Chapter 3

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Organizing, Displaying, and Interpreting Data

Chapter 3






Section 3a Frequency Distributions

Objectives:

• Learn how to construct a frequency distribution.• Know the characteristics of a frequency distribution.






Section 3.1 Frequency Distributions

Definition:

A frequency distribution is a summary technique that organizes data into classes and provides in tabular form a list of the classes along with the number of observations in each class.

The two steps in constructing frequency distributions are:• Choosing the classifications, and• Counting the number in each class.

Choosing the type of classification depends on whether the data is qualitative (nominal or ordinal) or quantitative (interval or ratio).






Section 3b Graphical Displays of Data: Pie Charts and Bar Graphs

Objectives:

• Create the basic types of pie charts and bar graphs.• Interpret data given in pie charts and bar graphs.






Section 3.2 The Value of Graphs

Graphs:

• A set of data can be graphically represented in many different ways.• Creating graphical displays requires a certain amount of artistic

judgment.• Development of graphical software has made graphing easy.• Types of graphs include:

• Bar charts• Pie charts• Line charts• Stem and leaf displays• Histograms






Section 3.2 The Value of Graphs

Types of Graphs:

• Bar charts

• Pie charts

• Stem and leaf displays

• Histograms

Stem Leaves0 97 99

1 08 10 11






Susan William Beth Rob0

50

100

150

200

250

Sales Performance

Sales PerformanceSales Person Total Dollars in Sales

(in thousands)Susan 187William 201

Beth 207Rob 193

The bar graphs below are both plots of the same data set.

What do you notice about the axis labels?Remember: When you see an axis that doesn’t start at zero, be a bit skeptical of the conclusions the author intends for you to make.

Misleading Graphs:


Section 3.3 Displaying Qualitative Data Graphically







Bar Chart:

The bar chart is a graphical display in which the length of each bar corresponds to the number of observations in a category.

Bar charts are:• used to illustrate a frequency distribution for qualitative data.• valuable as presentation tools.• effective at reinforcing differentials in magnitudes.• comprised of vertical or horizontal bars.







Bar Chart:As mentioned in the last slide, bar graphs represent qualitative data.

Can you tell the categories are qualitative?Specifically, what level of measurement are the categories an example of?

Solution: Ordinal







Conventions of Bar Chart Construction:

• Maintain order of categories• Miscellaneous or “other” should be listed at the bottom of horizontal

graphs or at the far right in vertical graphs• Effectively choose a scale to allow for desired comparison• Choose visually pleasing bar widths• Do not vary the bar width throughout the chart• Use shading, crosshatching, and color to help present data • The spacing between bars should be set at approximately one-half the

width of a bar• Source notes are placed below the chart • Gridlines are often used and increase readability• Label each axis if there is room







Stacked Bar Charts:

• Variation on the standard bar chart• Allows comparison of total quantity as well as the individual quantity of

several subcategories.

Example: Grandchildren living with their Grandparents

012345678

Under6 years

6-11years

12-14years

15-17years

Two parentsMother onlyFather onlyNeither parent

Num

ber o

f Chi

ldre

n







3-D Bar Charts: Below is an example of a 3-D bar chart.

The chart displays the following question from a survey by the Gallup poll:Do you think women should be permitted to sunbathe on public beaches, or

should it be banned?







Pie Charts:

A pie chart shows us how large each category is in relation to the whole.

• Can be used to express frequency distributions.

• The circle represents the total “pie” available.

• The slices are proportional to the amount in each category.

• Each slice of the pie represents the proportion of total observations belonging to the category.

• Easy to compare the total in each of the classifications to the total number of observations.







Pie Charts: Most commonly, pie charts are used to display how money is spent. The pie chart below tells an interesting story about how our tax dollars are spent.

