Scales of Measurement and Data Display
Spotlight 2.1 Rensis Likert
2.6 Introduction to Microsoft ® Excel and SPSS ®
The curves drawn to illustrate the concepts of skewness and kurtosis do not exhaust the myriad of ways in which scores can be patterned. A distribution can be shaped like aJ, a box, aU, or even anM; in fact, distributions can assume practically any shape. And these shapes matter. In Chapters 3 and 4 we will learn that different statistical concepts developed to describe data sets should be used for different shaped distributions.
and format. Information about the specific formatting expectations of the APA, including the use of graphs and tables, can be found in their publication manual and on various Internet sites.
40 35 30 25 20 15 10 5
Freshman Sophomore Junior Senior Graduate Figure 2.15 A table and graph generated using Microsoft Excel.
8 7 6
Count 5
4 3 2 1
0 5 10 15
Need for achievement
20 25 30
Figure 2.16 A histogram generated using the Chart Builder function in SPSS. The data comes from Table 2.2.
2.6 Introduction to Microsoft®Excel and SPSS® 63
Summary
Careful measurement is a necessity for conducting scientific research. This chapter introduces us to the 4 most useful scales of measurement for social and behavioral scientists. A nominal scale merely distinguishes one attribute, event, or thing from another. There is no quantity reflected in nominal num- bers, and any number can be assigned to any event as long as it is currently unas- signed. Variables measured on a nominal scale are qualitative.
Ordinal measures also categorize things, but in quantitative relation to one another. An ordinal scale is used to identify the relative position of an attribute, event, or object in comparison with others in terms of more or less. The concept of“ranking”is helpful when considering the ordinal scale.
More specific descriptions of quantity are found in the interval scale. Here all intervals between units on the scale are held constant. The amount of quantity needed to go from a five to a six on an interval scale is the same amount needed for a one-unit movement at any other position on the scale. This additional fea- ture is necessary for many statistical techniques since it allows averages to be found.
A ratio scale possesses all of the properties of an interval scale with the addi- tion of an absolute zero point. This allows ratio statements to be made. For example, on a ratio scale, 5 is actually half of 10 and 20 is one-third of 60.
A discontinuous or discrete variable is one that typically increments from one whole number to another whole number. Discrete variables are characterized by gaps between numbers that cannot be filled by any number. A continuous var- iable, on the other hand, does not have gaps between adjacent numbers. The upper and lower boundaries for a unit interval of a continuous variable are called the real limits. The upper real limit of the number is one-half the unit of measurement above the number, and the lower real limit of a number is one-half the unit of measurement below the number.
Using tables and graphs to organize data allows us to view a summary of the raw scores of the study. A simple frequency distribution lists all the possible scores and the frequency with which each score appears. A grouped frequency distribution indicates the number of scores that fall within each of several inter- vals. Class intervals are sets of equal-sized ranges used to organize data in grouped frequency distributions. A cumulative frequency distribution includes a column that shows the accumulation of the number of scores for a given inter- val as well as all of the preceding intervals.
The typical graph has two axes. The horizontal axis is called theXaxis or the abscissa. The vertical axis is called theYaxis or ordinate. Larger positive num- bers are to the right on the abscissa and upward on the ordinate.
A frequency polygon uses the midpoint to plot the number of scores in each of the intervals of a frequency distribution. A histogram represents the frequency of scores using the real limits of class intervals to create bars with common
borders. A bar graph is used to represent the frequency of scores associated with categories. A bar graph looks like a histogram except that the bars do not share a common border. Viewers of graphs should take note of the scaling on the axes to avoid misunderstanding the presented data.
Statisticians use terms to describe important features of the shape of distribu- tions. A normal curve is a symmetrical, bell-shaped line escalating gradually at first and then more sharply, inflecting at some point and then tapering to a peak.
A distribution that has scores that bunch at one end of the distribution is skewed. A positively skewed distribution has scores that bunch at the lower end of the distribution with an elongated tail pointing toward larger positive numbers. A negatively skewed distribution has scores that group around the upper end of the distribution with an elongated tail pointing toward smaller or negative numbers. Kurtosis is a term that refers to the quality of the peak of a distribution. A leptokurtic distribution features a narrow width and accen- tuated peak, while a platykurtic distribution features a wide width and muted peak.
