Describing and Summarizing Data: Shapes, Plots, and Graphical Integrity

Describing Data

Introduction

Describing data is a fundamental aspect of statistics, allowing us to summarize, visualize, and interpret information collected from observations or experiments. Effective data description involves understanding the distribution, central tendency, and variability of the data, as well as presenting it in clear graphical formats.

Shapes of Distributions

Types of Distributions

• Uniform Distribution: The frequency of each value of the variable is evenly spread out across the values. All outcomes are equally likely. Example: Rolling a fair die, where each face (1-6) has equal probability.

• Bell-Shaped Distribution (Symmetric): The highest frequency occurs in the middle, and frequencies tail off to the left and right. This is characteristic of the normal distribution. Example: Heights of adult humans.

• Skewed Right Distribution: The tail to the right of the peak is longer than the tail to the left. Most data are concentrated on the left. Example: Income distribution, where most people earn less and a few earn much more.

• Skewed Left Distribution: The tail to the left of the peak is longer than the tail to the right. Most data are concentrated on the right. Example: Age at retirement, where most retire at older ages but a few retire very young.

• Bimodal Distribution: The data have two distinct peaks. Example: Test scores from two groups of students with different preparation levels.

Note: Qualitative data are not described as skewed, uniform, or bell-shaped. Identifying the shape of a distribution can be subjective and may require flexibility, as real data may not perfectly match these categories.

Graphical Methods for Summarizing Data

Dot Plots

A dot plot is constructed by placing each observation horizontally in increasing order and placing a dot above the observation each time it is observed. Dot plots are useful for small datasets and for visualizing the frequency of individual values.

• Example: Number of customers arriving at a restaurant during a given period.

Stem-and-Leaf Plots

A stem-and-leaf plot uses the digits to the left of the rightmost digit to form the stem, and each rightmost digit forms a leaf. Stems are arranged vertically, and leaves are listed in rows next to their stems. This method allows for quick visualization and recreation of the original data.

• Example: Test scores: 15, 12, 7, 22, 24, 25, 27, 31, 39

Stem	Leaf
0	1 3 4 5 5 7
2	2 3 9
3	1 9

Additional info: Stem-and-leaf plots were popular between 1940-1990 due to their compatibility with monospaced fonts and early computer technology. Today, they are less common but still useful for data recreation and visualization.

Frequency Polygons

A frequency polygon is a graph that uses class midpoints (the average of consecutive lower class limits) connected by line segments to represent the frequencies for the classes.

• Formula for class midpoint: $\text{Class Midpoint} = \frac{\text{Lower Class Limit} + \text{Upper Class Limit}}{2}$

Class (X)	Count
80-89	3
90-99	4
100-109	14
110-119	17
120-129	11
130-139	9
140-149	2

Cumulative Frequency Tables

A cumulative frequency distribution displays the aggregate frequency of the category, showing the total number of observations less than or equal to the category for discrete data, or less than or equal to the upper class limit for continuous data.

A cumulative relative frequency distribution displays the proportion (or percentage) of observations less than or equal to the category or upper class limit.

Class Interval	Frequency	Cumulative Frequency	Cumulative Relative Frequency
80-89	3	3	0.05
90-99	4	7	0.117
100-109	14	21	0.35
110-119	17	38	0.633
120-129	11	49	0.817
130-139	9	58	0.967
140-149	2	60	1.0

Ogives

An ogive is a graph that represents the cumulative frequency or cumulative relative frequency for the class. Points are plotted with x-coordinates as the upper class limits and y-coordinates as the cumulative frequencies or cumulative relative frequencies. Line segments connect the points, and an additional segment connects the first point to the horizontal axis at the upper limit of the preceding class (if it existed).

Time Series Data

Definition and Plotting

Time series data are values of a variable measured at different points in time. A time-series plot is created by plotting time on the horizontal axis and the corresponding variable value on the vertical axis, connecting the points with line segments.

• Example: The Partisan Conflict Index (PCI) tracks political disagreement in the U.S. federal government, measured monthly.

Year	PCI
2004	76.87
2005	78.11
2006	78.67
2007	79.82
2008	80.67
2009	86.22
2010	108.67
2011	137.67
2012	134.67
2013	137.67
2014	133.67
2015	133.67
2016	163.67
2017	137.67
2018	133.67
2019	133.67
2020	133.67
2021	133.67
2022	133.67

Interpretation: Time-series plots help identify trends, cycles, and anomalies in data over time.

Misrepresentations of Data

Common Issues in Graphical Representation

• Incorrect Summation: The sum of percentages (relative frequencies) should equal 1 for a complete distribution. Misleading graphs may misrepresent this, as in the summer burglaries example.

• Scale Manipulation: Not starting the vertical axis at zero can exaggerate differences between groups, as seen in the tax rate example.

• Three-Dimensional Effects: 3D pie charts can distort the perceived size of categories, making some appear larger than they are.

• Truncated Scales: When the smallest observed value is large, truncating the scale can help focus on trends but must be clearly indicated to avoid misleading the reader.

Guidelines for Effective Graphs

• Title and Label Axes: Clearly label axes, include units, and provide data sources when appropriate.

• Avoid Distortion: Never misrepresent or lie about the data.

• Minimize White Space: Use available space to highlight data, and indicate any truncated scales.

• Avoid Clutter: Remove unnecessary gridlines, backgrounds, or pictures.

• Avoid Three Dimensions: Use two-dimensional graphs for clarity and accuracy.

• Consistent Design: Do not mix multiple designs in one graphic.

• Provide Data and Scales: Avoid graphs that lack data or scales, as they can mislead the reader.

Summary Table: Types of Graphs and Their Uses

Graph Type	Main Purpose	Best For
Dot Plot	Shows frequency of individual values	Small datasets
Stem-and-Leaf Plot	Displays data and preserves original values	Small to moderate datasets
Frequency Polygon	Shows distribution shape using midpoints	Continuous data
Cumulative Frequency Table	Shows running total or proportion	Discrete and continuous data
Ogive	Displays cumulative frequency visually	Continuous data
Time-Series Plot	Shows trends over time	Time-dependent data
Bar/Pie Chart	Compares categories	Qualitative/categorical data

Conclusion

Organizing and summarizing data through graphical methods is essential for effective statistical analysis. Understanding the shapes of distributions, choosing appropriate plots, and adhering to guidelines for graphical integrity ensures that data is communicated accurately and meaningfully.