Socia

l Secu

rity

National

DefenseMedica

re

Medicaid

0%

5%

10%

15%

20%

25%

Social Secu-rity 22%

Non-De-fense Discre-

tionary 19%National

Defense 17%

Other Enti-tlements

15%

Medicare 11% Net Interest 9%

Medicaid 7%






Section 3c Graphical Displays of Data: Histograms, Polygons,

Stem and Leaf Plots

Objectives:

• Understand how to read and interpret the information shown in line graphs, histograms, frequency polygons, ogives, and stem and leaf plots.

• Be able to perform appropriate operations related to the data shown in a line graph, histogram, frequency polygon, ogive, or stem and leaf plot.

• Construct histograms, frequency polygons, and ogives from the data given.






Section 3.4 Constructing Frequency Distributions for

Quantitative Data

Frequency Distributions:

The purpose of a frequency distribution is to condense a set of data into a meaningful summary form.

Remember there are two steps in the construction of a frequency distribution:

• choosing the classifications, and• counting the number in each class.







Quantitative Data

Types of Frequency Distributions:

Distributions used to organize data:

• Relative Frequency• Cumulative Frequency• Cumulative Relative Frequency







Quantitative Data

Selecting the Number of Classes:

The fundamental decision in constructing a frequency distribution is selecting the number of classes.

• The number of classes depends on the amount of data available.• Generally fewer than 4 classes compresses the data.• More than 20 classes provides too little summary information.• Once you determine the number of classes, the next step is to

specify the class width.







Quantitative Data

Determining the Class Width:

Usually, the class widths are equal widths, except for the beginning and ending of intervals.

There is no perfect formula for class width that will work for every data set. However a good starting point for determining class width is:

.largest value - smallest valueclass width = number of classes






Heart Rate Number of Students

57.50 to 67.5 367.51 to 77.5 1377.51 to 87.5 2987.51 to 97.5 497.51 to 107.5 1

77 84 79 90 67 84 82 7469 81 94 68 65 86 78 7983 83 84 82 93 80 81 8062 98 77 83 82 80 82 7377 79 81 70 72 85 84 8083 77 80 70 75 74 85 8779 88

Example: Create a frequency distribution with the following heart rate data:

If there are five classes, determine the class width.

largest value - smallest valueclass width = =number of classes

98 625 36

5 7.2



Quantitative Data

Class endpoints with fractional values will make the graph slightly difficult to digest. If possible, try a class width in the range of 8 to 10.An interval width of 10 is used in this example.







Quantitative Data

Relative Frequency:

The relative frequency represents the proportion of the total number of observations in a given class.

Relative frequency:

• Allows us to view the number in each category in relation to the total number of observations.

• Is a standardizing technique.• Enables us to compare data sets with different numbers of

observations.

number in classrelative frequency = total number of observations






Heart Rate Fraction of Students

57.50 to 67.567.51 to 77.577.51 to 87.587.51 to 97.597.51 to 107.5

number in classrelative frequency = total number of observations

77 84 79 90 67 84 82 7469 81 94 68 65 86 78 7983 83 84 82 93 80 81 8062 98 77 83 82 80 82 7377 79 81 70 72 85 84 8083 77 80 70 75 74 85 8779 88

3 .0650

.06

Fifty students had their heart rate checked. Find the relative frequency of each interval.

.26

13 .2650

.58

29 .5850

.08

4 .0850

.02

1 .0250

Example:



Quantitative Data






Heart Rate Frequency CumulativeFrequency

57.50 to 67.5 3

67.51 to 77.5 13

77.51 to 87.5 29

87.51 to 97.5 4

97.51 to 107.5 1

3

16

45

49

50

3 13 16

29 13 3 45 4 29 13 3 49

1 4 29 13 3 50

Cumulative Frequency:

The cumulative frequency is the sum of the frequency of a particular class and all preceding classes. Below is a cumulative frequency distribution for the heart rate data.



