(1) (2) (3)a (4)b
Political Affiliation f Prop f % f
Democrat 92 .42 42
Independent 33 .15 15
Republican 85 .38 38
Unspecified or other 11 .05 5
n221 Sum 1.00 Sum 100
aColumn 3 is obtained by dividing each fin column 2 by n221. bColumn 4 is obtained by multiplying column 3 by 100.
5. For each of the following, give (a) the number of class intervals, (b) the size of the class interval, and (c) the nominal limits of the class interval containing the smallest score.
Largest Score Smallest Score Number of Scores
a. 68 22 53
b. 260 106 21
c. 254 92 91
6. A test of mechanical aptitude was given to seniors at Middlecenter High School. Construct a grouped frequency distribution for the following data.
80 73 51 81 46 85 84 75 44 84 77 95 48 88 50 35 52 93 43 59 63 47 66 55 58 62 51 75 86 82 89 51 77 73 59
7. In a traffic safety project, the reaction time of 27 participants to the onset of a light was measured in milliseconds. For the following data, (a) construct two grouped frequency distributions having different i’s, and (b) discuss the relative merits of the two grouping schemes.
186 187 211 185 196 193 184 185 191 188 192 190 188 190 202 199 189 193 186 180 205 187 189 195 184 198 202
8. For the data in Exercise 6, construct a relative frequency distribution using Prop f. 9. Thirty-two college students participated in a paired-associates learning experi- ment in which they were shown 12 nouns written in hiragana (a Japanese writ- ing system) and asked to learn the corresponding English words. The number of trials each participant needed to be able to correctly anticipate the 12 English words on two consecutive trials is shown here. Construct a cumulative fre- quency distribution for the data.
10 9 11 12 6 14 10 12 11 10 12 10 9 11 16 8 9 7 8 11 10 8 12 12 13 10 10 9 11 13 7 11
10. For the data in Exercise 1, construct a cumulative proportionate frequency distribution.
11. Researchers asked a random sample of 29 students from each of the following clas- sifications—freshman, sophomore, junior, senior, and graduate student—whether they believed in extrasensory perception (ESP). The classifications of students who believed in ESP are listed here. Construct a frequency distribution for these data. junior senior junior sophomore junior
2.4 Graphs for Qualitative Variables
41
sophomore senior senior senior junior graduate sophomore junior freshman senior freshman junior junior senior sophomore junior senior sophomore senior
12. Under what condition is it meaningless to construct a cumulative frequency dis- tribution for a qualitative variable?
13. Terms to remember:
a. Equivalence class b. Frequency distribution
c. Class interval d. Ungrouped frequency distribution e. Grouped frequency distribution f. Nominal limits
d. Real limits e. Class interval size f. Proportionate frequency g. Percentage frequency
h. Relative frequency distribution i. Cumulative frequency distribution
2.3 INTRODUCTION TO GRAPHS
Frequency distributions present the main features of data succinctly, but they are still abstract numerical representations and require effort to interpret. Graphs can impart the same information and speak to us more directly. Their ease of interpretation makes them particularly useful when you want to present data to the general public. There are many ways to graph data. In fact, whole books have been devoted to the subject.5My presentation is limited to the six most common graphs: bar graphs,
pie charts, histograms, frequency polygons, cumulative polygons, and stem-and-leaf displays. Qualitative variables are usually represented by bar graphs and pie charts. Quantitative variables are usually represented by histograms, frequency polygons, cumulative polygons, and stem-and-leaf displays.
2.4 GRAPHS FOR QUALITATIVE VARIABLES
Bar Graph
Once a frequency distribution has been made, most of the work of constructing a
bar graphhas been done. The only step remaining is to represent the data in a two- dimensional figure, as illustrated in Figure 2.4-1 for the data in Table. 2.2-7. Class intervals are represented along the horizontal axis (abscissa, or Xaxis), and fre- quencies are represented along the vertical axis (ordinate, or Yaxis). The zero point or origin of the vertical axis is located at the Xand Yintercept—the point where the two axes cross. A vertical bar is erected over each class interval such that its height corresponds to the number of scores in the interval. The bars can be any width, but they should not touch. A space between the bars emphasizes the discrete, qualitative character of the class intervals. By convention, the height of the graph should be 66% to 75% of its width. This results in a rectangular figure whose proportions according to the ancient Greeks are the most aesthetically pleasing. Also, the Xand
Y axes of the graph should be labeled and a figure caption provided to help the reader interpret the graph.
