Statistical Graphs

6 Key excerpts on "Statistical Graphs"

• 

  eBook - ePub

  Statistical Testing Strategies in the Health Sciences

  • Albert Vexler, Alan D. Hutson, Xiwei Chen(Authors)
  • 2017(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  3 Statistical Graphics A picture is worth a thousand words.

  Barnard (1927)

  Graphs are essential to good statistical analysis.

  Anscombe (1973)

  The greatest value of a graph is when it forces us to see what we never expected.

  Tukey (1977)

  A picture may be worth a thousand words, but it may take a hundred words to do it.

  Tukey (1986)

  Visualization is critical to data analysis.

  Cleveland (1993)

  3.1    Introduction

  In this chapter, we introduce graphical statistical methods both as a powerful standalone exploratory data analysis and summary tool to complement and provide visual insight into more complex and formal statistical testing procedures. Statistical Graphs have various purposes, including but not limited to (1) providing a descriptive summary of the data, (2) providing a visualization tool to examine associations between variables, (3) providing insight for model selection based on the data under scrutiny, (4) graphically checking model assumptions about the behavior of the data as it pertains to a given statistical test, and (5) helping to determine if more complicated modeling is necessary. With increased computing speed and user-friendly graphics packages, implementation of graphical visualization as an exploratory data analysis tool is ever expanding, for example, heat maps for microarray data and various three-dimensional (3D) plots.

  For the purpose of this book, we focus on some commonly used statistical graphics displays, such as scatterplots and boxplots, as well as the implementation of probability plotting methods, including quantile–quantile (Q–Q) plots, probability–probability (P–P) plots, and modifications and hybrids of these. Heat maps commonly used in microarray data analysis are also introduced. Suggestions as to what may form the basis for the development of a general probability plotting procedure are also given. In general, graphical methods can be applied to the following areas: the comparison of samples, graphical estimates, and displays of distributions and summary measures; the presentation of results on sensitivity and specificity trade-offs of diagnostic methods; checking model assumptions; the analysis of collections of contrasts and sample variances; the assessment of multivariate contrasts; and the structuring of analysis of variance mean squares. Many of the objectives and techniques presented in this chapter are illustrated with examples. In addition, graphical methods may be useful in detecting data entry errors not easily seen when the data are in tabular form.

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  eBook - ePub

  Statistics for Exercise Science and Health with Microsoft Office Excel

  • J. P. Verma(Author)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  Chapter 3 Working with Graphs

  3.1 Introduction

  A graphical picture conveys a lot more than numerical data. It is often of great help in understanding facts at a glance. It also provides a comparison between two or more frequency distributions. A graph can be said to be a mathematical picture. It provides a way of analyzing a problem in visual terms. Many problems can be reduced to visual form and such reductions often facilitate the understanding and solving of problems. It is difficult to draw any conclusion about a player's performance by just looking at a cricket score sheet. However, if the score sheet contents are shown in terms of graphics, it conveys a lot more information in a simple format and grabs the attention of a large audience. During a live cricket match, you must have seen the vertical bars showing the runs scored per over or the hawk-eyed analysis that shows ball movement graphically. These graphics enhance the interest of the audience in the game as they convey a lot about match analysis in a simple format.

  Although graphic representation has many forms, only those forms that are useful in visualizing the important properties of frequency distribution and that facilitate comparison of two or more frequency distributions are discussed in this chapter. The latest software packages have helped students in creating different types of graphs with varied options, and therefore, it is recommended that students understand the concepts discussed in this chapter, which explains various graphical designs. Before discussing the construction of different types of graphs, the general guidelines common to most of the graphical presentation are discussed first.

  3.2 Guidelines for Constructing a Graph

  There are varieties of graphs that can be used to show the results of research data. The selection of a graph depends on the nature of the data. Irrespective of the graphic option that one may choose, there are some guidelines that are common in constructing most graphic display.

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  eBook - ePub

  Essential Mathematics and Statistics for Forensic Science

  • Craig Adam(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  This chapter will start by using examples to illustrate the variety of types of graph encountered within the forensic discipline. The main focus will be, however, on applications where an empirical or first-principles mathematical model may be found that explains the trends displayed by the experimental measurements. The basis of this technique is an understanding of the linearization of equations and the fitting of linear trend-lines to graphs. Once these topics are fully assimilated, you will be able to construct your own graphs and fit a variety of types of model to such data, including the use of calibration graphs for the chemical or spectroscopic analysis of evidence.

  5.1 Representing data using graphs

  Wherever possible, it is a good idea to display information, numerical data or experimental results using a graph. However, it is important to think carefully about the type of graph that best illustrates the implications of the data and how this graph aids your interpretation of it.

  1. Graphical representation may be used simply to display numerical information in order to con vey similarities and differences. For example, from a survey of the occurrence of the five Galton-Henry fingerprint classes in a sample of the UK population (Cummins and Midlo, 1961) it was found that some classes are far more common than others. As this form of data corresponds to a set of numbers related to particular items or classes, it is best represented by a bar chart or a pie chart (Figure 5.1 ).

  Figure 5.1 Distribution of the fingerprint classes amongst the UK population

  2. A further use of graphs is to display trends in a dataset, often over time. Here, there must be two sets of numbers such as a year and number relating to some measured quantity in that year. An important feature of this type of data is that we do not expect that there will be any correlation between the two sets of numbers, which would enable us to predict what the graph would look like if extended in either direction. For example, crime statistics that are published annually by the Home Office in England and Wales include data on the number of recorded offences per 100000 of the population. Retrospectively, these are of interest to determine trends but this data does not allow prediction of what the level of offences might be five years into the future. Either a bar chart or a line graph may be appropriate to display such data, but note here that as the data for earlier years is not annual, the bar chart gives a misleading impression as to the rate of change of these figures with time (Figure 5.2

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  eBook - ePub

  Statistical Literacy at School

  Growth and Goals

  • Jane M. Watson(Author)
  • 2013(Publication Date)
  • Routledge

    (Publisher)

  Some aspects of the cognitive demands of coordinate graphing are common to the needs of plotting algebraic functions as well as statistical representations, whereas other demands are different for the fields of algebra and statistics. Gaea Leinhardt and her colleagues provided an excellent summary in the general field in 1990, 7 whereas Fran Curcio set the stage for graphing of data 8 and continued updating valuable background information with Susan Friel and George Bright. 9 More specific research on statistical representations has focused very much on the purpose for which graphs are required to be created or interpreted. Sometimes the research has overlapped with the interests of science educators 10 and sometimes more generally with those of psychologists. 11 Lionel Pereira-Mendoza and his colleagues contributed to the understanding of how young children work with pictographs and bar graphs, 12 whereas John Ross and Bradley Cousins studied high school students struggling with graphing association and correlation. 13 Little research has focused on students’ interpretation of graphs in the media but Cliff Konold and his team have considered students’ work with stacked dot plots and how these representations assist in the discussion of variation in data sets. 14 Setting aside distinctions of terminology and details of particular graph types, the important links contributing to an understanding of data representation are shown in Fig. 3.1. Usually several types of representations could be used to illustrate a connection or connections. Keeping in mind the curriculum model suggested in Chapter 1 (cf. Fig. 1.1), data representation is often seen, as it is here, as the link between data collection, often through sampling, and data summary, often through analysis of central tendency and spread. The initial question that led to the data collection, however, may still impinge on the type of data representation selected, and hence the connection is featured

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  eBook - ePub

  Social Research Methods

  The Essentials

  • Nicholas Walliman(Author)
  • 2015(Publication Date)
  • SAGE Publications Ltd

    (Publisher)

  These are used to show correlation between two data sets. This chart type has two dependent variables: One is plotted along the x-axis, the other along the y-axis; the independent variable is the intersection of both dependent variables, realized as a data point in the diagram. It conveys an overall impression of the relationship between two variables. The way that the plots (or dots) are arranged can indicate whether there is some kind of a relationship between two variables. Statistical programs use these to display lines of regression, which show the mid-line between the points if they tend to bunch into a roughly linear fashion.

  Figure 13.8 Scatter plot charts showing different degrees of relationship

  Error bar and box and whisker charts

  Some data present variability in their measurements because they cannot be accurately measured.

  Charts with error bars are used to represent the variability of data and depict error or uncertainty in a reported measurement. They give a general idea of the range of values presented in the measurements and how far from the reported value the true (error free) value might be. Charts with box and whisker plots show the same thing but emphasize their quartiles by a box shape and lines (whiskers). Outliers (values that are exceptionally at extreme ends of the scale) may be plotted as individual points.

  Figure 13.9 Error bar chart

  Figure 13.10 Whisker and box plot

  Pictogram

  These are used to present data in a non-technical way for the general reader. Instead of bars on the charts, some form of relevant pictorial representation is used, for example, lines of cars, people, etc. It is best to use repeated images in a line to represent the amounts rather than the same image in different sizes, as the increase in size is in two dimensions making the area the measure rather than a line.

  Venn diagram

  This is an arrangement of overlapping circles, each circle representing a variable. It is used to plot how the variables relate to each other, overlapping one or more of each other to various degrees when they share properties. You can use the size of the circles to indicate the values of the variables.

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  eBook - ePub

  Basic Statistical Tools for Improving Quality

  • Chang W. Kang, Paul H. Kvam(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  numerical) data allows us to employ more powerful graphical tools and find out more about the collected information. With a numerical variable, order is important and differences in the ordered value can have spatial significance. The distance measure between the TCP recognition marks for the LCDs represents an example of numerical data. In this example, the distance measurement of 25.23 mm has a significant correspondence to the distance measurement of 25.21 mm, because both measurements are at the specification limit.

  The graphical procedures for quantitative data are featured under the Tools menu tab. We will initially focus on basic methods to graphically summarize quantitative data: bar graph, pie chart, Pareto chart, radar chart, histogram, box plot, and scatter plot. These graphical features allow you to discern how the selected data are distributed according average value, the diffuseness of the data and where observations seem to cluster.

  2.3 BAR CHART

  With data categorized into a fixed number of groups, a bar graph displays the frequency of observations corresponding to each category of some variable. To implement the bar graph, the data should be entered in two columns: one for the category headings on the horizontal axis, and the other for the category frequency on the vertical axis.

  Example 2-2: Returned Restaurant Orders

  Below are data collected by a restaurant in order to identify reasons for returned orders and use this data to implement process improvements. In a month-long study, the restaurant tabulated 101 returned food orders, and categorized the reason for returning the food into 5 major categories, with two return orders that did not fit in any of the five given categories. The data are found in the file Restaurant.ezs

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