4; 6, 3; 7, 3; 8, 2; 9, 2; 10, 1; 11, 1; 12,
1; 13, 0; 14, 0; 15, 0.

Question A
professor created a histogram showing the birth month of the students in one of
her classes. What is the shape of the histogram?

A histogram has a horizontal axis labeled
Month of Birth from 1 to 12 with the following tick marks from left to right:
1, 3, 6, 9, and 12. It has a vertical axis labeled Frequency from 0 to 10 in
increments of 2. Vertical bars of width 1 start at the horizontal axis value 1.
The heights of the bars are as follows, where the left horizontal axis label is
listed first and the frequency is listed second: 1, 7; 2, 6; 3, 7; 4, 6; 5, 6;
6, 5; 7, 6; 8, 6; 9, 7; 10, 6; 11, 6.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question A
student in a probability class rolled a six-sided die 1000 times. A histogram
of the results is shown below. What is the shape of the distribution?

A histogram has a horizontal axis labeled Die
Roll from 1 to 6 in increments of 1 and a vertical axis labeled Frequency from
0 to 200 in increments of 50. Vertical bars of width 1 are centered over a
horizontal axis label. The heights of the bars are as follows, where the
horizontal axis label is listed first and the approximate height is listed
second: 1, 170; 2, 150; 3, 155; 4, 150; 5, 170; 6, 165.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question Given
the following histogram for a set of data, how many values in the data set are
at least 5.5 and less than 8.5?

A histogram has a horizontal axis labeled
Values from 3.5 to 10.5 in increments of 1 and a vertical axis labeled
Frequency from 0 to 7 in increments of 1. The histogram has vertical bars of
width 1, starting at the horizontal axis value of 3.5. The approximate heights
of the bars are as follows, where the horizontal axis label is listed first and
the approximate height is listed second: 3.5, 5; 4.5, 6; 5.5, 7; 6.5, 5; 7.5,
5; 8.5, 5; 9.5, 6.

Question The
students in a statistics class record how many movies they have watched in the
previous month. The data are listed below.

1, 1, 1, 2,
2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6

Which of
the histograms below correctly represents this data?

A histogram
has a horizontal axis labeled Movies Watched from 0.5 to 6.5 in increments of 1
and a vertical axis labeled Frequency from 0 to 8 in increments of 1. The
histogram contains vertical bars of width 1, with one vertical bar centered
over each of the horizontal axis tick marks. The heights of the vertical bars
are as follows, where the value is listed first and the height is listed
second: 0.5, 2; 1.5, 8; 2.5, 3; 3.5, 2; 4.5, 1; 5.5, 1.

A histogram has a horizontal axis labeled
Movies Watched from 0.5 to 6.5 in increments of 1 and a vertical axis labeled
Frequency from 0 to 7 in increments of 1. The histogram contains vertical bars
of width 1, with one vertical bar centered over each of the horizontal axis
value tick marks. The heights of the vertical bars are as follows, where the
value is listed first and the height is listed second: 0.5, 3; 1.5, 7; 2.5, 2;
3.5, 3; 4.5, 1; 5.5, 1.

A histogram has a horizontal axis labeled
Movies Watched from 0.5 to 6.5 in increments of 1 and a vertical axis labeled
Frequency from 0 to 7 in increments of 1. The histogram contains vertical bars
of width 1, with one vertical bar centered over each of the horizontal axis
value tick marks. The heights of the vertical bars are as follows, where the
value is listed first and the height is listed second: 0.5, 3; 1.5, 7; 2.5, 3;
3.5, 2; 4.5, 1; 5.5, 1.

A bar graph has a horizontal axis titled
Values labeled from 0.5 to 6.5 in increments of 1 and a vertical axis titled
Frequency labeled from 1 to 6 in increments of 1. Six bars are plotted each
with a width of 1. From left to right, the heights of the bars are as follows:
2, 6, 3, 2, 1, 1.

A bar graph has a horizontal axis titled
Movies Watched labeled from 0.5 to 6.5 in increments of 1 and a vertical axis
titled Frequency labeled from 1 to 8 in increments of 1. Six bars are plotted
each with a width of 1. From left to right, the heights of the bars are as
follows: 3, 8, 3, 1, 2, 1.

Histograms

Constructing
Histograms

To
construct a histogram,

1. Decide how many bars or intervals
you need to clearly represent the data. (On average, most histograms consist of
5 to 15 bars.)

2. Choose
a starting point for the first interval. This value should be less than the
smallest data value. It is helpful to choose a starting point that is also
carried out to one more decimal placethan the data value with the most decimal
places. For example, if the value with the most decimal places is 6.1, and this
is the smallest value, a good starting point is 6.05 (6.1 – 0.05 = 6.05).

3. Choose
an ending point for the last interval. This value should be greater than the highest
data value. Like the starting point, It is helpful to choose an ending point
that is also carried out to one more decimal place than the data value with the
most decimal places.

*Note: When
these points and other boundaries are carried to one additional decimal place,
no data value will fall on a boundary.

4. Calculate
the width of the each bar or intervals. All intervals will be the same size. To
calculate this width, subtract the starting point from the ending value and
divide by the number of bars (the number of bars you chose).

5. Determine
the boundaries by adding the width to the starting point. Then add the width to
that value, and continue as such. Label the boundary values on the horizontal
axis.

6. Draw
bars in each interval with the height corresponding to the frequency of data
values that lie within each interval.

Example

Question Use the following data to construct a
histogram.

The
following data are the number of books bought by 50 part-time college students
at ABC College.

1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1,

2, 2, 2, 2,
2, 2, 2, 2, 2, 2,

3, 3, 3, 3,
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,

4, 4, 4, 4,
4, 4,

5, 5, 5, 5,
5,

6, 6

Eleven
students buy 1 book. Ten students buy 2 books. Sixteen students buy 3 books.
Six students buy 4 books. Five students buy 5 books. Two students buy 6 books.

Question Given
the following histogram for a set of data, how many values in the data set are
greater than 6.5and less than 9.5?

A histogram has a horizontal axis labeled
Values from 5.5 to 12.5 in increments of 1 and a vertical axis labeled
Frequency from 0 to 14 in increments of 2. The histogram contains vertical bars
of width 1, with one vertical bar centered over each of the horizontal axis
value tick marks. The heights of the vertical bars are as follows, where the
value is listed first and the height is listed second: 5.5, 5; 6.5, 3; 7.5, 6;
8.5, 5; 9.5, 4; 10.5, 3; 11.5, 13.

9

10

13

14

18

Histograms

Interpreting
Histograms

A histogram
is a graph that consists of contiguous (adjoining) boxes, and can show you the
shape, the center and the spread of the data. One advantage of a histogram is
that it can readily display large data sets. A histogram has both a horizontal
axis and a vertical axis. The horizontal axis is labeled with what the data
represents. The vertical axis is labeled with either the frequency or the
relative frequency.

A histogram has a horizontal axis labeled
Number of books from 0.5 to 6.5 in increments of 1 and a vertical axis labeled
Frequency from 0 to 16 in increments of 2. The histogram contains vertical bars
of width 1, with one vertical bar centered over each of the horizontal axis
value tick marks. The heights of the vertical bars are as follows, where the
value is listed first and the height is listed second: 0.5, 11; 1.5, 10; 2.5,
16; 3.5, 6; 4.5, 5; 5.5, 2.

For
example, in the histogram above, the horizontal axis represents the average
number of books read by Mr. Rucker’s students each month. The vertical axis
represents the number of students who read the corresponding number of books.
The tallest bar (3rd from the left) in
the histogram represents the number of students who read between 2.5 – 3.5
books on average each month. (These values can be found on the horizontal
axis.) The height of this bar is 16 (found on the vertical axis. This means
that 16 students read between 2.5 – 3.5 books on average each month.

Example

Question Given the histogram above, how many
students read between 4.5 – 5.5 books on average each month?

Question Describe the shape of the given
histogram.

A histogram has a horizontal axis from 0 to 16
in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in
increments of 2. The histogram contains vertical bars of width 1 starting at
the horizontal axis value 0. The heights of the bars are as follows, where the
horizontal axis label is listed first and the frequency is listed second: 0, 1;
1, 2; 2, 4; 3, 5; 4, 4; 5, 2; 6, 1; 7, 1; 8, 2; 9, 4; 10, 6; 11, 7; 12, 8; 13,
8; 14, 6; 15, 2.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question A
student surveys his class and creates a histogram showing the number of pets in
each student’s house. What is the shape of the distribution?

A histogram has a horizontal axis labeled Book
Price in dollars from 0 to 4 in increments of 1 and a vertical axis labeled
Frequency from 0 to 10 in increments of 2. Vertical bars of width 1 are
centered over a horizontal axis label. The heights of the bars are as follows,
where the horizontal axis label is listed first and the height is listed
second: 0, 5; 1, 6; 2, 3; 3, 2; 4, 1.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question Describe
the shape of the given histogram.

A histogram has a horizontal axis from 0 to 16
in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in
increments of 2. The histogram contains vertical bars of width 1 starting at
the horizontal axis value 0. The heights of the bars are as follows, where the
horizontal axis label is listed first and the frequency is listed second: 0, 0;
1, 1; 2, 1; 3, 2; 4, 3; 5, 5; 6, 6; 7, 6; 8, 5; 9, 4; 10, 2; 11, 1; 12, 1; 13,
0; 14, 0; 15, 0.

uniform

unimodal and symmetric

unimodal and left skewed

unimodal and right skewed

bimodal

Identify
and Labels Shapes of Histograms

Histogram
Shapes

A histogram
is a graph that helps show the distribution of values in a set of data. An
example of a histogram is shown below. The horizontal axis is labeled with data
values. It is divided into several sections that all have the same width. Then
a bar is drawn above each section, and the height of the bar is related to how
many of the data values are within the corresponding range on the horizontal
axis. The vertical axis (and the height of the bars) can be either counts of
data values (called a frequency) or the fraction of the data set in the range
(called the relative frequency).

