# Data analysis: Descriptive and inferential statistics

Susan Sullivan-Bolyai and Carol Bova

*After reading this chapter, you should be able to do the following:*

• Differentiate between descriptive and inferential statistics.

• State the purposes of descriptive statistics.

• Identify the levels of measurement in a research study.

• Describe a frequency distribution.

• List measures of central tendency and their use.

• List measures of variability and their use.

• Identify the purpose of inferential statistics.

• Explain the concept of probability as it applies to the analysis of sample data.

• Distinguish between a type I and type II error and its effect on a study’s outcome.

• Distinguish between parametric and nonparametric tests.

• List some commonly used statistical tests and their purposes.

• Critically appraise the statistics used in published research studies.

• Evaluate the strength and quality of the evidence provided by the findings of a research study and determine their applicability to practice.

*Go to Evolve at**http://evolve.elsevier.com/LoBiondo/**for review questions, critiquing exercises, and additional research articles for practice in reviewing and critiquing.*

As you read a research article you will find a discussion of the statistical procedures used in both the methods and results sections. In the methods section, you will find the planned statistical analyses. In the results section, you will find the data generated from testing the hypotheses or research questions. These data are the analyses using both descriptive and inferential **statistics**.

Procedures that allow researchers to describe and summarize data are known as **descriptive statistics**. Descriptive statistics include measures of central tendency, such as mean, median, and mode; measures of variability, such as range and standard deviation (SD); and some correlation techniques, such as scatter plots. For example, Alhusen and colleagues (2012; Appendix B) used descriptive statistics to inform the reader about the subjects’ personal characteristics (93% African American, 67% with less than a high school education) and clinical characteristics (84% initiated prenatal care by 14 weeks’ gestation, and 32% were first pregnancies).

This chapter describes how researchers use descriptive and inferential statistics in nursing research studies. This will help you to determine the appropriateness of the statistics used and to interpret the strength and quality of the reported findings, as well as the clinical significance and applicability of the research results for your evidence-based practice.

### Levels of measurement

**Measurement** is the process of assigning numbers to variables or events according to rules. Every variable in a study that is assigned a specific number must be similar to every other variable assigned that number. In other words, if you decide to assign the number 1 to represent all male subjects in a sample and the number 2 to represent all female subjects, you must use this same numbering scheme throughout your study.

The measurement level is determined by the nature of the object or event being measured. Understanding the **levels of measurement** is an important first step when you evaluate the statistical analyses used in a study. There are four levels of measurement: nominal, ordinal, interval, and ratio (Table 16-1). The level of measurement of each variable determines the type of statistic that can be used to answer a research question or test a hypothesis. The higher the level of measurement, the greater the flexibility the researcher has in choosing statistical procedures. Every attempt should be made to use the highest level of measurement possible so that the maximum amount of information will be obtained from the data. The following Critical Thinking Decision Path illustrates the relationship between levels of measurement and appropriate choice of descriptive statistics.

TABLE 16-1

LEVEL OF MEASUREMENT SUMMARY TABLE

MEASUREMENT | DESCRIPTION | MEASURES OF CENTRAL TENDENCY | MEASURES OF VARIABILITY |

Nominal | Classification | Mode | Modal percentage, range, frequency distribution |

Ordinal | Relative rankings | Mode, median | Range, percentile, frequency distribution |

Interval | Rank ordering with equal intervals | Mode, median, mean | Range, percentile, standard deviation |

Ratio | Rank ordering with equal intervals and absolute zero | Mode, median, mean | All |

**Nominal measurement** is used to classify variables or events into categories. The categories are mutually exclusive; the variable or event either has or does not have the characteristic. The numbers assigned to each category are only labels; such numbers do not indicate more or less of a characteristic. Nominal-level measurement can be used to categorize a sample on such information as gender, marital status, or religious affiliation. For example, Alhusen and colleagues (2012; Appendix B) measured marital status using a nominal level of measurement. Nominal-level measurement is the lowest level and allows for the least amount of statistical manipulation. When using nominal-level variables, typically the frequency and percent are calculated. For example, Alhusen and colleagues (2012) found that among their sample of pregnant women 54% were single, 34% were partnered, and 10% were married.

A variable at the nominal level can also be categorized as either a ** dichotomous** or

**variable. A**

*categorical***dichotomous**(nominal)

**variable**is one that has

*only two true values,*such as true/false or yes/no. For example, in the Thomas and colleagues (2012; Appendix A) study the variable gender (male/female) is dichotomous because it has only two possible values. On the other hand, nominal variables that are

**categorical**still have mutually exclusive categories but have

*more than two true values,*such as marital status in the Alhusen and colleagues study (single, partnered/not married, married, other).

**Interval measurement** shows rankings of events or variables on a scale with equal intervals between the numbers. The zero point remains arbitrary and not absolute. For example, interval measurements are used in measuring temperatures on the Fahrenheit scale. The distances between degrees are equal, but the zero point is arbitrary and does not represent the absence of temperature. Test scores also represent interval data. The differences between test scores represent equal intervals, but a zero does not represent the total absence of knowledge.

