Prob and Stats, Sep 4 Variations on the Frequency Histogram Book Sections: N/A Essential Questions: What are the methods for displaying data, and how can I build them? What are variations of the frequency histograms, how can I construct and use them? What is a Pareto Chart? Standards: PS.SPID.1, PS.SPMJ.6
Variations of Frequency Graphs There are three frequency graph variants that we will construct / understand: Pareto Chart Frequency Polygon Ogive Graph
Pareto Chart A Pareto chart is a graph used represent frequencies in categories, arranged in order from highest to lowest. The horizontal axis (x-axis) of a Pareto is qualitative (non-numerical) usually represented by categories. This graph is used to show an order of frequency by category. A Pareto is a bar graph.
Why a Pareto Chart? A Pareto chart has bars that are ordered from largest to smallest in terms of frequency, so this display can help you (or a decision maker) decide which categories are the critical few and which are the insignificant many.
Example of Pareto Chart
Example 2 of a Pareto Chart
Example 3 of a Pareto Chart
Pareto Chart Tidbits Some books show a Pareto with the bars touching, but most leave an even gap between the bars to distinguish this display from a frequency histogram (whether it be by numeric classes or qualitative classes). In here, we will leave a gap by skipping a column on our graph paper between categories.
Drawing Pareto Chart Step 1 Arrange data into categories, ranking their frequency from most to least (big to small). Step 2 Associate each category with a segment on the x-axis, labeling the position and leave a segment gap between each category. Make all bars the same width. Label that axis with the category names. Step 3 Scale y-axis according to the appropriate frequency range of the categories (Label with frequency values to scale) Step 4 Draw tower heights as they pertain to each category, remembering that there is a gap between each category. Step 5 Label graph purpose
Example 1 San Franciscans were asked how they got to work each day. Seventy five responded in the following way. Create a Pareto chart of this data.: Mode Number Bart 10 Bike 7 Bus 15 Cable Car 13 Car 25 Walk 5
Example 2 Create a Pareto chart for the number of registered taxi cabs in each of the following cities: City Number Baltimore 1151 Chicago 5300 New York 11,787 Philadelphia 1480 Washington DC 8348
Frequency Polygon The Frequency Polygon is a graph that displays the data using lines that connect points plotted for the frequencies at the midpoints of the classes. The frequencies are represented by the heights of the points. The frequency polygon is a line graph.
Why a Frequency Polygon? There is no advantage of the frequency polygon over the frequency graph. It is a preference to choose one over the other.
Example of Frequency Polygon 1
Example of Frequency Polygon 2
Example of Frequency Polygon 3
Example of an Ogive Graph
Example of an Ogive Graph 50 45 40 35 30 25 20 15 10 5 0
Cumulative Frequency Cumulative Frequency A running total of the frequency at the end of an interval of that interval plus the sum of all previous intervals. It starts at 0 and ends at the value of the number of data points in the sample. To compute cumulative frequency, start at 0, then add class frequency at each class.
Computing Cumulative Frequency Given the following frequency table, compute cumulative frequencies: Class Freq CumFreq 5.5-10.5 1 10.5-15.5 2 15.5-20.5 3 20.5-25.5 5 25.5-30.5 4 30.5-35.5 3 35.5-40.5 2
Ogive Graphs A cumulative frequency graph a running total frequency is plotted at each class, with the total number of data points being the endheight. It is represented as a line graph. This graph gives the total sample size at a glance, and the places of max and min slope on this graph indicate points of greatest and least change.
Drawing an Ogive Graph Step 0 Compute a tally table of the data with classes and cumulative frequencies (low is the beginning of a class, hi is the end of a class). Step 1 Set up graph paper with each class start point on x-axis Step 2 Set up cumulative frequency scale on y-axis Step 3 Plot the (first interval low, 0) Step 4 Starting at (first class hi, cum freq), plot each class cumulative frequency Step 5 Connect dots with straight lines, last point will be (last class hi, frequency total)
Example Class Freq CumFreq 5.5-10.5 1 1 10.5-15.5 2 3 15.5-20.5 3 6 20.5-25.5 5 11 25.5-30.5 4 15 30.5-35.5 3 18 35.5-40.5 2 20
Example The average ticket prices for the top 20 money earning concerts in the US for 2014 are summarized in the following frequency table. Draw a frequency polygon, an Ogive graph, and a Pareto chart for this data: Classes ($) Frequency Cum Freq 25-49 8 50-74 7 75-99 1 100-124 2 125-149 1 150-174 1
Problems You Need to be Able to Solve Given a frequency polygon or an Ogive graph, be able to tell the class width. Given an Ogive graph, be able to determine how many data values were used to created the graph. How can I do it?
How can I do it? A simple understanding of how each graph was created and why tell the tales. The distance between plots inside both graphs are the class width. The ending y-value on an Ogive graph is the total frequency.
Frequency Polygon Class Width
Class Width and Total Frequency?
Class Width and Total Frequency? 50 45 40 35 30 25 20 15 10 5 0
Class Width and Total Frequency? 100 80 60 40 20 0
Class work: Classwork 9/4/15, 1-5 Homework: None