A histogram has a horizontal axis labeled from
0 to 10 in increments of 1 and a vertical axis labeled Frequency from 0 to 10
in increments of 2. The histogram contains vertical bars of width 1 starting at
the horizontal axis value 0. The heights of the bars are as follows, where the
left horizontal axis label is listed first and the frequency is listed second:
0, 1; 1, 2; 2, 4; 3, 6; 4, 7; 5, 7; 6, 6; 7, 4; 8, 2; 9, 1.

The general
shape of a histogram can be described as uniform, unimodal, bimodal, or
multimodal. An example of each of these is given below. A uniform histogram has
bars that are all close to the same height. A unimodal histogram has a single
peak, and a multimodal histogram has more one peak. Sometimes the case with two
peaks is also called bimodal.

A histogram has a horizontal axis from 0 to 10
in increments of 1 and a vertical axis labeled Frequency from 0 to 10 in
increments of 2. The histogram contains vertical bars of width 1 starting at
the horizontal axis value 0. The heights of the bars are as follows, where the
left horizontal axis label is listed first and the frequency is listed second:
0, 6; 1, 6; 2, 6; 3, 6; 4, 6; 5, 6; 6, 6; 7, 6; 8, 6; 9, 6.

Uniform

A histogram
has a horizontal axis labeled from 0 to 10 in increments of 1 and a vertical
axis labeled Frequency from 0 to 10 in increments of 2. The histogram contains
vertical bars of width 1 starting at the horizontal axis value 0. The heights
of the bars are as follows, where the left horizontal axis label is listed
first and the frequency is listed second: 0, 1; 1, 2; 2, 4; 3, 6; 4, 7; 5, 7;
6, 6; 7, 4; 8, 2; 9, 1.

Unimodal

A histogram has a horizontal axis from 0 to 16
in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in
increments of 2. The histogram contains vertical bars of width 1 starting at
the horizontal axis value 0. The heights of the bars are as follows, where the
left horizontal axis label is listed first and the frequency is listed second:
0, 1; 1, 2; 2, 4; 3, 6; 4, 7; 5, 7; 6, 6; 7, 4; 8, 4; 9, 6; 10, 7; 11, 7; 12,
6; 13, 4; 14, 2; 15, 1.

Bimodal

A histogram
has a horizontal axis from 0 to 20 in increments of 4 and a vertical axis
labeled Frequency from 0 to 10 in increments of 2. The histogram contains
vertical bars of width 1 starting at the horizontal axis value 0. The heights
of the bars are as follows, where the left horizontal axis label is listed
first and the frequency is listed second: 0, 1; 1, 2; 2, 4; 3, 5; 4, 4; 5, 3;
6, 4; 7, 6; 8, 7; 9, 8; 10, 7; 11, 6; 12, 5; 13, 4; 14, 4; 15, 5; 16, 6; 17, 4;
18, 2; 19, 1.

Multimodal

A histogram can also be described by it symmetry
or skewness. A symmetric histogram has two halves that are approximately mirror
images of each other. The uniform, unimodal, and bimodal histograms shown above
are all symmetric. A histogram is skewed left when the tail of bars extending
towards smaller data values is longer than the tail extending towards larger
data values. A histogram is skewed right when the tail of bars extending
towards larger data values is longer than the tail extending towards smaller
data values. Examples of skewed histograms are shown below.

A histogram
has a horizontal axis from 0 to 16 in increments of 2 and a vertical axis
labeled Frequency from 0 to 10 in increments of 2. The histogram contains
vertical bars of width 1 starting at the horizontal axis value 0. The heights
of the bars are as follows, where the left horizontal axis label is listed
first and the frequency is listed second: 0, 0; 1, 1; 2, 1; 3, 1; 4, 2; 5, 2;
6, 3; 7, 3; 8, 4; 9, 5; 10, 6; 11, 7; 12, 7; 13, 5; 14, 3; 15, 1.

Skewed Left

A histogram
has a horizontal axis from 0 to 16 in increments of 2 and a vertical axis
labeled Frequency from 0 to 10 in increments of 2. The histogram contains
vertical bars of width 1 starting at the horizontal axis value 0. The heights
of the bars are as follows, where the left horizontal axis label is listed
first and the frequency is listed second: 0, 1; 1, 3; 2, 5; 3, 7; 4, 7; 5, 6;
6, 5; 7, 4; 8, 3; 9, 3; 10, 2; 11, 2; 12, 1; 13, 1; 14, 1; 15, 0.