Quantitative Data






The cumulative relative frequency is the proportion of observations in a particular class and all preceding classes. Below is a cumulative relative frequency distribution for the heart rate data.

Heart Rate RelativeFrequency

CumulativeRelative

Frequency

57.50 to 67.5 0.06 0.06

67.51 to 77.5 0.26

77.51 to 87.5 0.58

87.51 to 97.5 0.08

97.51 to 107.5 0.02

0.32

0.90

0.98

1.00

.06 .26 .32 .06 .26 .58 .90

.06 .26 .58 .05 .98

.06 .26 .58 .08 .02 1.00



Quantitative Data

Cumulative Relative Frequency:






Section 3.5 Histograms

Histograms:

A histogram is a bar graph of a frequency or relative frequency distribution in which the height of each bar corresponds to the frequency or relative frequency of the class.

A histogram:

• is one of the most frequently used statistical tools.• reveals the structure of the data.• is easy to interpret.






Ch 3. Organizing, Displaying, and Interpreting Data

3.5 Histograms

57.5 67.5 77.5 87.5 97.50

5

10

15

20

25

30

Histogram of Student Heart Rate Data

Beats per Minute

Freq

uenc

y

57.5 67.5 77.5 87.5 97.50

5

10

15

20

25

30

3-D Histogram of Student Heart Rate Data

Beats Per Minute

Freq

uenc

y


Section 3.5 Histograms

Examples of Histograms:






Section 3.6 The Stem and Leaf Display

Stem and Leaf Display:

The stem and leaf display is a hybrid graphical method.

• The display is similar to a histogram, but the data remains visible.

• Useful in ordering and detecting patterns in the data.

• One of the few graphical methods in which raw data is not lost in the construction.

• As the name implies there will be a “stem” to which “leaves” will be attached in some pattern.






Consider the following data: 97, 99, 108, 110, 111.

Here we are interested in the variation of the last digit. Make a table first, then construct the stem and leaf display.

Data Value Stem Leaf

97 09 7

99 09 9

108 10 8

110 11 0

111 11 1

Stem and Leaf DisplayStem Leaves

09 7 910 811 0 1

Notice the leaves are the ones digit and the stems are the tens digit.



Example:








Example:Suppose that now we are interested in the last two digits. Let’s make the table first. Since we are looking for the last two digits we know what to put in the leaf column. Now simply put what's left (if anything) in the stem column. Now construct the stem and leaf display.

Data Value Stem Leaf

97 0 97

99 0 99

108 1 08

110 1 10

111 1 11

Stem and Leaf DisplayStem Leaves

0 97 99

1 08 10 11






Section 3.7 The Ordered Array

Ordered Array:

An ordered array is a listing of all the data in either increasing or decreasing magnitude.

• Data listed in increasing order is said to be listed in rank order.

• If listed in decreasing order, data is listed in reverse order.

• Listing the data in an ordered way can be very helpful.

By ordering the data it enables you to scan the data quickly for the largest and smallest values, for large gaps in data, and for concentrations or clusters in values.






Section 3.7 The Ordered Array

Example:

The personnel records for a clothing department store located in the mall are examined and all the current ages are noted. There are 25 employees, and their ages are all listed below.

Place the ages in rank order.

Solution:

Ages (raw)

32 21 24 19 61 18 18 16 16 35 39 17 22

21 60 18 53 18 57 63 28 20 29 35 45

Ages (ordered)

16 16 17 18 18 18 18 19 20 21 21 22 24

28 29 32 35 35 39 45 53 57 60 61 63






Section 3.8 Dot Plots

Dot Plot:

A dot plot is a graph where each of the data values is plotted as a point on the horizontal axis.

If there is a multitude of entries of the same data value, they are plotted one above the other.






Section 3.9 Plotting Time Series Data

Time Series Plot:

A time series plot graphs data using time as the horizontal axis.

Time series data can be represented in many different ways including bar graphs, line graphs, or 3-D line graphs.

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