Data Visualization and Analysis

Central Tendency and Dispersion

Most teachers and administrators have a good idea of measures of central tendency: mean (=average), median (=50th percentile, or the score that marks the central point above which and below which half of the scores occur), and mode (most frequently occurring score). In addition, there are measures of dispersion, which are the range (lowest to highest score) and standard deviation (the average distance of all the scores from the mean). This latter statistic is important in understanding the ‘normal distribution’ or bell curve. In a normal distribution, one standard deviation above and below accounts for 34.1% of the data on each side of the mean:

Figure 3.1 The Normal Curve (c.f. Brown, 1988)

Source: https://commons. Wikimedia.org/w/index.php?curid=l 788643

Figure 3.2 Normal and Non-Normal Distributions in a Boxplot

Source-. https://c0mm0ns.wikimedia.0rg/wiki/File:N0rmality_b0x-pl0t.png

Based on this notion of distribution of scores, many other different ways of visualizing data exist that will help administrators understand the distribution of their data. One important and useful way of visualizing data is the boxplot (Larson-Hall, 2017). A boxplot is informative in that it shows the group center and the spread (dispersion). In addition, the shape of the boxplot is very useful, as the central line shows the 50th percentile: a normal distribution should have equal portions above and below the central line. The shaded part of the box shows the interquartile range from the 25th to the 75th percentiles. Boxplots also identify outliers, which are data points that are very far from the average. These scores could be truly extreme or the result of data entry error, which can be checked. The lines extend to the minimum and max of data set, with outliers being mapped separately. A boxplot is illustrated in Figure 3.2, showing the 50th percentile on the right boxplot, with many scores higher than the 50th percentile, including outliers.

Cluster Analysis

The statistics of central tendency and dispersion allow an administrator to see descriptive patterns. Cluster analysis uses a statistical procedure that goes beyond description, however (Staples & Biber, 2015).

Jain (2010) identified three main goals of cluster analysis. The first is to understand ‘underlying structure’: to gain insight into data, generate hypotheses, detect anomalies, and identify salient features. The second is ‘natural classification’: to identify the degree of similarity among forms or organisms (phylogenetic relationships). The third reason has the most potential for use in IEP contexts: this is ‘compression’, which is a method for ‘organizing the data and summarizing it through cluster prototypes’.

One of the problems that lEPs face is that learners may have strengths in one area but weaknesses in another. For example, some learners may have some oral fluency with a limited range of vocabulary and syntax but be less competent in reading and/or writing. However, the strengths and weaknesses can potentially be summed together to form a central point; this ‘summation’ is the goal of a cluster analysis. The analysis will take account of the strengths and weaknesses and will group students based on their most similar characteristics.

Most statistical software packages have a k-means (or kernel) cluster analysis function, which is the specific statistical procedure that looks at many data points from a large group and then tries to establish subgroups that are as different from each other as possible based on these multiple data points. In other words, ‘high dimensional data are transformed into lower dimensional data’ (Ding & He, 2004, p. 29). An important first step is to convert the scores to Z scores (that is, scores with a mean of 0 and standard deviation of 1.) This conversion is important because the scores must be on the same scale/range for the program to make the correct calculations. The package SPSS, which is available to most university faculties, will carry out these calculations, under the ‘Classify’ menu and choosing k-means. (In statistics, ‘k’ stands for ‘group’.) This method allows the researcher to specify the number of desired groups that reflect levels in an IEP. A second method is called the ‘two step’ cluster, which does not allow the researcher to specify the number of groups but instead provides a strictly statistical look at how the data may best be classified.