Microsoft Excel and SPSS are two programs students may have access to when learning about statistics. Tables and graphs can be constructed within both of these programs. While sophisticated, both programs are easy to use with the help of tutorials. Most behavioral and social science manuscripts require APA formatting of figures and tables. The APA Pub- lication Manual is a recommended resource for all behavioral and social science students.
Key Terms
Measurement Cumulative frequency distribution
Nominal scale Abscissa
Ordinal scale Ordinate
Interval scale Frequency polygon
Ratio scale Histogram
Discontinuous (discrete) variable Bar graph
Continuous variable Normal distribution (or normal curve)
Midpoint Skewed distribution
Real limits Positively skewed distribution Raw (original) scores Negatively skewed distribution Simple frequency distribution Kurtosis
Grouped frequency distribution Leptokurtic
Class intervals Platykurtic
Key Terms 65
Questions and Exercises
1 Indicate whether each of the following scales of measurement are nominal, ordinal, interval, or ratio.
a Amount of change in attitudes.
b Attitudes toward nuclear disarmament (for/against).
c Ratings of popularity.
d Amount of time tolerating a painful stimulus.
e Heart rate under stress.
f A measure of need for approval.
g Amount of weight lifted.
h Numbers assigned based on political affiliation.
i Numbers assigned to different types of diagnostic categories.
j Mood disorder versus no mood disorder.
k A listing of tennis players from best to worst.
2 Think of different ways to measure the following concepts using as many different scales as possible.
a Academic proficiency b Athletic prowess c Creativity
d Daily food consumption e Size of extended family
3 For each class interval, specify the width, the real limits, and the midpoint.
a 1–3 b 5–10 c –4––8 d −2–+2 e 1.50–3.50 f 25–50
4 The data in the following table are from a midterm examination. Set up fre- quency distributions with:
a i= 1 (simple frequency distribution) b i =3
c i= 10 d i =20
e Includecum fcolumns for each one
Midterm examination scores
40 98 63 90 70 60 45 43 78
67 56 54 78 87 43 90 81 81
77 80 79 80 81 66 75 88 84
49 63 78 79 80 92 89 84 77
5 For each of the frequency distributions in Problem 4, specify the real limits of each class interval.
6 Construct histograms for each of the frequency distributions of Prob- lem 4.
7 Based on the histograms of Problem 6, draw frequency polygons.
8 Think of two variables that may be normally distributed; defend the rationale.
9 Think of two variables that may be negatively skewed; defend the rationale.
10 Think of two variables that may be positively skewed; defend the rationale.
11 The amount of sugar per serving in breakfast cereal might be misrepre- sented on the side of a cereal box. Draw two different bar graphs, both using the data in the table below: one graph faithfully representing the rela- tionship between the cereals and the other misrepresenting the relation- ship in such a way as to suggest Cereal A is far superior to these other brands in terms of sugar content.
Cereal type A B C
Sugar in grams/serving 12 14 15
Computer Work
12 Use a software package to establish simple, grouped, and cumulative fre- quency distributions for the following numbers. Also, generate a graphic of
Questions and Exercises 67
the following numbers. If the program allows for various graphics, repre- sent the data in each graphic form (e.g. polygon, histogram, etc.).
15 12 13 14 10 15 30 12 17 15 15 30
16 17 28 19 22 25 10 19 32 11 22 32
14 43 32 20 25 29 19 18 29 10 18 39
30 35 19 29 47 25 25 45 16 75 60 25
74 55 18 70 50 20 40 50 45 60 40 62
62 89 61 72 90 65 85 80 60 45 22 49
35 18 49 25 30 59 50 78 35 60 75 39
60 70 25 53 74 74 43 74 72 70 90 75
75 99 77 75 89 60 67 80 80 64 77 82
68 85 80 63 82 75 48 34 16 17 22 25