Frequency Distributions and Graphs

Organizing Data 1st: Data has to be collected in some form of study. When the data is collected in its’ original form it is called raw data. 2 nd : Simply looking at raw data usually does not show an abundant amount of information so we will construct a frequency distribution. 3 rd : Depending on the style of data collected there are different styles of frequency distributions.

Categorical Distributions Used for data that can be placed into categories. Examples ( Political Affiliation, Blood type, Eye Color) Since data is categorical discrete classes can be used. Normal categorical distributions are created with 4 columns ( Class, Tally, Frequency, and Percent) What’s your favorite pizza?

Ungrouped Frequency Distributions Discrete Data Consists of numerical data. Normally used when the range of the data is smaller. Range: Highest value – lowest value. Most frequency distributions should only have anywhere from 5 to 20 classes. Normally contain 3 columns. But may contain 4. The 3 columns are class, tally, and frequency. Ever play Black Jack?

Ungrouped Continuous Data Since numerical data can be discrete or continuous in nature their frequency distributions are somewhat different. When dealing with continuous data we need to add in 2 other columns. They are the class boundaries column and the cumulative frequency column. Boundaries are created using addition/subt. Cumulative freq. is created through the addition of previous frequencies. The order therefore becomes (class, class boundaries, tally, frequency, cumulative freq.)

Grouped Frequency Distributions Again consists of numerical data. This time the range of the data can be rather large so data must be grouped together. Most columns stay exactly the same except the first becomes the Class Limit column. Lower class limit: The lower number in a class limit. Upper class limit: The higher number in a class limit. Lower Boundary: The lower number in a class boundary. Upper Boundary: The higher number in a class boundary.

Rules of thumb for grouped dist. #1: There should be between 5 and 20 classes. #2: The class width should be an odd number. Class width: created by taking the range of the data and dividing by # of classes desired. If a remainder occurs ALWAYS round up to next whole number. Doing this will help to ensure that the midpoint of each class has the same place value as the data. Class midpoint: obtained by adding the lower and upper boundaries/limits and dividing by 2. Midpoint are going to used later in Chapters 2 and 3. #3: Classes should be mutually exclusive. Used so that the data can not be placed into 2 classes. #4: Classes must be continuous. Even if there are no data values in a class it must be included. #5: Classes must be exhaustive. There should be enough for all of the data. #6: Classes must be equal in width. Ex.10-15, 16-21, 22-27

Example Create a grouped freq. distribution from the following set of data. Pretend data represents # of minutes a senior spent doing homework over the past week. 114, 125, 95, 110, 143, 105, 99, 138, 87, 109, 128, 121, 137, 140, 129, 101, 90, 135, 125, 80, 141, 118, 120, 100, 97

Graphs Purpose of a graph, in statistics, is to present the data to a viewer in pictorial form. Usually easier for people to understand the meaning of the data graphically than in tables or freq. distributions. 3 most commonly used graph in research are: The histogram The frequency polygon The cumulative freq. graph or Ogive

Histogram Displays data by using vertical bars of various heights to represent the frequencies. Uses class boundaries to create each bar. No space in between bars unless there is a boundary that has no data in it. Going to create histogram of data from example.

Frequency polygon Displays the data by using lines that connect points plotted for the frequencies at the midpoints of the classes. The frequencies represent the height of the midpoints. Place the midpoints, of the boundaries, along x- axis and frequencies along the y-axis. When all points are plotted need to make sure that the beginning and end of the graph are attached to the x-axis. Create a freq. polygon of the data from the example.

Cumulative freq. graph (Ogive) Cumulative freq.: The sum of the frequencies accumulated up to the upper boundary of a class in the distribution. Ogive: Graph that represents the cumulative freq. for the classes in a freq. distribution. Place the first lower boundary and all other upper boundaries along the x-axis. This line graph will start on the x-axis at the lower boundary of the first class. Create an Ogive for the data from the example.

Other Graphs Along with the 3 graphs just discussed there are several others often used. The Pareto Chart Time Series Graph Pie Graph

Pareto Chart Used to represent a freq. distribution for a categorical variable, and the frequencies are displayed by the heights of the vertical bars. The bars DO get attached to one another. Major difference between histogram and Pareto Chart is that in a Pareto Chart the data goes from the highest frequency to the lowest frequency. The qualitative variables are placed along the x- axis and frequency along the y-axis.

Pareto Example Use the table below to create a Pareto Chart of the data.

Time Series Graph Represents data that occurs over a specific period of time. The following data represents the high temps. in Somerville from Sept. 1st – Sept 11 th of this year. 90, 92, 94, 84, 84, 90, 95, 99, 94, 77, 83. Create a time series graph from the data.

Pie Graph A circle that is divided into sections or wedges according to the percentage of frequencies in each category of the distribution. How is each section created? Create a pie graph for the following data: