Scatter Graphs

11 Key excerpts on "Scatter Graphs"

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  eBook - ePub

  Compassionate Statistics

  Applied Quantitative Analysis for Social Services (With exercises and instructions in SPSS)

  • Vincent Faherty(Author)
  • 2007(Publication Date)
  • SAGE Publications, Inc

    (Publisher)

  Chapter 11 , will introduce correlations used for inferential purposes.

  Scattergrams/Scatterplots

  It would be perfectly legitimate to portray the existence of a numerical correlation between two variables in a standard table, as in Table 10.1 .

  Unfortunately, the true nature of this correlation is not easily apparent in a table format. Only partially revealed in Table 10.1 is the fact that the data indicate (surprisingly) that the longer clients stayed in treatment at this agency, the worse became their attitude about that agency’s effectiveness. It is for this reason that most researchers create scattergrams, rather than tables, to present correlations used solely for descriptive purposes.

  Table 10.1   Table of the Length of Client Contact (in Weeks) and Client Attitude Toward Agency Effectiveness (10 = Very High , 1 = Very Low )

  Client #	Contact in Weeks	Attitude Toward Agency
  01	2	10
  02	4	8
  03	7	2
  04	6	3
  05	4	6
  06	2	10
  07	7	2
  08	10	1
  09	5	5
  10	7	3
  11	3	8
  12	2	8
  13	8	3
  14	9	1
  15	4	10
  16	8	2
  17	4	10

  A scattergram , also called a scatterplot, offers a clear visual image of the intersection of the values contained in the two variables. Scattergrams are conceptually based on the image of the X axis and Y axis. You are undoubtedly familiar with this image since it is commonly used as a template in economics and business administration classes to display economic trends and forecasts.

  A scattergram literally pinpoints where individual cases (usually people) are placed on a grid bounded by the possible values of the two variables being analyzed, then scatters those points, thereby forming some variation of a pattern. These possible values start at an absolute zero point, where the X axis crosses the Y axis, and then continue to increase vertically and horizontally out from that zero point, as illustrated in Figure 10.4

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  eBook - ePub

  Advanced Statistics for Physical and Occupational Therapy

  • Thomas Gus Almonroeder(Author)
  • 2022(Publication Date)
  • Routledge

    (Publisher)

  As you progress through this chapter, you’ll learn more about how to both qualitatively and quantitatively examine the relationship between variables, as well as other key information that can be gleaned from a scatter plot. However, as a primer, let’s start by simply describing some general things that can be observed from the scatter plot in Figure 8.1. First, you’ll notice that there appears to be a general pattern to the data points, as they tend to start in the bottom-left corner and move toward the top-right corner of the scatter plot. This indicates that athletes with stronger lower bodies (i.e. greater one repetition maximum weights) tended to be able to jump higher. You’ll also notice that most data points appear to be located fairly closely to the line of best fit but that very few data points lie directly on the line. Finally, you’ll see that some of the data points are located farther from the line of best fit than others. For example, the athlete whose data point is circled didn’t appear to fit the general pattern observed for the sample, as his lower body strength was slightly below average, but he was able to jump higher than almost all of the other athletes in the study.

  As you can already see, scatter plots provide a tremendous amount of information. Examining a scatter plot is often done as a preliminary step to qualitatively visualize the nature of the relationship (or lack thereof) between variables before performing a more quantitative analysis. Scatter plots can also help us determine whether our data meets some of the key assumptions associated with bivariate correlation (discussed at the end of this chapter). Fortunately, scatter plots are fairly easy to generate and are often standard output when conducting a bivariate correlation analysis using most statistical analysis software packages. Scatter plots are also commonly included as figures in research articles that report the results of a bivariate correlation analysis.

  An introduction to correlation coefficients

  While examining a scatter plot allows us to qualitatively assess the relationship between two variables, correlation coefficients help to quantitatively characterize the nature of the relationship. The Pearson product-moment correlation coefficient, which is typically represented by a lowercase, italicized r and often referred to as the ‘r value’, is commonly used to quantitatively describe the relationship between two continuous variables. Throughout this chapter, any reference to a correlation coefficient, or r

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  eBook - ePub

  Understanding Statistics

  • Bruce J. Chalmer(Author)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  11

  Describing Relationships Between Two Variables

  11.1  A scatterplot shows the shape of a relationship between two variables.

  Relationships between variables

  Before we discuss ways of describing relationships between variables, we need to consider why we should bother. The answer is the same as it was when we considered group differences on a single variable: Many scientific hypotheses can be stated in terms of the relationship between two variables. In fact, the issue of group differences can itself be considered in terms of a relationship between two variables and some of the same techniques apply, as we will see.

  What do we mean by a “relationship” between variables? To say that two variables are related means that knowledge of an individual's score on one variable changes our best guess about the individual's score on the other variable.

  For example, yield of com per hectare and amount of rainfall during the growing season are presumably related to each other. If we knew nothing else about a given farm (besides the fact that com was planted there), our best guess about the yield we might expect from that farm would simply be the average com yield. But if we were given information about the rainfall at the farm, our guess would probably be affected. If we were told that there is almost no rain at the farm, then, in the absence of irrigation, we would expect a very low yield. If we were told that there is too much rain, we would similarly expect a low yield. If we were told that the amount of rain is just right for com, we would expect a high yield.

  Figure 11.1 shows a graphical representation of this type of relationship. By drawing such a picture, we can characterize the “shape” of a relationship. In this case it is curvilinear: High com yields are associated with moderate rainfall, with lower yields for very high or very low rainfall.

  Scatterplots

  Of course, Figure 11.1 is unrealistic. Even though rainfall undoubtedly does affect com yield, it is not the only factor. Many other things matter also. Knowing the amount of rainfall would change our best guess about com yield, but we still would not be able to predict com yield perfectly

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  eBook - ePub

  Introduction to Bayesian Statistics

  • William M. Bolstad, James M. Curran(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  RMS, root mean square. It is not as affected by outliers as the variance is, but it is still quite affected. It inherits good mathematical properties and good combining properties from the variance. The standard deviation is the most widely used measure of spread. It is in the same units as mean, so its size is directly comparable to the mean.

  3.5 Displaying Relationships Between Two or More Variables

  Sometimes our data are measurements for two variables for each experimental unit. This is called bivariate data. We want to investigate the relationship between the two variables.

  Scatterplot

  The scatterplot is just a two-dimensional dotplot. Mark off the horizontal axis for the first variable, the vertical axis for the second. Each point is plotted on the graph. The shape of the “point cloud” gives us an idea as to whether the two variables are related, and if so, what is the type of relationship.

  When we have two samples of bivariate data and want to see if the relationship between the variables is similar in the two samples, we can plot the points for both samples on the same scatterplot using different symbols so we can tell them apart.

  EXAMPLE 3.3

  The Bears.mtw file stored in Minitab contains 143 measurements on wild bears that were anesthetized, measured, tagged, and released. Figure 3.11 shows a scatterplot of head length versus head width for these bears. From this we can observe that head length and head width are related. Bears with large width heads tend to have heads that are long. We can also see that male bears tend to have larger heads than female bears.

  Figure 3.11

  Head length versus head width in black bears.

  Scatterplot Matrix

  Sometimes our data consists of measurements of several variables on each experimental unit. This is called multivariate data. To investigate the relationships between the variables, form a scatterplot matrix

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  eBook - ePub

  Multivariate Analysis for the Behavioral Sciences, Second Edition

  • Kimmo Vehkalahti, Brian S. Everitt(Authors)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  When there are many variables measured on all the individuals in a study, an initial examination of all the separate pairwise scatterplots becomes difficult. For example, if 10 variables are available, there are 45 possible scatterplots. But all these scatterplots can be conveniently arranged into a scatterplot matrix that then aids in the overall comprehension and understanding of the data.

  A scatterplot matrix is defined as a square, symmetric grid of bivariate scatterplots. The grid has q rows and columns, each one corresponding to a different variable. Each of the grid’s cells shows a scatterplot of two variables. Variable j is plotted against variable i in the ij th cell, and the same variables appear in cell ji with the x - and y -axes of the scatterplots interchanged. The reason for including both the upper and lower triangles of the grid, despite the seeming redundancy, is that it enables a row and a column to be visually scanned to see one variable against all others, with the scales for the one variable lined up along the horizontal or the vertical.

  To illustrate the use of a scatterplot matrix, we shall use the data shown in Table 2.6 . These data arise from an experiment in which five different types of electrode were applied to the arms of 16 subjects and the resistance measured (in kilohms). The experiment was designed to see whether all electrode types performed similarly. The scatterplot matrix for the data is shown in Figure 2.26 ; each of the scatterplots in the diagram has been enhanced by the addition of the linear fit of the y variable on the x variable. The diagram suggests the presence of several outliers, the most extreme of which is subject 15; the reason for the two extreme readings on this subject was that he had very hairy arms. Figure 2.26

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  eBook - ePub

  R in Action, Third Edition

  Data analysis and graphics with R and Tidyverse

  • Robert I. Kabacoff(Author)
  • 2022(Publication Date)
  • Manning

    (Publisher)

  Scatter plots and scatter plot matrices allow you to visualize relationships between quantitative variables two at a time. The plots can be enhanced with linear and loess fit lines showing trends.
• When you’re creating a scatter plot based on a large volume of data, methods that plot densities rather than points are particularly useful.
• The relationships among three quantitative variables can be explored using 3D scatter plots or 2D bubble charts.
• Change over time can be described effectively with line charts.
• Large correlation matrices are difficult to understand in table form, but easily explored via corrgrams—visual plots of correlation matrices.
• The relationships between two or more categorical variables can be visualized with mosaic charts.