The Yaxis also can be used to represent proportionate frequency or percentage frequency, depending on the questions of interest to the researcher. You saw in Section 2.2 that these transformations are useful in determining whether a frequency is large in a relative rather than an absolute sense and in comparing frequency distri- butions with different total numbers of scores.
Pie Chart
Perhaps the most easily interpreted graph is a pie chart,which is merely a circle divided into sectors representing the proportionate frequency or percent- age frequency of the class intervals.
A pie chart is illustrated in Figure 2.4-2 for the data in Table 2.2-7. To construct a pie chart, think of the pie chart as a circle that has 60 minutes like the face of a clock. To determine the size of a pie sector corresponding to one of the class intervals, convert its Prop for % finto minutes. This is accomplished using the following formulas:
Prop f 60 or (% f/100) 60
For Figure 2.4-2, the minutes corresponding to the four percentage frequencies are as follows:
Democrat (42%/100) 60 25.2 min Independent (15%/100) 60 9.0 min Republican (38%/100) 60 22.8 min Unspecified or other (5%/100) 60 3.0 min
Thus, 42% corresponds to 25.2 minutes after 12 o’clock; the next 15% corresponds to 25.2 9.0 34.2 minutes after 12 o’clock; the next 38% corresponds to
100 90 80 70 60 50 40 30 20 10 0 Unspecified or other Republican Frequency Independent Democrat
Figure 2.4-1. Political affiliation of a random sample of n221 students at Ohio State University. (Data from Table 2.2-7.)
2.4 Graphs for Qualitative Variables
43
25.2 9.0 22.8 57 minutes; and the final 5% corresponds to 25.2 9.0 22.8 3.0 60 minutes or 12 o’clock. By visualizing the face of a clock, you can mark off the four pie sectors on the pie chart. The last steps in constructing the pie chart are to label the sectors and provide an appropriate figure caption.
CHECK YOUR UNDERSTANDING OF SECTION 2.4
14. College students were asked to name their favorite leisure-time activity. The five most commonly mentioned activities were rapping with friends (RF), read- ing (R), watching television (TV), participating in a sport (PS), and drinking (D). Construct a bar graph for the following data.
RF PS D RF R TV RF D PS RF RF R TV RF D TV RF TV D TV RF RF D RF R R RF R R TV D TV D D RF TV TV RF PS TV RF TV TV D D D TV RF PS RF RF D
15. A study was conducted in an Arizona nursing school to determine whether stu- dents would have a positive attitude toward research after conducting a re- search project of their own. After completing a required research course and project, students were asked to indicate which one of four statements best rep- resented their attitude. Of the 230 student nurses who responded, 31 checked the statement that said they would like to be involved in research after gradua- tion. Seventy-three checked the statement that said nurses should understand research as a part of their professional responsibility. Sixty checked the state- ment that said they felt confident in their ability to evaluate research in nursing. Sixty-six checked the statement that said the required project was responsible
Unspecified or other 5% Republican 38% Independent 15% Democrat 42%
Figure 2.4-2. Political affiliation in percentage frequency of a random sample of
for their improved understanding of the research process. Construct a bar graph for these data. (Suggested by Van Bree, Nancee S. [1981]. Undergraduate re- search. Nursing Outlook,29, 39–41.)
16. Construct a pie chart for the data in Exercise 14. 17. Construct a pie chart for the data in Exercise 15. 18. Terms to remember:
a. Bar graph b. Abscissa c. Xaxis d. Ordinate e. Yaxis f. Intercept g. Pie chart