Summary
Descriptive Statistics
16: Descriptive Statistics
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Statistical analysis enables researchers to organize, interpret, and communicate numeric information. Mathematic skill is not required to grasp statistics—only logical thinking ability is needed. In this book, we underplay computation. We focus on explaining which statistics to use in different situations and on how to understand what statistical results mean.
Statistics can be descriptive or inferential. Descriptive statistics are used to describe and synthesize data (e.g., a percentage). When a percentage or other descriptive statistic is calculated from population data, it is called a parameter . A descriptive index from a sample is a statistic . Research questions are about parameters, but researchers calculate statistics to estimate them and use inferential statistics to make inferences about the population. This chapter discusses descriptive statistics, and Chapter 17 focuses on inferential statistics. First we discuss levels of measurement because the analyses that can be performed depend on how variables are measured.
LEVELS OF MEASUREMENT
Scientists have developed a system for classifying measures. The four levels of measurement are nominal, ordinal, interval, and ratio.
Nominal Measurement
The lowest level of measurement is nominal measurement , which involves assigning numbers to classify characteristics into categories. In previous chapters, we referred to nominal measurement as categorical. Examples of variables amenable to nominal measurement include gender, blood type, and marital status.
Numbers assigned in nominal measurement have no quantitative meaning. If we code males as 1 and females as 2, the number 2 does not mean “more than” 1. The numbers are only symbols representing different values of gender. We easily could use 1 for females, 2 for males.
Nominal measurement provides no information about an attribute except equivalence and nonequivalence. If we were to “measure” the gender of Nate, Alan, Mary, and Anna by assigning them the codes 1, 1, 2, and 2, respectively, this means Nate and Alan are equivalent on the gender attribute but are not equivalent to Mary and Anna.
Nominal measures must have categories that are mutually exclusive and collectively exhaustive. For example, if we were measuring marital status, we might use these codes: 1 = married, 2 = separated or divorced, 3 = widowed. Each person must be classifiable into one and only one category. The requirement for collective exhaustiveness would not be met if there were people in a sample who had never been married.
Numbers in nominal measurement cannot be treated mathematically. It is not meaningful to calculate the average gender of a sample, but we can compute percentages. In a sample of 50 patients with 30 men and 20 women, we could say that 60% were male and 40% were female.
Ordinal Measurement
Ordinal measurement involves sorting people based on their relative ranking on an attribute. This measurement level goes beyond categorization: Attributes are ordered according to some criterion. Ordinal measurement captures not only equivalence but also relative rank.
Consider this ordinal scheme for measuring ability to perform activities of daily living: (1) completely dependent, (2) needs another person’s assistance, (3) needs mechanical assistance, (4) completely independent. The numbers signify incremental ability to perform activities of daily living. People coded 4 are equivalent to each other with regard to functional ability and, relative to those in the other categories, have more of that attribute.
Ordinal measurement does not, however, tell us anything about how much greater one level is than another. We do not know if being completely independent is twice as good as needing mechanical assistance. Nor do we know if the difference between needing another person’s assistance and needing mechanical assistance is the same as that between needing mechanical assistance and being completely independent. Ordinal measurement tells us only the relative ranking of the attribute’s levels.
As with nominal measures, mathematic operations with ordinal-level data are restricted—for example, averages are usually meaningless. Frequency counts, percentages, and several other statistics to be discussed later are appropriate for ordinal-level data.
Interval Measurement
Interval measurement occurs when researchers can assume equivalent distance between rank ordering on an attribute. The Fahrenheit temperature scale is an example: A temperature of 60°F is 10°F warmer than 50°F. A 10°F difference similarly separates 40°F and 30°F, and the two differences in temperature are equivalent. Interval-level measures are more informative than ordinal ones, but interval measures do not communicate absolute magnitude. For example, we cannot say that 60°F is twice as hot as 30°F. The Fahrenheit scale uses an arbitrary zero point: Zero degrees does not signify an absence of heat. Most psychological and educational tests are assumed to yield interval-level data.
Interval scales expand analytic possibilities—in particular, interval-level data can be averaged meaningfully. It is reasonable, for example, to compute an average daily body temperature for hospital patients. Many statistical procedures require interval measurements.
Ratio Measurement
Ratio measurement is the highest measurement level. Ratio measures provide information about ordering on the critical attribute, the intervals between objects, and the absolute magnitude of the attribute because they have a rational, meaningful zero. Many physical measures provide ratio-level data. A person’s weight, for example, is measured on a ratio scale. We can say that someone who weighs 200 pounds is twice as heavy as someone who weighs 100 pounds.
