Reflection paper
Hnno
Chapter2.pptChapter 2
Graphical Descriptions of Data
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc.
All rights reserved.
Graphical Descriptions of Data
2.1 Frequency Distributions
HAWKES LEARNING SYSTEMS
math courseware specialists
Ordered array – an ordered list of the data from largest to smallest or vice versa.
Distribution – displays data values that occur and how often they occur. It can be a chart or a table.
Frequency Distribution – table that divides data into groups, called classes, and shows how many data values occur in each group.
Frequency, f – number of data values in a class.
Organizing Data:
Graphical Descriptions of Data
2.1 Frequency Distributions
HAWKES LEARNING SYSTEMS
math courseware specialists
- Decide on the number of classes
- Between 5 and 20
- Choose an appropriate class width
- Find the class limits
- Start with the lowest value and add the class width to get the next class limit.
- Determine the frequency of each class
- Count the number of data values in each class.
Creating frequency tables:
Graphical Descriptions of Data
2.1 Frequency Distributions
HAWKES LEARNING SYSTEMS
math courseware specialists
- Classes boundaries
- Split the difference in the gap between the upper limit of one class and the lower limit of the next class.
- Midpoints
- Relative Frequency
- Cumulative Frequency
- The sum of the frequency for a given class and all previous classes.
Other characteristics can be calculated once the basic frequency table has been constructed:
Graphical Descriptions of Data
2.1 Frequency Distributions
HAWKES LEARNING SYSTEMS
math courseware specialists
Create a frequency distribution using 5 classes:
Solution – First place the data in an ordered array:
| Quiz Grades | ||||
| 9 | 3 | 5 | 4 | 7 |
| 8 | 10 | 8 | 6 | 7 |
| 4 | 5 | 2 | 7 | 8 |
| 10 | 7 | 9 | 10 | 1 |
| 8 | 6 | 10 | 9 | 8 |
| Quiz Grades – Ordered Array | ||||
| 1 | 2 | 3 | 4 | 4 |
| 5 | 5 | 6 | 6 | 7 |
| 7 | 7 | 7 | 8 | 8 |
| 8 | 8 | 8 | 9 | 9 |
| 9 | 10 | 10 | 10 | 10 |
Graphical Descriptions of Data
2.1 Frequency Distributions
HAWKES LEARNING SYSTEMS
math courseware specialists
Since we have the smallest and largest values, we can find the class width.
Round 1.8 up to a sensible value, 2.
Next begin building the class limits with the smallest data value in the set.
Solution – continued:
Graphical Descriptions of Data
2.1 Frequency Distributions
HAWKES LEARNING SYSTEMS
math courseware specialists
The frequency distribution:
| Quiz Grades | |||||
| Class | f | Class Boundaries | Midpoint | Relative Frequency | Cumulative Frequency |
| 1 – 2 | 2 | 0.5 – 2.5 | 1.5 | 2 | |
| 3 – 4 | 3 | 2.5 – 4.5 | 3.5 | 5 | |
| 5 – 6 | 4 | 4.5 – 6.5 | 5.5 | 9 | |
| 7 – 8 | 9 | 6.5 – 8.5 | 7.5 | 18 | |
| 9 – 10 | 7 | 8.5 – 10.5 | 9.5 | 25 |
Graphical Descriptions of Data
2.1 Frequency Distributions
HAWKES LEARNING SYSTEMS
math courseware specialists
Create a frequency distribution using 6 classes:
Solution – First place the data in an ordered array:
| GPA’s | ||||
| 3.2 | 2.6 | 2.9 | 2.0 | 3.1 |
| 3.5 | 1.8 | 1.3 | 3.8 | 3.0 |
| 1.1 | 2.0 | 2.5 | 3.1 | 3.4 |
| GPA’s – Ordered Array | ||||
| 1.1 | 1.3 | 1.8 | 2.0 | 2.0 |
| 2.5 | 2.6 | 2.9 | 3.0 | 3.1 |
| 3.1 | 3.2 | 3.4 | 3.5 | 3.8 |
Graphical Descriptions of Data
2.1 Frequency Distributions
HAWKES LEARNING SYSTEMS
math courseware specialists
Since we have the smallest and largest values, we can find the class width.
Round 0.45 up to a sensible value, 0.5.
Next begin building the class limits with the smallest data value in the set.
