2. The histogram. class limits class boundaries frequency cumulative frequency

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1 MA 115 Lecture 03 - Some Standard Graphs Friday, September, 017 Objectives: Introduce some standard statistical graph types. 1. Some Standard Kinds of Graphs Last week, we looked at the Frequency Distribution table. For today, we will take that same information, and put them into three kinds of graphs: the histogram, the frequency polygon, and the ogive ( O jive ). There are many other types of graphs (like pie charts), of course, but I think we can cover the main issues with these three.. The histogram In Lecture 0, we were working with 30 scores from an exam. One of the frequency distributions we discussed used groups as in the following table. class limits class boundaries frequency cumulative frequency (1) One thing that I pointed out was that scores in the 0 s and 50 s were more common than in the other classes. Observations like these are often more important than the exact numbers, so in these cases, a visual presentation can be useful. A histogram presents the information found in the class boundaries and frequency columns. In a histogram, we use rectangular blocks to represent the frequencies in each class. The height of each block tells us the frequency for that class, and the left and right edges mark the class boundaries. It follows that the width of each block is the same as the class width (in this case, the class width is ). Constructing the histogram so that there are no gaps between the blocks allows us to reconstruct the frequency distribution table from the histogram. We can read off the class boundaries and frequencies directly, and once we know these, we can figure out the class limits and cumulative frequencies. 1

2 Figure 1. A histogram for table (1)..1. Example using a different frequency distribution. Suppose we have a collection of scores from a dexterity test given to a group of people. The scores were recorded in whole seconds measuring the time it took to complete the task. The data is grouped, and the frequency distribution is presented in Figure Figure. The histogram for the dexterity test times. Scores were recorded in whole seconds between and 3 seconds. We can see from the histogram that there were scores between.5 seconds and 11.5 seconds. Since the recorded times were measured in whole seconds, we also know that these scores lie in the 9-11 second class. We can read from the histogram, therefore, class limits, class boundaries, and frequencies.

3 MA 115 Lecture 03 - Some Standard Graphs 3 3. Practice Problem Find the values of A through T in Table () below using the histogram in Figure. class limits class boundaries frequency cumulative frequency () A I O B J P C K Q D L R E G M S F H N T 3.1. Answers. A-F: -, 9-11, 1-1, 15-17, 1-0, 1-3. G,H: , I-N:,,9,,,3. O-T:,,15,3,7,30.. Frequency Polygons Frequency polygons are very similar to histograms, except instead of using bars to represent the frequency of each class, a single plotted point over the midpoint of the class is used. These points are then joined by line segments. In Figure 3, the frequency polygon for Table (1) is shown superimposed on the histogram for Table (1) Figure 3. A frequency polygon is superimposed on the histogram of Figure 1. Taking the histogram away, and marking the midpoints along the horizontal axis, we see the frequency polygon on its own in Figure. Since each point is plotted over the midpoint, we need to know what the midpoints are. These are exactly halfway between the class boundaries. One way to do this is to add the class boundaries together and divide by two.

4 The midpoint for the first class would be (3) = 5.5 Of course, the midpoint is also halfway between the class limits, so we could have found the midpoint for the first class with () = Figure. The frequency polygon shown in Figure 3 is shown here on its own. 5. Ogives Ogives ( O jives ) are similar to frequency polygons, except an ogive presents the cumulative frequencies instead of the class frequencies. Since the cumulative frequency is the total number of scores in a particular class and the classes below, it makes the most sense to plot the point at the right-hand class boundary. The cumulative frequency is then the number of scores smaller than the right-hand class boundary Figure 5. This is an ogive for the data in Table (1). Points are determined by cumulative frequency and the right-hand class boundary.

5 MA 115 Lecture 03 - Some Standard Graphs 5 Figure 5 is an ogive based on the data from the frequency distribution shown in Table (1). Note that the right-hand class boundary for the first class is 59.5, and the cumulative frequency is 7. The second class has right-hand class boundary 9.5 and cumulative frequency 11. Cumulative frequencies only get bigger as we add in the frequencies from each class, so ogives always go up, and the highest point is equal to the total number of scores in the data. The largest class is the class 0-9, and this is reflected as the steepest line segment in the ogive. It is the shape of the ogive that we care most about. At first glance, the information provided by an ogive is less clear than in the other two graphs we ve seen, but note that the steeper line segments go with the larger classes. At the very least, an ogive presents the data in a different way, and that alone might make you see something that you wouldn t have otherwise.. Homework 03 For problem 1-, a dexterity test is given to a set of subjects. The times were recorded in whole seconds, and the results are presented in the following histogram. Complete the table below (Table (5)). (A ruler might be useful in reading the histograms accurately.) Figure. The histogram for the scores on a dexterity test. Scores were recorded in whole seconds between 1 and 35 seconds. class limits class boundaries frequency cumulative frequency (5) D A E B C F G

6 1. The value for A in Table (5).. The value for B in Table (5). 3. The value for C in Table (5).. The value for D in Table (5). 5. The value for E in Table (5).. The value for F in Table (5). 7. The value for G in Table (5). For problems -, We re using the histogram from Figure to create the equivalent frequency polygon. We would plot points over the midpoints of the classes.. Find the midpoint of the class corresponding to the first bar. 9. Find the midpoint of the class corresponding to the second bar.. Find the midpoint of the class corresponding to the third bar. For problems 11-1, data is collected from a small dog club. Breeders in the club reported the number of litters they had produced. This data is presented in the ogive shown in Figure 7. (You might want your ruler again, and remember that an ogive shows cumulative frequencies.) Figure 7. This is an ogive for data collected from a small dog club. 11. How many breeders were included in the survey? 1. How many breeders produced between and 9 litters? 13. How many breeders produced between 1 and 17 litters?

7 MA 115 Lecture 03 - Some Standard Graphs 7 1. How many breeders produced between and 5 litters? Answers on next page.

8 HW Answers: 1) 1-19 ) -7 3) 3-35 ) 3 5) 9 ) 5 7) 37 (You ll need to add up all the frequencies to get this.) ) = 7 is one way to get this. 9) ) 13 11) (The ogive gives cumulative frequencies, so the last cumulative frequency is the same as the number of values in the data set.) 1) 1 (The first class increases the cumulative frequency from 0 to 1.) 13) 0 (The cumulative frequency went from 3 to 3, so there must not have been any breeders in this class.) 1)