M110 SECTION 14.1 ORGANIZING & VISUALIZING DATA For many years, the word statistics referred to numerical information about state or political territories. The word itself comes from the Latin statisticus, meaning “of the state”. We now live in an information age and the study of statistics is more important than ever before. In today’s world, much of statistics involves making sense of data. Visual illustrations are an important way to depict information from statistics. These visual illustrations are simply pictures that display data – which may then tell us a story about the data. In this section, we will be exploring the following visual depictions of information: Stem and Leaf Plots Frequency Tables Grouped Frequency Tables Histograms Bar Graphs I. Frequency Table A frequency table shows the number of times a certain piece of data occurs. Let’s make one which describes the number of siblings we have: Number of Siblings Tally Frequency Relative Frequency 0 1 2 3 4 5 6 TOTALII. Grouped Frequency Tables Whenever you have a large set of data, you can put your measurements into groups, called classes or intervals. The following classes were formed from the president data: 40-49, 50-59, 60-69, 70-79, 80-89, and 90-99. What is the size or width of each class? Use these classes and the presidential death data to fill in the following grouped frequency table. Ages at Death Tally Frequency Relative Frequency 40 - 49 Total In the above example, the groups were already calculated for you. However, when you collect your own data you would have to do this yourself. Next is a systematic way of doing so.President Age at Death President Age at Death President Age at Death Washington Adams Jefferson Madison Monroe Adams Jackson Van Buren Harrison Tyler Polk Taylor 67 90 83 85 73 80 78 79 68 71 53 65 Fillmore Pierce Buchanan Lincoln Johnson Grant Hayes Garfield Arthur Cleveland Harrison McKinley 74 64 77 56 66 63 70 49 57 71 67 58 Roosevelt Taft Wilson Harding Coolidge Hoover Roosevelt Truman Eisenhower Kennedy Johnson Nixon 60 72 67 57 60 90 63 88 78 46 64 81SETTING UP A GROUPED FREQUENCY TABLE To set up a grouped frequency table for a given set of data, sometimes you will be given the exact number and size of intervals to use and other times you will need to decide this for yourself. The following are instructions on creating a grouped frequency table so that it will accommodate all of your measurements: EXAMPLE: Construct a grouped frequency table for these 20 test scores, using the lowest score to start the first class (interval) and making 5 classes or uniform width. 31 30 23 27 19 26 28 38 17 29 26 34 21 23 23 22 12 26 39 25 1. Decide on the INTERVAL OR CLASS SIZE a. Subtract the smallest data point from the largest data point ________________________________ b. Divide this number by the number of classes or intervals you plan to have in your table ________________________________ c. Round this UP to a convenient value. This is your INTERVAL SIZE. ________________________________ 2. Start your first interval with either your smallest data point or some other convenient value (that will, of course, include your smallest value). 3. Add the interval size to this first value. This will be the lower limit for the SECOND INTERVAL! Continue this process until you have the desired number of intervals. 4. Fill in the UPPER LIMITS for each interval.III. Histogram We can look at a graphical representation of this data as well using a histogram. Using the same intervals as in the above frequency table, let’s create a histogram. (We use a squiggle to indicate that part of the scale has been omitted; therefore the scale is not accurate at this area.) IV. Bar Graph A Bar Graph is just like a histogram, but typically has spaces between the bars. Let’s make a Bar Graph with our sibling data. (YOU CAN USE EITHER THE FREQUENCIES OR THE RELATIVE FREQUENCIES FOR THE VERTICAL AXIS) Frequency PercentV. Double Bar Graph This can be used to make comparisons between two sets of data. Use the presidential data, but divide the group into early presidents (the first 18) and later presidents (the last 18). Our double bar graph will be comparing the death ages between the early and late presidents. VI. Stem and Leaf Plot To construct a stem and leaf plot for the president data, let’s assign the ten’s digit as the “stems” and the one’s digits as the “leaves”. Now let’s make an Ordered Stem and Leaf (Just arrange the leaves least to greatest) Ages of Presidents at Death 4 | 5 | 6 | 7 | 8 | 9 | If you turn a stem and leaf plot sideways, you can see how a histogram is formed. The advantage that the stem and leaf has over the histogram is that you can actually see each data point whereas on the histogram you can only see the number of occurrences of each outcome. EARLY PRESIDENTS Age at Death Tally Frequency 40-49 50-59 60-69 70-79 80-89 90-99 Total LATER PRESIDENTS Age at Death Tally Frequency 40-49 50-59 60-69 70-79 80-89 90-99 TotalA Back-to-Back Stem and Leaf (Use the frequency tables about the early and late presidents) Advantages of a Stem and Leaf Plot?