Solution – continued:
Graphical Descriptions of Data
2.1 Frequency Distributions
HAWKES LEARNING SYSTEMS
math courseware specialists
The frequency distribution:
| Quiz Grades | |||||
| Class | f | Class Boundaries | Midpoint | Relative Frequency | Cumulative Frequency |
| 1.0 – 1.4 | 2 | 0.95 – 1.45 | 1.2 | 2 | |
| 1.5 – 1.9 | 1 | 1.45 – 1.95 | 1.7 | 3 | |
| 2.0 – 2.4 | 2 | 1.95 – 2.45 | 2.2 | 5 | |
| 2.5 – 2.9 | 3 | 2.45 – 2.95 | 2.7 | 8 | |
| 3.0 – 3.4 | 5 | 2.95 – 3.45 | 3.2 | 13 | |
| 3.5 – 3.9 | 2 | 3.45 – 3.95 | 3.7 | 15 |
Graphical Descriptions of Data
2.2 Graphical Displays of Data
HAWKES LEARNING SYSTEMS
math courseware specialists
- Should be able to stand alone without the original data.
- Must have a title and labels for both axes.
- When appropriate, a legend, a source, and a date should be included.
Graphs:
Graphical Descriptions of Data
2.2 Graphical Displays of Data
HAWKES LEARNING SYSTEMS
math courseware specialists
Shows how large each category is in relation to the whole.
Pie Chart:
Create a pie chart from the following information:
| Types of Housing | |
| Types of Housing | Number of Students |
| Apartment | 20 |
| Dorm | 15 |
| House | 9 |
| Sorority/Fraternity House | 5 |
Solution:
Graphical Descriptions of Data
2.2 Graphical Displays of Data
HAWKES LEARNING SYSTEMS
math courseware specialists
100%
100˚
Angles
Frequency
*
Graphical Descriptions of Data
2.2 Graphical Displays of Data
HAWKES LEARNING SYSTEMS
math courseware specialists
Create a bar graph from the following information:
Notice that the bar graph shown is in descending order of largest to smallest.
This type of bar graph is called a Pareto chart and is typically used with nominal data.
| Types of Housing | |
| Types of Housing | Number of Students |
| Apartment | 20 |
| Dorm | 15 |
| House | 9 |
| Sorority/Fraternity House | 5 |
Graphical Descriptions of Data
2.2 Graphical Displays of Data
HAWKES LEARNING SYSTEMS
math courseware specialists
Create a side-by-side bar graph from the following information:
| Types of Housing | ||
| Types of Housing | Number of Students from Class A | Number of Students from Class B |
| Apartment | 20 | 13 |
| Dorm | 15 | 24 |
| House | 9 | 6 |
| Sorority/Fraternity House | 5 | 7 |
Graphical Descriptions of Data
2.2 Graphical Displays of Data
HAWKES LEARNING SYSTEMS
math courseware specialists
Create a stacked bar graph from the following information:
With the stacked bar graph, it is easier to see that more students live in the dorms than in apartments.
| Types of Housing | ||
| Types of Housing | Number of Students from Class A | Number of Students from Class B |
| Apartment | 20 | 13 |
| Dorm | 15 | 24 |
| House | 9 | 6 |
| Sorority/Fraternity House | 5 | 7 |
Graphical Descriptions of Data
2.2 Graphical Displays of Data
HAWKES LEARNING SYSTEMS
math courseware specialists
- A bar graph of a frequency distribution.
- The horizontal axis is a real number line.
- The width of the bars represent the class width from the frequency table and should be uniform.
- The bars should touch.
- The height of each bar represents the frequency of the class it represents.
Histograms:
Graphical Descriptions of Data
2.2 Graphical Displays of Data
HAWKES LEARNING SYSTEMS
math courseware specialists
Create a histogram from the following information:
Although we use the class boundaries to draw the histograms, it is appropriate to use either the class boundaries (shown in the figure to the left) or midpoints (shown in the figure to the right) when labeling the x-axis.
| Plasma Screen TV Prices | |||
| Class | f | Midpoint | Boundaries |
| $1500 – $1599 | 2 | 1549.5 | 1499.5 – 1599.5 |
| $1600 – $1699 | 5 | 1649.5 | 1599.5 – 1699.5 |
| $1700 – $1799 | 4 | 1749.5 | 1699.5 – 1799.5 |
| $1800 – $1899 | 5 | 1849.5 | 1799.5 – 1899.5 |
| $1900 – $1999 | 4 | 1949.5 | 1899.5 – 1999.5 |
Graphical Descriptions of Data
2.2 Graphical Displays of Data
HAWKES LEARNING SYSTEMS
math courseware specialists
- A visual display of the frequencies of each class using the midpoints from a frequency table.
