Section 13.3 Frequency Distribution and Statistical Graphs

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:

Example 1: Frequency Distribution

Example 1: Frequency Distribution Eight families had no children, 11 families had one child, 18 families had two children, and so on. Note that the sum of the frequencies is equal to the original number of pieces of data, 64.

Rules for Data Grouped by Classes 1. 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.

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

Example 3: A Frequency Distribution of 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

Example 3: A Frequency Distribution of Family Income Construct a frequency distribution with a first class of 31.5–37.6. Solution 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

Example 3: A Frequency Distribution of Family Income Solution Class width is 37.6 – 31.5 = 6.2.

Example 3: A Frequency Distribution of Family Income Solution The modal class is 43.9–50.0. The class mark 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.

Constructing a Histogram 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.

Example 4: Construct a Histogram Solution Vertical scale: 0 – 20. Horizontal scale: 0 – 9. Bar above 0 extends to 8. Above 1, bar extends to 11. Bar above 2 extends to 18. Continue this procedure for each observed value to get the histogram on the next slide.

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 Histogram Construct a frequency polygon of the frequency distribution in Example 1, found on the next slide.

Example 5: Construct a Histogram Solution Vertical scale: 0 – 20. Horizontal scale: 0 – 9, plus one at each end. Place a mark above 0 at 8. Place a mark above 1 at 11. And so on. Connect the dots, bring the end points down to the horizontal axis.

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.

Constructing a Stem-and-Leaf Display 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

Example 8: Constructing a Stem-and-Leaf Display Solution Stem 2 3 4 5 6 7 Leaves 9 7 8 1 9 2 3 7 4 5 4 6 9 8 0 0 2 8 1 2

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 9: Circus Performances

Example 9: Circus Performances Solution To determine the number of respondents in a category, we multiply the percentage for each category, written as a decimal number, by the total number of people, 800.

Example 9: Circus Performances Solution Tigers 38% Elephants 26% Acrobats 17% Jugglers 14% Other 5% Create the table on the next slide.

Example 9: Circus Performances Solution

Example 9: Circus Performances Solution 304 people indicated tigers were their favorite performance, 208 indicated elephants, 136 people indicated the acrobats, 112 people indicated the jugglers, and 40 people indicated some other performance.