A distribution is bimodalif it has two humps, each with the same maximum frequency.
Bimodal distributions often result when two distinct samples are represented on a single graph. For example, a graph like that shown in Figure 2.6-1(f) would result if you plotted the masculinity scores of 50 men and 50 women.
A graph with three or more humps, each with the same maximum frequency, is multimodal.
Technically, a distribution is bimodal or multimodal only if its humps have the same frequency. Nevertheless, distributions with pronounced but slightly unequal humps are commonly described as bimodal or multimodal.
J,U, and Rectangular Distributions
J and Udistributions are so named because their shapes resemble those letters.
A J-shaped curve like the one in Figure 2.6-1(g) is obtained, for example, if the probability of coming to a complete stop at a stop sign is plotted on the vertical axis
2.6 Shapes of Distributions
51
and the number of passengers in the car is plotted on the horizontal axis. A reversed
Jcurve is obtained if the number of people arriving for church is plotted on the ver- tical axis and the number of minutes that they are late is plotted on the horizontal axis. Similar results are obtained in most studies of conforming social behavior— most people conform to social conventions and laws, so fewer and fewer people exhibit larger degrees of nonconformity.
An inverted Ucurve like the one in Figure 2.6-1(h) is obtained, for example, if performance on a difficult task is plotted on the vertical axis and level of motivation of the participants is plotted on the horizontal axis.
A rectangularor uniformdistribution is one in which each class interval has the same frequency.
A rectangular distribution is produced when test scores are converted to percentiles (see Section 4.2) and the number of scores in the class intervals 0–10th percentile, 10th–20th percentile, . . . , 90th–100th percentile is graphed. It follows that the resulting graph will be rectangular because each of the 10 class intervals by defini- tion must contain 10% of the scores.
This section described some common distributions, and in the process introduced four important characteristics of distributions: (1) central tendency, (2) dispersion, (3) symmetry or lack of symmetry (skewness), and (4) kurtosis. In Chapters 3 and 4 you will learn how to compute numbers that represent each of these important characteristics.
CHECK YOUR UNDERSTANDING OF SECTION 2.6
28. Indicate whether the following statements are true or false. a. A normal distribution is symmetrical and mesokurtic.
b. If the upper half of a distribution is not the mirror image of the lower half, the distribution is asymmetrical.
c. A distribution that is more peaked than the normal distribution is called platykurtic.
d. The tail of a positively skewed distribution extends away from theXand Y
intercept.
e. A distribution with two maximum humps, each with the same frequency, is said to be multimodal.
29. Draw the shape of a frequency polygon that would occur in each of the follow- ing experiments. Identify each distribution.
a. Miss America contestants take a masculinity test.
b. An intelligence test is given to a large sample of sixth-grade children. c. Students at Curtis Institute of Music take a test of musical aptitude.
d. Students are surprised with a pop quiz immediately after the Christmas vacation.
30. Terms to remember:
a. Normal distribution b. Kurtosis c. Mesokurtic d. Platykurtic
e. Leptokurtic f. Central tendency g. Dispersion h. Symmetrical distribution i. Skewness (negative and positive) j. Bimodal
k. Multimodal l. Jdistribution
m.Udistribution n. Rectangular (uniform) distribution
2.7 MISLEADING GRAPHS
Graphs should be constructed so that they accurately portray the essential character- istics of data. Not all graphs do this—some even defy correct interpretation. Two graphs of the same data can convey entirely different impressions, as shown in Figures 2.7-1(a) and (b), which report crime statistics for three similar neighbor- hoods. In neighborhood A, cruising patrol cars were eliminated during a three- month trial period; neighborhood B had five cruising cars during the period; and C was flooded with 15 cars. Your conclusions about the effects of patrol cars would probably depend on which graph you saw. Figure 2.7-1(a) gives the impression that the presence or absence of patrol cars is associated with a dramatic difference in crime rate. Note, however, that the largest difference—1000 versus 970—is only 3%. Such a small difference could just as easily be attributed to chance factors or to differences in crime reporting procedures. The graph is misleading because it vio- lates the 66% to 75% height-width rule mentioned in Section 2.4 and because the
Yaxis begins with a frequency of 960 crimes instead of 0 crimes.7The use of such
misleading graphing procedures is contrary to the aim of statistics, which is to help the user make sense out of data.
1000 990 980 970 960 a. A No patrol cars B 5 patrol cars C 15 patrol cars Number of crimes 1000 900 800 700 600 500 400 300 200 100 0 b. A No patrol cars B 5 patrol cars C 15 patrol cars Number of crimes
Figure 2.7-1. Number of reported crimes in three similar neighborhoods during a three-month test period. Note how graph (a) gives the false impression of a great difference in crime rate across the three conditions.
7Huff (1954) and Tufte (1983) illustrate other misleading techniques and provide examples of outstand-
2.7 Misleading Graphs
53
A more subtle form of misrepresentation can occur in pictograms.
A pictogram represents quantity by presenting pictures of the objects being compared.
Pictograms are often used in the mass media in place of bar graphs and histograms to enliven a presentation. Consider Figure 2.7-2, in which sales for three brands of computers are represented by two types of pictograms. Figure 2.7-2(a) is inherently misleading because our perception of the sales of the three brands is influenced not only by the heights of the pictures but also by their areas, and area is an irrelevant dimension. For example, sales for brand C are approximately twice those for brand B, but the area of brand C’s picture is 4.3 times larger than that of brand B. The pic- togram is Figure 2.7-2(b) provides a more realistic representation of sales.
CHECK YOUR UNDERSTANDING OF SECTION 2.7
31. Prepare two bar graphs for the following data. Design one to deliberately sug- gest that government spending has been stable, the other to suggest a dramatic increase in government spending.
Month Spending Month Spending
June $29,400,000 October $29,500,000 July 29,200,000 November 29,600,000 August 29,300,000 December 29,800,000 September 29,600,000 January 30,200,000 32. Term to remember: a. Pictogram 270 240 210 180 150 120 90 60 30 0 a. A B
Brand of computer Brand of computer
30,000 units C A B C Sales in tho u sands of u nits 270240 210 180 150 120 90 60 30 0 b. Sales in tho u sands of u nits
Figure 2.7-2. Pictograms representing sales of three popular computers. Pictogram (a) is misleading because our perception of sales is influenced by the heights of the pictures and by their areas, and area is an irrelevant dimension.
2.8 LOOKING BACK: WHAT HAVE YOU LEARNED?
You have learned about two descriptive devices that make data easier to compre- hend: frequency distributions and graphs. A frequency distribution is a first and sometimes final step in summarizing data. It organizes data into a number of equiv- alence classes called class intervals and shows the number of observations that fall into each class interval. The distribution is ungrouped if each class interval is a sin- gle score value; if the classes contain two or more score values, the distribution is grouped. Grouping simplifies the interpretation of data by assigning scores to a lim- ited number of class intervals, usually between 10 and 20.
A graph is a pictorial representation of a frequency distribution and hence is eas- ier to interpret. The most common graphs for qualitative variables are bar graphs and pie charts. Histograms, frequency polygons, cumulative polygons, and stem-and- leaf displays are used to represent quantitative variables.
A graph should present data accurately, unambiguously, and in such a way that its main characteristics can be seen at a glance. To achieve this end, certain conven- tions are followed: (1) frequency is plotted on the Yaxis, and equivalence classes are plotted on the Xaxis; (2) the zero point (or origin) of the Yaxis is placed at the X
and Yintercept; (3) the height of the graph is 66% to 75% of its width; (4) the Xand
Yaxes are labeled; and (5) a figure caption is provided.