Frequency Polygons:
- Mark the class boundaries on the x-axis and the frequencies on the y-axis. Note that extra classes at the lower and upper ends will be added, each having a frequency of 0. From the previous example of plasma TV prices, these classes will be 1400 – 1499 at the lower end and 2000 – 2099 at the upper end.
Steps for creating a frequency polygon:
Graphical Descriptions of Data
2.2 Graphical Displays of Data
HAWKES LEARNING SYSTEMS
math courseware specialists
- Add the midpoints to the x-axis and plot a point at the frequency of each class directly above its midpoint. Notice that the class boundaries have been lightened. This is due to the fact that a frequency polygon represents the midpoints of the data.
Steps for creating a frequency polygon (continued):
- Join each point to the next with a line segment.
Graphical Descriptions of Data
2.2 Graphical Displays of Data
HAWKES LEARNING SYSTEMS
math courseware specialists
- A line graph which depicts the cumulative frequency of each class from a frequency table.
Ogives:
- Begin by tabulating the cumulative frequencies for each class.
Steps for creating an ogive:
| Plasma Screen TV Prices | |||
| Class | f | Cumulative Frequencies | Boundaries |
| $1500 – $1599 | 2 | 2 | 1499.5 – 1599.5 |
| $1600 – $1699 | 5 | 7 | 1599.5 – 1699.5 |
| $1700 – $1799 | 4 | 11 | 1699.5 – 1799.5 |
| $1800 – $1899 | 5 | 16 | 1799.5 – 1899.5 |
| $1900 – $1999 | 4 | 20 | 1899.5 – 1999.5 |
Graphical Descriptions of Data
2.2 Graphical Displays of Data
HAWKES LEARNING SYSTEMS
math courseware specialists
- Unlike a frequency polygon where two classes are added, we only include an extra class at the lower end for this graph, giving it a frequency of 0.
- Next, plot a point at the cumulative frequency for each class directly above its upper class boundary.
Steps for creating an ogive (continued):
- Finally, join the points together with line segments.
Graphical Descriptions of Data
2.2 Graphical Displays of Data
HAWKES LEARNING SYSTEMS
math courseware specialists
- Retain the original data.
- The leaves are usually the last digit in each data value and the stems are the remaining digits.
Stem and Leaf Plots:
- Create two columns, one on the left for stems and one on the right for leaves.
- List each of the stems that occur in the data set in numerical order.
- List each leaf next to its stem.
- Create a key to guide interpretation of the stem and leaf plot.
- The leaves may then be put in order, if desired, to create an ordered stem and leaf plot.
Steps for creating a stem and leaf plot:
Graphical Descriptions of Data
2.2 Graphical Displays of Data
HAWKES LEARNING SYSTEMS
math courseware specialists
Create a steam and leaf plot from the following information:
Key: 1|8 = 18
ACT Scores
Leaves
Stem
1
2
3
8
3
1
9
4
2
8
7
5
8
6
7
2
7
9
4
0
1
6
5
Key: 1|8 = 18
Ordered Array
ACT Scores
Leaves
Stem
1
2
3
7
0
1
8
1
2
8
2
5
8
3
9
4
4
5
6
6
7
9
7
| ACT Scores | ||||
| 18 | 23 | 24 | 31 | 19 |
| 27 | 26 | 22 | 32 | 18 |
| 35 | 27 | 29 | 24 | 20 |
| 18 | 17 | 21 | 25 | 26 |
Graphical Descriptions of Data
2.3 Analyzing Graphs
HAWKES LEARNING SYSTEMS
math courseware specialists
- Uniform – the frequency of each class is relatively the same.
- Symmetrical – the data lies evenly on both sides of the distribution.
- Skewed to the Right – the majority of the data falls on the left of the distribution.
- Skewed to the Left – the majority of the data falls on the right of the distribution.
Shapes of Distribution:
- Time-series – a picture of how data changes over time.
- Cross-sectional study – a picture of the data at a given moment of time.
- Outlier – a data value that falls outside the normal shape of the graph.
Definitions:
Graphical Descriptions of Data
2.3 Analyzing Graphs
HAWKES LEARNING SYSTEMS
math courseware specialists
Look at the two graphs shown below depicting the same data on people’s satisfaction level with their local shopping mall. Which graph is more accurate and why?
Graphical Descriptions of Data
2.3 Analyzing Graphs
HAWKES LEARNING SYSTEMS
math courseware specialists
*
2
0.08
25
=
3
0.12
25
=
4
0.16
25
=
9
0.36
25
=
7
0.28
25
=
2
0.133
15
»
1
0.067
15
»
3
0.200
15
=
5
0.333
15
»
2
0.133
15
»
