What You Will Learn Frequency Distributions Histograms Frequency Polygons Stem-and-Leaf Displays Circle Graphs

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.

Example 1: Frequency Distribution The number of children per family is recorded for 64 families surveyed. Construct a frequency distribution of the following data: Families had no children: ____ Families had one child: _____ Families had two children: _____

Rules for Data Grouped by Classes 1. The classes should be of the same “width.” 2. The classes should not overlap. Each piece of data should belong to only one class. Often suggested that there be 5 – 12 classes.

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.

Example 3: Frequency Distribution Family Income The following set of data represents the family income (in thousands of dollars, rounded to the nearest hundred) of 15 randomly selected families. 50.7 56.3 39.8 40.3 35.5 48.8 53.7 44.6 52.4 65.2 40.9 44.7 45.8 31.8 46.5 Rearrange data from lowest to highest. 65.2 52.4 46.5 44.6 39.8 56.3 50.7 45.8 40.9 35.5 53.7 48.8 44.7 40.3 31.8 Construct a frequency distribution with a first class of 31.5–37.6. Class width is 37.6 – 31.5 = 6.2.

Example 3: Frequency Distribution Family Income 65.2 52.4 46.5 44.6 39.8 56.3 50.7 45.8 40.9 35.5 53.7 48.8 44.7 40.3 31.8 First class of 31.5–37.6. Class width is 37.6 – 31.5 = 6.2. The modal class is 43.9–50.0. The class mark (midpoint) of the first class is (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.

Constructing a Histogram A 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.

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

Frequency Polygon Frequency 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. 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 Polygon When 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.

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

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.

Constructing a Stem-and-Leaf Display To construct a stem-and-leaf display each value is represented with two different groups of digits. The left group of digits is called the stem. 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.

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 56 60 62 59 58 32 47 27 50 28 71 72 44 45 44 68 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.

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.

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 93 76 68 84 54 88 77 79 80 94 92 94 80 94 66 81 67 75 73 66 87 76 45 43 56 57 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. 19

Example 4 continued 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. Class Frequency 40-49 50-59 60-69 70-79 80-89 90-99 Total: n = ? 20

Use the frequency distribution to determine: Practice HW: Use the frequency distribution to determine: The total number of observations. The width of each class The midpoint of the second class The modal class(or classes) The class limits of the next class if an additional class were to be added. 25 10 54.5 40-49, 80-89 100-109 Class Frequency 40-49 7 50-59 5 60-69 3 70-79 2 80-89 90-99 1

HW: Constructing a Stem-and-Leaf Plot Use the data below for a stem-and-leaf plot: 82 47 75 64 57 82 63 93 76 68 84 54 88 77 79 80 94 92 94 80 94 66 81 67 75 73 66 87 76 45 43 56 57 74 50 78 71 84 59 76 22