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Chapter 1: Picturing Distributions with Graphs

Knowledge Prerequisites

none

Learning Goals

1. declarative knowledge (definitions)
   1. individuals
   2. categorical variable
   3. quantitative variable
   4. distribution
   5. exploratory data analysis
   6. pie chart
   7. bar graph
   8. histogram
   9. class width
   10. shape, center and spread of a distribution
   11. outlier of a distribution
   12. skewness of a distribution
   13. stem and leaf of a stemplot
   14. split stems
   15. time plot
2. procedural knowledge
   1. how to construct a pie chart from given data
   2. how to construct a bar graph from given data
   3. how to construct a histogram from given data
   4. how to construct a stemplot from given data
   5. how to construct a timeplot from given data
3. conditional knowledge
   1. be able to distinguish between types of data.
   2. know which graph is most appropriate for given data.
   3. know how to analyze a histogram or stemplot:
      1. describe overall pattern and any deviations (outliers)
      2. describe shape, center and spread

Questions to Ponder

1. What is the best way to study/prepare for the concepts in this section? Are there are other methods to study these concepts?
2. What questions do you need to ask the instructor?

Purpose of this Section

Chapter 1 considers single variable statistics (i.e., one measured aspect of an individual).

This section includes:

• distribution of a variable - describes what values the variable takes and how often it takes these values.
• exploratory data analysis - the use of graphs and numerical summaries to describe the variables in a data set and the relations among them.

This information will be very important for later Chapters

General Notes

For several good description of statistics, see the resources page: miscellaneous topics.

Strategies for exploratory data analysis:

1. Examine each variable individually, then examine relationships among them.
   1. Graph the data.
      1. Types:
         1. Pie charts (for categorial/qualitative variables)
            1. Here is a previous class separated into groups by zip code (Note that the area of each slice is proportional to the percentage of people in each zip code):
               1. jpeg or MS excel
         2. Bar Graphs (for categorial/qualitative variables)
            1. Here is a previous class separated into groups by zip code (
               1. Note that the height (or area) of each bar represents the number of people in each zip code:
                  1. jpeg or MS excel
               2. Or the height (or area) of each bar can represent the percentage of people in each zip code:
                  1. jpeg or MS excel
         3. Histograms (for quantitative variables)
            1. Formal steps to construct a histogram by hand:
               1. Divide the range of values of the variable into classes of equal width: $\frac{(\max -\min )}{\#\mathrm{of}\mathrm{classes}}$
               2. Count the number of observations in each class
               3. Make a histogram
            2. Luckily, the TI83 constructs histograms in a more efficient manner (see ti83hist.htm).
            3. NOTE: choose classes wisely. The class width affects the graph of the data. To see this effect, here is a java demonstration of how class width affects a histogram.
            4. Percentages are preferred for large data sets.
         4. Stemplots (for quantitative variables)
            1. Stemplots work best for small numbers of observations.
            2. Steps in constructing a stemplot:
               1. Separate each observation into stem and leafs.
               2. Write the stems in a vertical column with smallest value at top. Do not skip numbers in the stem.
               3. Write each leaf in a row to the right of the stem in increasing order.
            3. Luckily, the STEMPLOT program on the TI83 constructs stemplots in a more efficient manner (see ti83stem.htm).
            4. You can split the stems (i.e., each stem appears twice). The upper stem will have leafs 0-4 and the lower stem will have leafs 5-9. For an example, see the top of page 17 of your text.
            5. Back-to-back stemplots are useful when we wish to compare two related distributions.
         5. Time plots (when data is measured over time - they can reveal trends)
            1. Luckily, the TI83 constructs timeplots (see ti83time.htm).
   2. Preliminary examination of data sets:
      1. What kind of individuals?
      2. How many variables are present?
      3. What is the exact definition of the variable? What type? In what units? How is thevariable measured?
   3. Numerical summaries of specific aspects of the data.
      1. description of the shape of the distribution (from one of the graphs above)
      2. identification of possible outliers (from histogram) or actual outliers (from modified boxplot to be discussed in Chapter 2).
      3. identify appropriate measure of center
      4. identify appropriate measure of spread
      5. Is there a trend? (time plots only)

Skewed distributions

1. Skewed to right: if the right side of the distribution extends further out than the left side.
2. Skewed to left: if the left side of the distribution extends further out than the right side.

Steps to determine the skewness of a distribution.

NOTE: when constructing graphs, make sure to appropriately label the graphs.

Resources

• Histogram Applet - http://www.amstat.org/publications/jse/v6n3/applets/Histogram.html - This applet is designed to teach students how bin widths (i.e., class widths) affect a histogram.
• Histogram - http://www.shodor.org/interactivate/activities/histogram/index.html - This activity allows the user to view and make their own histograms.
• Histogram Explorer - http://www.ratrat.com/histogram_explorer/he.html - Histogram Explorer is a tool for understanding histograms and their statistics.

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