In many areas in the social sciences, including nursing, the classification of the level of measurement of scales that use Likert-type response options to measure concepts such as quality of life, depression, functional status, or social support is controversial, with some regarding these measurements as ordinal and others as interval. You need to be aware of this controversy and look at each study individually in terms of how the data are analyzed. Interval-level data allow more manipulation of data, including the addition and subtraction of numbers and the calculation of means. This additional manipulation is why many argue for classifying behavioral scale data as interval level. For example, Alhusen and colleagues (2012) used the Maternal-Fetal Attachment Scale (MFAS), which rates 24 items related to attachment on a 5-point Likert scale from 1 (definitely no) to 5 (definitely yes) with higher scores indicating greater maternal-fetal attachment. They reported the mean MFAS score as 84.1 with a standard deviation (SD) of 14.2.

### Descriptive statistics

#### Frequency distribution

One way of organizing descriptive data is by using a frequency distribution. In **a frequency distribution** the number of times each event occurs is counted. The data can also be grouped and the frequency of each group reported. Table 16-2 shows the results of an examination given to a class of 51 students. The results of the examination are reported in several ways. The columns on the left give the raw data tally and the frequency for each grade, and the columns on the right give the grouped data tally and grouped frequencies.

TABLE 16-2

INDIVIDUAL | GROUP | ||||

SCORE |
TALLY |
FREQUENCY |
SCORE |
TALLY |
FREQUENCY |

90 | | | 1 | >89 | | | 1 |

88 | | | 1 | |||

86 | | | 1 | 80-89 | ||||| ||||| ||||| | 15 |

84 | ||||| | | 6 | |||

82 | || | 2 | 70-79 | ||||| ||||| ||||| ||||| ||| | 23 |

80 | ||||| | 5 | |||

78 | ||||| | 5 | |||

76 | | | 1 | 60-69 | ||||| ||||| | 10 |

74 | ||||| || | 7 | |||

72 | ||||| |||| | 9 | <59 | || | 2 |

70 | | | 1 | |||

68 | ||| | 3 | |||

66 | || | 2 | |||

64 | |||| | 4 | |||

62 | | | 1 | |||

60 | 0 | ||||

58 | | | 1 | |||

56 | 0 | ||||

54 | | | 1 | |||

52 | 0 | ||||

50 | 0 | ||||

Total | 51 | 51 |

Mean, 73.1; standard deviation, 12.1; median, 74; mode, 72; range, 36 (54-90).

When data are grouped, it is necessary to define the size of the group or the interval width so that no score will fall into two groups and each group will be mutually exclusive. The grouping of the data in Table 16-2 prevents overlap; each score falls into only one group. The grouping should allow for a precise presentation of the data without serious loss of information.

Information about frequency distributions may be presented in the form of a table, such as Table 16-2, or in graphic form. Figure 16-1 illustrates the most common graphic forms: the histogram and the frequency polygon. The two graphic methods are similar in that both plot scores, or percentages of occurrence, against frequency. The greater the number of points plotted, the smoother the resulting graph. The shape of the resulting graph allows for observations that further describe the data.

#### Measures of central tendency

**Measures of central tendency** are used to describe the pattern of responses among a sample. Measures of central tendency include the mean, median, and mode. They yield a single number that describes the middle of the group and summarize the members of a sample. Each measure of central tendency has a specific use and is most appropriate to specific kinds of measurement and types of distributions.

The **mean** is the arithmetical average of all the scores (add all of the values in a distribution and divide by the total number of values) and is used with interval or ratio data. The mean is the most widely used measure of central tendency. Most statistical tests of significance use the mean. The mean is affected by every score and can change greatly with extreme scores, especially in studies that have a limited sample size. The mean is generally considered the single best point for summarizing data when using interval- or ratio-level data. You can find the mean in research reports by looking for the symbols *M =* or .

The **median** is the score where 50% of the scores are above it and 50% of the scores are below it. The median is not sensitive to extremes in high and low scores. It is best used when the data are skewed (see Normal Distribution in this chapter), and the researcher is interested in the “typical” score. For example, if age is a variable and there is a wide range with extreme scores that may affect the mean, it would be appropriate to also report the median. The median is easy to find either by inspection or by calculation and can be used with ordinal-, interval-, and ratio-level data.

The **mode** is the most frequent value in a distribution. The mode is determined by inspection of the frequency distribution (not by mathematical calculation). For example, in Table 16-2 the mode would be a score of 72 because nine students received this score and it represents the score that was attained by the greatest number of students. It is important to note that a sample distribution can have more than one mode. The number of modes contained in a distribution is called the **modality** of the distribution. It is also possible to have no mode when all scores in a distribution are different. The mode is most often used with nominal data but can be used with all levels of measurement. The mode cannot be used for calculations, and it is unstable; that is, the mode can fluctuate widely from sample to sample from the same population.