2. Dr. Serhat Eren 1 3.3.1.3 Other Uses for Bar Charts Bar charts are used to display data for different categories where the data are some kind of quantitative measure for each category. For example, a bar chart could be used to display and compare sales revenues for a sample of different software products. In this case, the y axis of the chart would represent the value of the variable being studied. 3.3 GRAPHICAL DISPLAYS OF DATA

3. Dr. Serhat Eren 2 3.3.1.4 Creating a Pie Chart for Qualitative Data A Pie chart represents data in the form of slices or sections of a circle. Each slice represents a category and the size of the slice is proportional to the relative frequency of the category 3.3 GRAPHICAL DISPLAYS OF DATA

4. Dr. Serhat Eren 3 3.3.2 Graphical Displays for quantitative Data The tool used to display quantitative data is called a histogram. A histogram is very similar to a bar chart, but since numbers are naturally ordered, the x axis of the graph must be scaled to reflect this. There are slight differences for dealing with integer and continuous data. 3.3 GRAPHICAL DISPLAYS OF DATA

5. Dr. Serhat Eren 4 3.3.2.1 Histograms for Integer Data As in a bar chart, you use a rectangle to represent each possible data value, with the height of the bar corresponding to the frequency or relative frequency for that value. The x axis must accommodate all of the possible values, whether or not there were any observations of the value. The rectangles are centered on the data values as in a bar chart, but the bars are contiguous; that is, they touch each other. 3.3 GRAPHICAL DISPLAYS OF DATA

6. Dr. Serhat Eren 5 A histogram for continuous data differs from the one for integer data in that each rectangle represents a class interval, which is a range of values. For this reason, the rectangles are not centered on values, but begin and end at each of the class boundaries. 3.3 GRAPHICAL DISPLAYS OF DATA

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9. 8 3.3.3 Displaying Small Data Sets The rules for creating histograms are really not suitable for data sets with less than 25 observations. Often when we collect data we do not have more than 25 observations. Is there a way to display these types of data sets graphically? The answer is a graphical method called a dot-plot. 3.3 GRAPHICAL DISPLAYS OF DATA

10. Dr. Serhat Eren 9 In a dot-plot, each observation is plotted as a point on a single, horizontal axis. The axis is scaled so that each of the data points can be located uniquely on the axis. When there is more than one observation with the same value the points are “stacked” on top of each other. 3.3 GRAPHICAL DISPLAYS OF DATA

11. Dr. Serhat Eren 10 3.4 DESCRIBING AND COMPARING DATA 3.4.1 Describing Quantitative Data In statistics, the features of interest for a set of numerical data can be classified as center, shape, and variability. The center of a set of data describes where, numerically, the data are centered or concentrated. The shape of a set of data describes how the data are spread out around the center with respect to the symmetry or skewness of the data.

12. Dr. Serhat Eren 11 The variability of a set of data describes how the data are spread out around the center with respect to the smoothness and magnitude of the variation. Together these three features describe the distribution of the data. To describe the data it is useful to picture the distribution of the data as being represented by a smooth curve that captures the "shape" of the histogram, Figure 3.5. 3.4 DESCRIBING AND COMPARING DATA

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14. Dr. Serhat Eren 13 If you smooth out the plotted line you get a curve that looks like the one shown in Figure 3.6. 3.4 DESCRIBING AND COMPARING DATA

15. Dr. Serhat Eren 14 The shape of a distribution concerns how the data are spread out on either side of the center, that is, whether they are symmetric or skewed. When data are evenly spread out on both sides of the center, we describe the distribution of the data as symmetric. 3.4 DESCRIBING AND COMPARING DATA

16. Dr. Serhat Eren 15 When the data are not evenly spread out on either side of the center then we refer to the distribution as being skewed. Skewness has a direction associated with it, either left (negative) or right (positive). The direction of the skew describes the side on which the distribution of the data covers a larger distance, that is, the direction in which the distribution "tails off" more slowly. Figure 3.8 shows both right and left skewed distributions. 3.4 DESCRIBING AND COMPARING DATA

17. Dr. Serhat Eren 16 When data are skewed, either left or right, the tailing off of the data is continuous and gradual as shown in Figure 3.9a. When the tailing off involves a gap in the data, a place where classes in the frequency histogram have no observations, as shown in Figure 3.9b, the data are not really skewed. More likely the data you have contain some extreme or unusual observations. 3.4 DESCRIBING AND COMPARING DATA

18. Dr. Serhat Eren 17 In addition to center and shape we would like to describe how much the data differ from the center or typical values. At this point we can describe the variability of the distribution in two ways: –the "smoothness" of the curve and –the total spread of the data 3.4 DESCRIBING AND COMPARING DATA

19. Dr. Serhat Eren 18 When data are not very variable, the frequency of observations decreases steadily as you move away from the center. Sometimes when data are highly variable, the distribution will be jagged. Figures 3.10 and 3.11 show histograms with different degrees of variability. 3.4 DESCRIBING AND COMPARING DATA

20. Dr. Serhat Eren 19 3.4.2 Comparing Data Distributions One of the major reasons for doing statistical analyses of data is to obtain facts for making informed decisions. As a result, we often need to make comparisons between or among samples taken from different populations. When comparing different data sets, we make those comparisons based on the qualities of the data you just learned: center, shape, and variability. 3.4 DESCRIBING AND COMPARING DATA