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10-1 Displaying Data: Part I

Data: Categorical and Numerical Pictographs Dot Plots Stem and Leaf Plots Back-to-Back Stem and Leaf Plots Grouped Frequency Tables Histograms and Bar Graphs Other Bar Graphs Circle Graphs (Pie Charts)

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Graphs are used to try to tell a story.

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A fundamental idea in prekindergarten through grade 2 is that data can be organized or ordered and that this “picture” of the data provides information about the phenomenon or question.

In grades 3–5, students should develop skill in representing their data, often using bar graphs, tables, or line plots. . . .

Students in grades 6–8 should begin to compare the effectiveness of various types of displays in organizing the data for further analysis or in presenting the data clearly to an audience. (p. 49)

NCTM Standard: Graphs

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Data

Data are observations (such as measurements, genders, survey responses) that have been collected. Statisticians often collect data from small portions of a large group in order to determine information about the group. This information is then used to make conjectures about the entire group.

Describing data frequently involves reading information from graphical displays, tables, lists, and so on.

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Data: Categorical and Numerical

Categorical data are data that represent characteristics of objects or individuals in groups (or categories), such as black or white, inside or outside, male or female.

Numerical data are data collected on numerical variables. For example, in grade school, students may ask whether there is a difference in the distance that girls and boys can jump. The distance jumped is a numerical variable and the collected data is numerical.

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A pictograph is used to represent tallies of categories. For example, categorical data might be seen in determining the month that the most automobiles were sold. The month is the category. A symbol, or icon, is used to represent a quantity of items.

Pictographs

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Pictographs

A symbol or an icon is used to represent a quantity of items. A legend tells what the symbol represents.

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Pictographs

All graphs need titles. If needed, legends should be shown.

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Pictographs

The number of students in each teacher’s fifth-grade class at Hillview is depicted in tabular representation in the table and in a pictograph.

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A dot plot, or line plot, provides a quick and simple way of organizing numerical data. They are typically used when there is only one group of data with fewer than 50 values.

Dot Plots

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Suppose the 30 students in Abel’s class received the following test scores:

Dot Plots

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A dot plot for the class scores consists of a horizontal number line on which each score is denoted by a dot, or an x, above the corresponding number-line value. The number of x’s above each score indicates how many times each score occurred.

Dot Plots

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The score 52 is an outlier

Scores 97 and 98 form a cluster

A gap occurs between scores 88

and 97.

Dot plots

Two students scored 72.

Four students scored 82.

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Outliera data point whose value is significantly greater than or less than other values

Cluster an isolated group of points

Gapa large space between data points

Modedata value(s) that occur most often.

Dot Plots

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Dot Plots

If a dot plot is constructed on grid paper, then shading in the squares with x’s and adding a vertical axis depicting the scale allows the formation of a bar graph.

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Stem and Leaf Plots

The stem and leaf plot is similar to the dot plot, but the number line is usually vertical, and digits are used rather than x’s.

9 | 7 represents 97.

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Stem and Leaf Plots

In an ordered stem and leaf plot, the data are in order from least to greatest on a given row.

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Advantages of stem-and-leaf plots:

They are easily created by hand.

Do not become unmanageable when volume of data is large.

No data values are lost.

Disadvantage of stem-and-leaf plots:

We lose information – we may know a data value exists, but we cannot tell which one it is.

Stem and Leaf Plots

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How to construct a stem-and-leaf plot:

1. Find the high and low values of the data.

2. Decide on the stems.

3. List the stems in a column from least to greatest.

4. Use each piece of data to create leaves to the right of the stems on the appropriate rows.

Stem and Leaf Plots

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5. If the plot is to be ordered, list the leaves in order from least to greatest.

6. Add a legend identifying the values represented by the stems and leaves.

7. Add a title explaining what the graph is about.

Stem and Leaf Plots

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Back-to-back stem-and-leaf plots can be used to compare two sets of related data. In this plot, there is one stem and two sets of leaves, one to the left and one to the right of the stem.

Back-to-Back Stem-and-Leaf Plots

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Example 10-1Group the presidents into two groups, George Washington to Rutherford B. Hayes and James Garfield to Ronald Reagan.

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Example 10-1 (continued)

a. Create a back-to-back stem and leaf plot of the two groups and see if there appears to be a difference in ages at death between the two groups.

b. Which group of presidents seems to have lived longer?

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Example 10-1 (continued)

Because the ages at death vary from 46 to 93, the stems vary from 4 to 9. The first 19 presidents are listed on the left and the remaining 19 on the right.

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Example 10-1 (continued)

The early presidents seem, on average, to have lived longer because the ages at the high end, especially in the 70s through 90s, come more often from the early presidents.

The ages at the lower end come more often from the later presidents.

For the stems in the 50s and 60s, the numbers of leaves are about equal.

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Stem and Leaf Plots

A stem and leaf plot shows how wide a range of values the data cover, where the values are concentrated, whether the data have any symmetry, where gaps in the data are, and whether any data points are decidedly different from the rest of the data.

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Frequency Tables

A grouped frequency table shows how many times data occurs in a range.

The data for the ages of the presidents at death are summarized in the table.

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Each class interval has the same size.

The size of each interval can be computed by subtracting the lower endpoint from the higher and adding 1, e.g., 49 – 40 +1 = 10.

We know how many data values occur within a particular interval but we do not know the particular data values themselves.

Frequency Tables

Characteristics of Frequency Tables

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As the interval size increases, information is lost.

Classes (intervals) should not overlap.

Frequency Tables

Characteristics of Frequency Tables

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Histograms and Bar Graphs

A histogram is made up of adjoining rectangles, or bars.

The bars are all the same width. The scale on the vertical axis must be uniform.

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Uni

form

sca

leHistograms and Bar Graphs

The death ages are shown on the horizontal axis and the numbers along the vertical axis give the scale for the frequency.

Frequencies are shown by the heights of vertical bars each having same width.

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Bar Graphs

A bar graph typically has spaces between the bars and is used to depict categorical data.

The bars representing Tom, Dick, Mary, Joy, and Jane could beplaced in any order.

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Histograms and Bar Graphs

A distinguishing feature between histograms and bar graphs is that there is no ordering that has to be done among the bars of the bar graph, whereas there is an order for a histogram.

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A double bar graph can be used to make comparisons in data.

Double-Bar Graphs

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Other Bar Graphs

The table shows various types of shoes worn by students in one class and the approximate percentages of students who wore them.

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Other Bar Graphs

The figure depicts a percentage bar graph with that same data.

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Other Bar Graphs

The table shows information about the expenditures of a business over a period of years.

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Other Bar Graphs

The figure depicts a stacked bar graph with that same data.

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Circle Graphs (Pie Charts)

A circle graph, or pie chart, is used to represent categorical data. It consists of a circular region partitioned into disjoint sections, with each section representing a part or percentage of the whole.

A circle graph shows how parts are related to the whole.

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Example 10-2

Construct a circle graph for the information in the table, which is based on information taken from a U.S. Bureau of the Census Report (2006).

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Example 10-2 (continued)

The entire circle represents the total 299 million people.

The measure of the central angle (an angle whose vertex is at the center of the circle) of each sector of the graph is proportional to the fraction or percentage of the population the section represents.

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Example 10-2 (continued)

For example, the measure of the angle for the

sector for the under-5 group is or approximately 7% of the circle.

Because the entire circle is 360°, of 360°, or

about 24°, should represent the under-5 group.

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Example 10-2 (continued)

The table shows the number of degrees for each age group.

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Example 10-2 (continued)