Because ratio measures have an absolute zero, all arithmetic operations are permissible. Statistical procedures suitable for interval-level data are also appropriate for ratio-level data. In previous chapters, we called variables that were measured on either the interval or ratio scale as continuous.
Example of Different Measurement Levels: Grønning and colleagues (2014) tested the effect of a nurse-led education program for patients with chronic inflammatory polyarthritis. Gender (male/female) and diagnosis (psoriatic, rhematoid, or unspecified arthritis) were measured as nominal-level variables. Education (10 years, 11–12 years, 13+ years) was operationalized as an ordinal measurement in this particular study. Many outcomes (e.g., self-efficacy, coping, pain, hospital anxiety and depression) were measured on an interval-level scale. Several variables were measured on a ratio level (e.g., age, number of hospital admissions).
Comparison of the Levels
The four levels of measurement form a hierarchy, with ratio scales at the top and nominal measurement at the base. Moving from a higher to a lower level of measurement results in an information loss. For example, if we measured a woman’s weight in pounds, this would be a ratio measure. If we categorized the weights into three groups (e.g., under 125, 125 to 175, and 176+), this would be an ordinal measure. With this scheme, we would not be able to differentiate a woman who weighed 125 pounds from one who weighed 175 pounds—we have much less information with the ordinal information. This example illustrates another point: With information at one level, it is possible to convert data to a lower level, but the converse is not true. If we were given only the ordinal measurements, we could not reconstruct actual weights.
It is not always easy to identify a variable’s level of measurement. Nominal and ratio measures usually are discernible, but the distinction between ordinal and interval measures is more problematic. Some methodologists argue that most psychological measures that are treated as interval measures are really only ordinal measures. Although instruments such as Likert scales produce data that are, strictly speaking, ordinal, many analysts believe that treating them as interval measures results in too few errors to warrant using less powerful statistical procedures.
TIP: In operationalizing variables, it is best to use the highest measurement level possible because they are more powerful and precise. Sometimes, however, group membership is more informative than continuous scores, especially for clinicians who need “cut points” for making decisions. For example, for some purposes, it may be more relevant to designate infants as being of low versus normal birth weight (nominal level) than to use actual birth weight values (ratio level). But it is best to measure at the higher level and then convert to a lower level, if appropriate.
FREQUENCY DISTRIBUTIONS
When quantitative data are unanalyzed, it is not possible to discern even general trends. Consider the 60 numbers in Table 16.1 , which are fictitious scores of 60 preoperative patients on a six-item measure of anxiety—scores that we will consider as interval level. Inspection of the numbers does not help us understand patients’ anxiety.
TABLE 16.1: Patients’ Anxiety Scores
22
27
25
19
24
25
23
29
24
20
26
16
20
26
17
22
24
18
26
28
15
24
23
22
21
24
20
25
18
27
24
23
16
25
30
29
27
21
23
24
26
18
30
21
17
25
22
24
29
28
20
25
26
24
23
19
27
28
25
26
A set of data can be described in terms of three characteristics: the shape of the distribution of values, central tendency, and variability. Central tendency and variability are dealt with in subsequent sections.
Constructing Frequency Distributions
Frequency distributions are used to organize numeric data. A frequency distribution is a systematic arrangement of values from lowest to highest, together with a count of the number of times each value was obtained. Our 60 anxiety scores are shown in a frequency distribution in Table 16.2 . We can readily see the highest and lowest scores, the most common score, where the bulk of scores clustered, and how many patients were in the sample (total sample size is typically depicted as N ). None of this was apparent before the data were organized.
Frequency distributions consist of two parts: observed score values (the Xs) and the frequency of cases at each value (the f s). Scores are listed in order in one column, and corresponding frequencies are listed in another. The sum of numbers in the frequency column must equal the sample size. In less verbal terms, ∑f = N, which means the sum of (signified by Greek sigma, ∑) the frequencies (f) equals the sample size (N).
It is useful to display percentages for each value, as shown in column 3 of Table 16.2 . Just as the sum of all frequencies should equal N, the sum of all percentages should equal 100.
Frequency data can also be displayed graphically. Graphs for displaying interval- and ratio-level data include histograms and frequency polygons , which are constructed in a similar fashion. First, score values are arrayed on a horizontal dimension, with the lowest value on the left, ascending to the highest value on the right. Frequencies or percentages are displayed vertically. A histogram is constructed by drawing bars above the score classes to the height corresponding to the frequency for that score. Figure 16.1 shows a histogram for the anxiety score data. Frequency polygons are similar, but dots connected by straight lines are used to show frequencies. A dot corresponding to the frequency is placed above each score ( Figure 16.2 ).
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