Skewed
Right

Real world
distributions tend to be right skewed when there is a lower bound for the
values, most values are clustered in a range, but it is not impossible for very
large values to occur. A classic example of this is income distribution is some
countries. The lower bound is zero, most of the population have incomes within
a few standard deviations of the mean, but there are people with much larger
incomes.

Similarly,
left skewed distributions tend to occur for quantities that have a natural
upper bound when most of the population tend to be near that bound. For
example, student scores on a easy test will tend to all be fairly high and
create a peak near 100%. However, it is possible for a few students to create a
tail that extends down to much lower scores.

Uniform
distributions are less common. One simple example is rolling a fair dice. Every
number from 1to 6 has the same probability of appearing. Bimodal distributions
occur when there is a reason for two different peaks. For example, the
distribution of the people attending Disney Land at different timings could be
bimodal. It could have a peak at 11 am and another peak at 2 pm.

The
unimodal symmetric distribution, also called bell shaped is a very common
distribution. For example, the distribution of the mean wages of many random
samples of 30 people will be a unimodal symmetric distribution.

Example

Question Describe
the shape of the histogram shown below.

A histogram has a horizontal axis from 0 to 16
in increments of 2 and a vertical axis labeled Frequency from 0 to 10 in
increments of 2. The histogram contains vertical bars of width 1 starting at
the horizontal axis value 0. The heights of the bars are as follows, where the
left horizontal axis label is listed first and the frequency is listed second:
0, 2; 1, 7; 2, 8; 3, 7; 4, 6; 5, 4; 6, 3; 7, 3; 8, 2; 9, 2; 10, 1; 11, 1; 12,
1; 13, 0; 14, 0; 15, 0.

Question The
histogram shows the income of the families of the students in a statistics
class. What is the shape of the histogram?

A histogram has a horizontal axis labeled
Income in thousands from 0 to 200 in increments of 40 and a vertical axis
labeled Frequency from 0 to 10 in increments of 2. Vertical bars of width 20
start at the horizontal axis value 0. The heights of the bars are as follows,
where the left horizontal axis label is listed first and the frequency is
listed second: 0, 6; 20, 8; 40, 9; 60, 8; 80, 6; 100, 4; 120, 3; 140, 2; 160,
1; 180, 1.

uniform

unimodal and symmetric

unimodal and left-skewed

unimodal and right-skewed

bimodal

Question The
kindergarten students in a school were asked to reach into a bag of candy and
pull out as many pieces as they could with one hand. The number of candies for each student was
counted, and the results are displayed in the following frequency table.

Which
histogram accurately summarizes the data?

Value

Frequency

8

2

9

6

10

2

11

4

12

6

13

7

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A histogram
has a horizontal axis labeled Values from 5.5 to 11.5 in increments of 1 and a
vertical axis labeled Frequency from 0 to 7 in increments of 1. The histogram
contains vertical bars of width 1, with one vertical bar centered over each of
the horizontal axis value tick marks. The heights of the vertical bars are as
follows, where the value is listed first and the height is listed second: 5.5,
2; 6.5, 6; 7.5, 2; 8.5, 4; 9.5, 6; 10.5, 7.

A histogram has a horizontal axis labeled
Values from 9.5 to 15.5 in increments of 1 and a vertical axis labeled
Frequency from 0 to 7 in increments of 1. The histogram contains vertical bars
of width 1, with one vertical bar centered over each of the horizontal axis
value tick marks. The heights of the vertical bars are as follows, where the
value is listed first and the height is listed second: 9.5, 2; 10.5, 6; 11.5,
2; 12.5, 4; 13.5, 6; 14.5, 7.

A bar graph has a horizontal axis titled
Values labeled from 7.5 to 13.5 in increments of 1 and a vertical axis titled
Frequency labeled from 0 to 7 in increments of 1. Six bars are plotted each
with a width of 1. From left to right, the heights of the bars are as follows:
2, 6, 2, 4, 6, 7.

A bar graph has a horizontal axis titled
Values labeled from 3.5 to 9.5 in increments of 1 and a vertical axis titled
Frequency labeled from 0 to 7 in increments of 1. Six bars are plotted each
with a width of 1. From left to right, the heights of the bars are as follows:
2, 6, 2, 4, 6, 7.

Question Several
people were asked to report the number of hours of sleep they average per
night. The results are shown in the
histogram below. How many of those
people average greater than 4.5 and less than 6.5 hours of sleep per night?

A histogram has a horizontal axis labeled
Values from 3.5 to 8.5 in increments of 1 and a vertical axis labeled Frequency
from 0 to 12 in increments of 2. The histogram contains vertical bar