Graphical Descriptions of Data Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. HAWKES LEARNING SYSTEMS math courseware specialists Chapter 2 Graphical Descriptions of Data

Frequency, f – number of data values in a class. HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.1 Frequency Distributions Organizing Data: 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.

Decide on the number of classes Choose an appropriate class width HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.1 Frequency Distributions Creating frequency tables: 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.

Classes boundaries Midpoints Relative Frequency Cumulative Frequency HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.1 Frequency Distributions Other characteristics can be calculated once the basic frequency table has been constructed: 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.

Quiz Grades – Ordered Array HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.1 Frequency Distributions Create a frequency distribution using 5 classes: Quiz Grades 9 3 5 4 7 8 10 6 2 1 Solution – First place the data in an ordered array: Quiz Grades – Ordered Array 1 2 3 4 5 6 7 8 9 10

Round 1.8 up to a sensible value, 2. HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.1 Frequency Distributions Solution – continued: 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.

HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.1 Frequency Distributions The frequency distribution: Quiz Grades Class f Class Boundaries Midpoint Relative Frequency Cumulative Frequency 1 – 2 2 0.5 – 2.5 1.5 3 – 4 3 2.5 – 4.5 3.5 5 5 – 6 4 4.5 – 6.5 5.5 9 7 – 8 6.5 – 8.5 7.5 18 9 – 10 7 8.5 – 10.5 9.5 25

HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.1 Frequency Distributions Create a frequency distribution using 6 classes: GPA’s 3.2 2.6 2.9 2.0 3.1 3.5 1.8 1.3 3.8 3.0 1.1 2.5 3.4 Solution – First place the data in an ordered array: GPA’s – Ordered Array 1.1 1.3 1.8 2.0 2.5 2.6 2.9 3.0 3.1 3.2 3.4 3.5 3.8

Round 0.45 up to a sensible value, 0.5. HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.1 Frequency Distributions Solution – continued: 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.

HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.1 Frequency Distributions 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 1.5 – 1.9 1 1.45 – 1.95 1.7 3 2.0 – 2.4 1.95 – 2.45 2.2 5 2.5 – 2.9 2.45 – 2.95 2.7 8 3.0 – 3.4 2.95 – 3.45 3.2 13 3.5 – 3.9 3.45 – 3.95 3.7 15

Should be able to stand alone without the original data. HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.2 Graphical Displays of Data Graphs: 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.

Sorority/Fraternity House HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.2 Graphical Displays of Data Pie Chart: Shows how large each category is in relation to the whole. Create a pie chart from the following information: Types of Housing Number of Students Apartment 20 Dorm 15 House 9 Sorority/Fraternity House 5

HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.2 Graphical Displays of Data Solution: Frequency Angles 100% 100˚

Sorority/Fraternity House HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.2 Graphical Displays of Data Create a bar graph from the following information: Types of Housing Number of Students Apartment 20 Dorm 15 House 9 Sorority/Fraternity House 5 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.

Number of Students from Class A Number of Students from Class B HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.2 Graphical Displays of Data Create a side-by-side bar graph from the following information: 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

Number of Students from Class A Number of Students from Class B HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.2 Graphical Displays of Data Create a stacked bar graph from the following information: 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 With the stacked bar graph, it is easier to see that more students live in the dorms than in apartments.

A bar graph of a frequency distribution. HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.2 Graphical Displays of Data Histograms: 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.

Plasma Screen TV Prices HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.2 Graphical Displays of Data Create a histogram from the following information: 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 1849.5 1799.5 – 1899.5 $1900 – $1999 1949.5 1899.5 – 1999.5 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.

HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.2 Graphical Displays of Data Frequency Polygons: A visual display of the frequencies of each class using the midpoints from a frequency table. Steps for creating a frequency polygon: 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.

HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.2 Graphical Displays of Data Steps for creating a frequency polygon (continued): 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. Join each point to the next with a line segment.

Plasma Screen TV Prices Cumulative Frequencies HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.2 Graphical Displays of Data Ogives: A line graph which depicts the cumulative frequency of each class from a frequency table. Steps for creating an ogive: Begin by tabulating the cumulative frequencies for each class. Plasma Screen TV Prices Class f Cumulative Frequencies Boundaries $1500 – $1599 2 1499.5 – 1599.5 $1600 – $1699 5 7 1599.5 – 1699.5 $1700 – $1799 4 11 1699.5 – 1799.5 $1800 – $1899 16 1799.5 – 1899.5 $1900 – $1999 20 1899.5 – 1999.5

HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.2 Graphical Displays of Data Steps for creating an ogive (continued): 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. Finally, join the points together with line segments.

Retain the original data. HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.2 Graphical Displays of Data Stem and Leaf Plots: Retain the original data. The leaves are usually the last digit in each data value and the stems are the remaining digits. Steps for creating a stem and leaf plot: 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.

HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.2 Graphical Displays of Data Create a steam and leaf plot from the following information: ACT Scores 18 23 24 31 19 27 26 22 32 35 29 20 17 21 25 ACT Scores ACT Scores Stem Leaves Stem Leaves 1 8 9 8 8 7 1 7 8 8 8 9 2 3 4 7 6 2 7 9 4 1 5 6 2 1 2 3 4 4 5 6 6 7 7 9 3 1 2 5 3 1 2 5 Key: 1|8 = 18 Key: 1|8 = 18 Ordered Array

HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.3 Analyzing Graphs Shapes of Distribution: 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.

Time-series – a picture of how data changes over time. HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.3 Analyzing Graphs Definitions: 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.

HAWKES LEARNING SYSTEMS math courseware specialists Graphical Descriptions of Data 2.3 Analyzing Graphs 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?