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Section 2.3: Additional Displays of Quantitative Data

Objectives

By the end of this section, you will be able to...

  1. construct frequency polygons*
  2. create cumulative frequency and relative frequency tables
  3. construct ogives*
  4. draw time-series graphs

* You will not be tested on these objectives.

For a quick overview of this section, feel free to watch this short video summary:

In addition to histograms, stem-and-leaf plots, and dot plots, there are some other, section common plots. We'll introduce a couple in this section. The first type, frequency polygons, are not a type of plot that will be expected of you on exams, though you will be asked questions about them on homework.

Frequency Polygons

A frequency polygon is drawn by plotting a point above each class midpoint and connecting the points with a straight line. (Class midpoints are found by average successive lower class limits.)

Example 1

To illustrate the idea, let's look at the average commute data from the last section.

average commute midpoint frequency relative frequency
16-17.9 17 1 1/15 ≈ 0.07
18-19.9 19 2 2/15 ≈ 0.13
20-21.9 21 1 1/15 ≈ 0.07
22-23.9 23 6 6/15 = 0.40
24-25.9 25 2 2/15 ≈ 0.13
26-27.9 27 1 1/15 ≈ 0.07
28-29.9 29 1 1/15 ≈ 0.07
30-31.9 31 1 1/15 ≈ 0.07

The three images below show the relationship between the histogram and the frequency polygon.

frequency polygon

frequency polygon

frequency polygon

Note: No technology section this time, since you won't be asked to do this for exams.

Cumulative Tables

Cumulative tables are just what they imply - they show the sum of values up to and including that particular category. As with regular tables, we can have both cumulative frequency and relative frequency.

Example 2

To illustrate the idea, let's look at the average commute data from the last section.

average commute frequency cumulative frequency
16-17.9 1 1
18-19.9 2 3
20-21.9 1 4
22-23.9 6 10
24-25.9 2 12
26-27.9 1 13
28-29.9 1 14
30-31.9 1 15
average commute relative
frequency
cumulative
relative
frequency
16-17.9 1/15 ≈ 0.07 1/15 ≈ 0.07
18-19.9 2/15 ≈ 0.13 3/15 ≈ 0.20
20-21.9 1/15 ≈ 0.07 4/15 ≈ 0.27
22-23.9 6/15 = 0.40 10/15 ≈ 0.67
24-25.9 2/15 ≈ 0.13 12/15 = 0.80
26-27.9 1/15 ≈ 0.07 13/15 ≈ 0.87
28-29.9 1/15 ≈ 0.07 14/15 ≈ 0.93
30-31.9 1/15 ≈ 0.07 15/15 = 1.00

Technology

  1. Enter or import the data.
  2. Select Stat > Tables > Frequency.
  3. Select the column(s) you want to summarize.
  4. Select Cumulative frequency or Cumulative relative frequency as the Statistic(s).
  5. Click Compute.

Creating cumulative tables from a frequency table.

Unfortunately, there is no easy way to create cumulative tables in StatCrunch. You actually need to write a custom function to do this.

  1. Go to Data > Compute > Expression.
  2. Enter cumsum([column name]) (Where [column name] is the column where the frequencies or relative frequencies are stored.)
  3. If desired, enter a Column label.
  4. Click Compute.

Ogives

Ogives are pretty funky graphs, and rarely used except in specific areas. We'll just give a quick example here, but like frequency polygons, you won't be expected to create these on an exam. (Though it may come up in homework.)

An ogive (read as "oh jive") is a graph that represents the cumulative frequency or cumulative relative frequency for the class. It is constructed by plotting points - the x-coordinates are the upper class limits and the y-coordinate is the corresponding cumulative frequency or cumulative relative frequency.

Example 3

To illustrate the idea, let's again use the average commute data from the last section.

average commute relative
frequency
cumulative
relative
frequency
16-17.9 1/15 ≈ 0.07 1/15 ≈ 0.07
18-19.9 2/15 ≈ 0.13 3/15 ≈ 0.20
20-21.9 1/15 ≈ 0.07 4/15 ≈ 0.27
22-23.9 6/15 = 0.40 10/15 ≈ 0.67
24-25.9 2/15 ≈ 0.13 12/15 = 0.80
26-27.9 1/15 ≈ 0.07 13/15 ≈ 0.87
28-29.9 1/15 ≈ 0.07 14/15 ≈ 0.93
30-31.9 1/15 ≈ 0.07 15/15 = 1.00

ogive

Note: No technology section this time, since you won't be asked to do this for exams.

 

time series

Time-Series Graphs

Time series graphs are much more common than the last couple times we've looked at. It's common to see stock prices and daily temperature graphs in the news - both are time series plots.

A time series plot is obtained by plotting the time in which a variable is measured on the horizontal axis and the corresponding value of the variable on the vertical axis.

The example above is from the Chicago Tribune and reflects the price of uranium from 2001-2006.

Example 4

Here's another example, using the daily high temperature in Elgin, IL, for the month of June, 2008.

date daily high
temperature
6/1 80
6/2 86
6/3 72
6/4 81
6/5 89
6/6 89
6/7 86
6/8 85
6/9 73
6/10 80
6/11 84
6/12 91
6/13 82
6/14 84
6/15 81
6/16 72
6/17 77
6/18 78
6/19 81
6/20 85
6/21 82
6/22 81
6/23 78
6/24 81
6/25 80
6/26 85
6/27 82
6/28 83
6/29 75
6/30 81

And the time series plot would look something like this:

time series

Technology

Here's a quick overview of how to create a time series plot in StatCrunch.

  1. Enter or import the data.
  2. Select Graphics > Index Plot
  3. Select the column(s) you want to plot and click Next.
  4. Set any desired options and click Create Graph!

 

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