Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Frequency Distributions...

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

What You Will Learn

Frequency DistributionsHistogramsFrequency PolygonsStem-and-Leaf DisplaysCircle Graphs

13.3-1


Frequency Distribution

A piece of data is a single response to an experiment.

A frequency distribution is a listing of observed values and the corresponding frequency of occurrence of each value.

13.3-2


Example 1: Frequency DistributionThe number of children per family is recorded for 64 families surveyed. Construct a frequency distribution of the following data:

13.3-3

Families had no children: ____

Families had one child: _____

Families had two children: _____


Rules for Data Grouped by Classes1. The classes should be of the same

“width.”2. The classes should not overlap.3. Each piece of data should belong to

only one class.

Often suggested that there be 5 – 12 classes.

13.3-4


Definitions

The Modal class of frequency distribution is the class with the greatest frequency.

Midpoint of a class (class mark) is found by adding the lower and upper class limits and dividing the sum by 2.

Classes

Lower class limits

0 4

5 9

10 14

15 19

20 24

25 29

Upper class limits

13.3-5


Example 3: Frequency Distribution Family IncomeThe following set of data represents the family income (in thousands of dollars, rounded to the nearest hundred) of 15 randomly selected families.

50.756.339.840.335.5

48.853.744.652.465.2

40.944.745.831.846.5

13.3-6

Construct a frequency

distribution with a

first class of 31.5–37.6.

Rearrange data from lowest

to highest.

65.252.446.544.639.8

56.350.745.840.935.5

53.748.844.740.331.8

Class width is 37.6 – 31.5 = 6.2.


13.3-7

Example 3: Frequency Distribution Family Income

65.252.446.544.639.8

56.350.745.840.935.5

53.748.844.740.331.8 First class of 31.5–37.6.

Class width is 37.6 – 31.5 = 6.2.

The modal class is

The class mark (midpoint) of the first class is

43.9–50.0.

(31.5 + 37.6)÷2 = 34.55.


Histograms

A histogram is a graph with observed values on its horizontal scale and frequencies on its vertical scale.

Because histograms and other bar graphs are easy to interpret visually, they are used a great deal in newspapers and magazines.

13.3-8


Constructing a HistogramA bar is constructed above each observed value (or class when classes are used), indicating the frequency of that value (or class).

The horizontal scale need not start at zero, and the calibrations on the horizontal and vertical scales do not have to be the same.

The vertical scale must start at zero. To accommodate large frequencies on the vertical scale, it may be necessary to break the scale.

13.3-9


Example 4: Construct a HistogramThe frequency distribution developed in Example 1 is shown on the next slide. Construct a histogram of this frequency distribution.

13.3-10


Frequency PolygonFrequency polygons are line graphs with scales the same as those of the histogram; that is, the horizontal scale indicates observed values and the vertical scale indicates frequency.

13.3-11

Constructing a Frequency Polygon

Place a dot at the corresponding frequency above each of the observed values.

Then connect the dots with straight-line segments.


Constructing a Frequency PolygonWhen constructing frequency polygons, always put in two additional class marks, one at the lower end and one at the upper end on the horizontal scale.

Since the frequency at these added class marks is 0, the end points of the frequency polygon will always be on the horizontal scale.

13.3-12


Example 5: Construct a Frequency PolygonConstruct a frequency polygon of the frequency distribution in Example 1, found on the next slide.

13.3-13


Stem-and-Leaf Display

A stem-and-leaf display is a tool that organizes and groups the data while allowing us to see the actual values that make up the data.

13.3-14


Constructing a Stem-and-Leaf Display1. To construct a stem-and-leaf display each value is

represented with two different groups of digits.

2. The left group of digits is called the stem.

3. The remaining group of digits on the right is called the leaf.

• There is no rule for the number of digits to be included in the stem.

• Usually the units digit is the leaf and the remaining digits are the stem.

13.3-15


Example 8: Constructing a Stem-and-Leaf Display

The table below indicates the ages of a sample of 20 guests who stayed at Captain Fairfield Inn Bed and Breakfast. Construct a stem-and-leaf display.

29 31 39 43 5660 62 59 58 3247 27 50 28 7172 44 45 44 68

13.3-16

Stem Leaves


Circle Graphs

Circle graphs (also known as pie charts) are often used to compare parts of one or more components of the whole to the whole.

13.3-17


Example 9: Circus Performances

Eight hundred people who attended a Ringling Bros. and Barnum & Bailey Circus were asked to indicate their favorite performance. The circle graph shows the percentage of respondents that answered tigers, elephants, acrobats, jugglers, and other. Determine the number of respondents for each category.

13.3-18


Example: Constructing a Grouped Frequency Distribution

Here are the statistics test scores for a class of 40 students:

82 47 75 64 57 82 63 9376 68 84 54 88 77 79 8094 92 94 80 94 66 81 6775 73 66 87 76 45 43 5657 74 50 78 71 84 59 76

Group the frequencies into classes that are meaningful for the data.

Since letter grades are given based on 10-point ranges, use the classes 40−49, 50−59, 60−69, 70−79, 80−89, 90−99.

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Example 4 continuedClass Frequency40-4950-5960-6970-7980-8990-99

Total: n = ?

The class 40–49 has 40 as the lower class limit and 49 as the upper class limit.The class width is 10. It is sometimes helpful to vary the width of the first or last class to allow for items that fall above or below most data.

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Practice HW: Use the frequency distribution to determine:

a) The total number of observations.b) The width of each classc) The midpoint of the second classd) The modal class(or classes)e) The class limits of the next class if an

additional class were to be added.

13.3-21

Class Frequency

40-49 7

50-59 5

60-69 3

70-79 2

80-89 7

90-99 1

2510

54.540-49, 80-89

100-109


HW: Constructing a Stem-and-Leaf Plot

Use the data below for a stem-and-leaf plot:

82 47 75 64 57 82 63 9376 68 84 54 88 77 79 8094 92 94 80 94 66 81 6775 73 66 87 76 45 43 5657 74 50 78 71 84 59 76

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Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Frequency Distributions...

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