The distribution of a variable shows its pattern of variation, as given by the values of the variables and their frequencies. The following data set, , data from College Board contains the mean SAT scores for each of the 50 US states and Washington D. The data patterns however are not yet clear. To get an idea of the pattern of variation of a categorical variable such as region, we can display the information with a bar graph or pie chart. This will bring up the list of categorical variables in the data set. This should place the variable Region in the Categorical Variables window. For example, place your mouse over the blue colored slice again this has to be done in Minitab not on the notes! To produce a bar graph or bar chart, return to the menu bar in Minitab and from the Graph options select Bar Chart then Simple. The steps will proceed similar from Step 3 above. In the Minitab Bar Chart, however, placing your mouse over a bar produces the number within that category. For example, if you place your mouse over the region labeled MA again this has to be done in Minitab not on the notes! For these variables we should use histograms, boxplots, or stem-and-leaf plots. Stem-and-leaf plots are sometimes referred to as stemplots. Histograms differ from bar graphs in that the represent frequencies by area and not height. A good display will help to summarize a distribution by reporting the center, spread, and shape for that variable. For now the goal is to summarize the distribution or pattern of variation of a single quantitative variable. For the math SAT scores the range is from 478 to 608. This range of 130 when divided by 10 comes to 13. For ease, Minitab goes to widths of 15. That is, count the number from 473 to 488, 489 to 504, etc. This will bring up the list of quantitative variables in the data set. This should place the variable Math SAT 2005 in the Graph Variables window. For example, if you place your mouse over the bar for 510 Minitab will display a value of 14 and bin of 502. The heights of the bars in the histogram also give you the count frequency for each class. Notice that the height for the class centered at 510 is also 14. The distribution appears to be centered near 530, meaning that of the 51 observations half would be at or below 530 and the other half would be at or above 530. The values are spread from 478 to 608 giving a range of 130. The shape appears to be somewhat skewed right or positively skewed meaning that a bulk of the data gathers on the left of the graph with the remainder of the data filling in, or trailing, to the right. The advantage of a histogram is that the construction is easy for a large data set. A disadvantage is that individual observations are not specified. For example, from the histogram you cannot tell what the mean Math SAT 2005 score is for Alabama. For a relatively small or medium size data sets, a quick graphical display that includes each observation we can construct a stem-and-leaf plot. The stem-and-leaf plot consists of a vertical list of stems after which are recorded a horizontal list of one-digit leafs. Example: We can construct a stem-and-leaf plot for the mean Math SAT 2005 scores. Now we fill in the leafs by entering the single last digit horizontally for its respective stem: 47 8 48 49 689 50 223558. This will bring up the list of quantitative variables in the data set. This should place the variable Math SAT 2005 in the Graph Variables window. When we construct these graphs by hand this column is not required, but Minitab has the machinery capability to do this quickly. The interpretation of this left-hand column is quite simple. The row with the parentheses indicates the row that contains the median observation for this data set and the number in the parentheses is the number of leafs i. Preceding this point, the numbers indicate how many observations are in that row and prior. This continues to the row containing the median observation. After the median, the interpretation continues except the number indicates how many observations are in that row and any rows following that row. If you could flip this graph so the stems are place horizontally you would see that the tail goes off to the right with the bulk of the data based to the left symbolizing a distribution shape that is skewed to the right or positively skewed. Stem-and-leaf plots can be constructed using what is called split stems. In split stems, the stems are either divided into groups of two or groups of five. The stems could also be split by creating 5 of each stem and then placing in leafs and observations of 0,1: 2,3: 4,5; 6,7; 8,9. Minitab will automatically create split stems if the data warrant a splitting. Shape The shape of a dataset is usually described as either symmetric, meaning that it is similar on both sides of the center, or skewed, meaning that the values are more spread out on one side of the center than on the other. If it is skewed to the right, the higher values toward the right on a number line are more spread out than the lower values. If it is skewed to the left, the lower values toward the left on a number line are more spread out than the higher values. A symmetric dataset may be bell-shaped or another symmetric shape. The shape is called unimodal if there is one prominent peak in the distribution, and is called bimodal if there are two prominent peaks. Figures 1, 2 and 3 show examples of three different shapes. Figure 1: A symmetric shape. Figure 2: Data skewed to the right. Figure 3: Data skewed to the left.
