Week 5: 2

I’m thinking of middle school level and about 2 days duration for the following lesson.

Lesson on Graphing and Misleading Graphs

Graphing of data is often used to visually show trends in the data.  In this activity, you will learn about what makes a good graph, and what makes a graph misleading.  You will learn to identify several types of misleading graphs.  You will have an opportunity to find data on your own and use it to create a well made graph and a misleading graph.

Introduction: Some basic types of graphs
The graphs we will be working with will be of data with two variables, an independent (free) variable (often time) and a dependent variable whose value changes depends on the free variable.  By practice, the free variable is put on the x-axis increasing from left to right.  This makes it possible to visually read trends in the data by glancing from left to right.

Line Graphs
Some of the strengths of line graphs are that:
They are good at showing specific values of data: Given one variable the other can easily be determined.

They show trends in data clearly: They visibly show how one variable is affected by the other as it increases or decreases.

They enable the viewer to make predictions about the results of data not yet recorded.

Unfortunately, it is possible to alter the way a line graph appears to make data look a certain way. This is done by either not using consistent scales on the axes, meaning that the value in between each point along the axis may not be the same, or when comparing two graphs using different scales for each. It is important that we all be aware of how graphs can be made to look a certain way, when that might not be the way the data really is.

Let’s take a look at an example.  Here are two graphs of the same data, about the hourly minimum wage.  The first example is not well made, because the free variable is not scaled in a constant way.  The second graph fixes this problem

Questions:
1. What was the minimum wage in January, 1978?
2. When did the minimum wage reach $3.35?
3. Between what time periods was the largest increase in minimum wage?
4. Based on your observations of the graph, make a prediction about what the wage might be in the year 2000.
5. What about the scales used on the graph might make the data appear differently than how it really is?

These next two graphs demonstrate another problem that can occur with scaling. Can you tell what it is?  There is also second problem with this pair of graphs, do you see it?

Bar Graphs

Bar graphs are used to display data in a similar way to line graphs. However, rather than using a point on a plane to define a value, a bar graph uses a horizontal or vertical rectangular bar that levels off at the appropriate level.

Scatter Plots

Scatter plots are similar to line graphs in that they use horizontal and vertical axes to plot data points. Scatter plots are used to show how much one variable is affected by another. The relationship between two variables is called their correlation .

Scatter plots usually consist of a large body of data. The closer the data points come when plotted to making a straight line, the higher the correlation between the two variables, or the stronger the relationship.

If the data points make a straight line going from the origin out to high x- and y-values, then the variables are said to have a positive correlation . If the line goes from a high-value on the y-axis down to a high-value on the x-axis, the variables have a negative correlation.

However, perfect linear correlation rarely happens in the real world.  There is more likely to be correlation that isn’t perfect and correlation that isn’t linear. Here is an example

Unfortunately, I do not have a better copy of this graph.  What it shows is the standard deviation of batting averages of major league baseball players (plotted on the y-axis) over time (year plotted on the x-axis).  There is a clear downward trend, but the trend is a curve rather than linear.

At the other extreme from perfect correlation is no correlation:

Pie Charts

Pie charts, or circle graphs as they are sometimes known, are very different from the other three types of graphs that we’ve looked at. They don’t use a set of axes to plot points.Pie charts  display percentages. Therefore, they are used to compare different parts of something.

The circle of a pie graph represents 100%. Each portion that takes up space within the circle stands for a part of that 100%. In this way, it is possible to see how something is divided among different groups.

Good graphing practice, and mistakes to watch out for!

Use the appropriate graph for the appropriate purpose.

Trend graphs. If you wish to emphasize the trend in a time series, a line chart is better than a series of side-by-side bars.

Relative size graphs. Here side-by-side bar graphs are best, but all bars must be anchored at zero. All bars should be equal width, otherwise, readers of the graph will be confused by differences in area, rather than difference in lengths of the bars.

Make sure that the graph is complete. All axes must be labeled. There should be a title on the graph.

Think about the overall presentation of the graph. The points on a plot should be spread over the area of the graph without being shoved into one corner. The axes scales should be appropriate.

Where is the 0 point on a graph. In particular, bar charts should always be anchored at zero. Use different plotting symbols or line-types to differentiate among groups on the graph. The independent variables is usually plotted on the X-axis; the dependent variable usually on the Y-axis.

The best graph is one that is self-explanatory!

There are many common errors that are made in poor graphs. Here are some of the most common errors:

Wrong graph type. Think about what you want to present. Trends are best displayed using lines. Compositions best displayed using segmented-bar-charts.

Missing text. All tick-marks and axes must be labeled. The graph needs a title.

Inconsistent scale. The scale must be constant across the graph; don’t change the increments between tick marks.. Most people read increasing scales from left to right and from bottom to top. Comparative graphs must be plotted on the same axes to facilitate comparisons.

Misplaced zero point. Most people assume that the zero point is at the bottom of the graph. This can give a very misleading impression of the amount of change present in a data series.

Overloading your graph with too much information.  You want just what is necessary to clearly see a trend.

Confusing of area and length. If you make a picture twice as large, it looks as if it has four times the area!.

No adjustment for inflation. Dollar amounts must be adjusted for inflation. Otherwise, any comparison is misleading.

Quiz Activity
Check your understanding about creating graphs.  Try the following worksheets and quizzes:
http://www.bbc.co.uk/schools/gcsebitesize/maths/data/representingdata2rev5.shtml
http://go.hrw.com/resources/go_sc/hst/HSTSW281.PDF
http://www.ms.uky.edu/algebracubed/lessons/Graphs.pdf

What’s wrong with the following graph?

Here’s a site with some great examples of good and bad graphs:
http://www.math.yorku.ca/SCS/Gallery.

Graphing Activity
Create two sets of misleading and well made graphs with data you discover on your own.  Possibilities for topics include: temperature and global warming, employment and unemployment, per capita income, U. S. casualties in Iraq or Afghanistan, Popularity of a political figure, Average age at the onset of puberty, or any other topic you choose.  Make sure to reference the source for your data.

References
http://mste.illinois.edu/courses/ci330ms/youtsey/intro.html
http://www.math.sfu.ca/~cschwarz/Stat-301/Handouts/node7.html
http://www.bbc.co.uk/schools/gcsebitesize/maths/data/representingdata2rev5.shtml
http://go.hrw.com/resources/go_sc/hst/HSTSW281.PDF
http://www.ms.uky.edu/algebracubed/lessons/Graphs.pdf
http://www.math.yorku.ca/SCS/Gallery

5 Responses to Week 5: 2

  1. I will be using this for reference when I next teach “How to lie with statistics.” Thank you detailed resources!

    I know it already was a lot of work, with research and making graph illustrations, but there was a piece I would add: some examples of non-linear correlations. “School math” always uses linear fit to stand for the notion of correlation, and I find it unfortunate. As we used to joke about the resulting undergraduate math “rules” when I taught college calculus: “Every function is linear!” Here is a reference to a non-linear correlation measure.

  2. esivel says:

    Thanks for the feedback.
    I probably bit off more than I should have as far as the extent of material covered with this lesson, so I had cut some corners…
    I have updated the part on correlation.

    • Neat graph of baseball players! Actually, poor quality media (or media copyrighted too viciously) is an excellent invitation for students to create their own graphs and release them freely for everybody to use.

  3. peterhorn33 says:

    Wow, another excellent task. I am at a loss as to how to improve it. My only question is would it be possible or even feasible to break this up into multiple lessons?

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