Descriptive Statistics

Description of the data collected in research is an important component for both the researcher and the reader. In both quantitative and qualitative analysis, the reduction of a large amount of data to an easily digestible summary is an important function. In qualitative research, descriptive statistics are typically observed in mixed method, action research, or other qualitative designs. More important, description lays the foundation for later analyses and interpretation of collected data.

When numerical data are collected, the description of these data is termed descriptive statistics. Descriptive statistics constitute a mathematical summarization of the data where a large number of observed values are mathematically converted to a few numbers. This is a variable-oriented approach where typically a large number of cases are involved versus a case-oriented approach where typically a few cases are involved. In qualitative research, descriptive statistics allow researchers to provide another context, a richer picture or enhanced representation, in which to examine the phenomenon of interest. The inclusion of quantitative data can also enhance legitimacy (e.g., validity, credibility, trustworthiness, transferability), although this might not be appropriate for many qualitative projects.

The simplest ways to categorize descriptive statistics are (a) numerical, such as measures of central tendency and variability; and (b) graphical, such as histograms, bar charts, and scatter plots. Descriptive statistics are different from inferential statistics, where the purpose is to infer from the sample to the population of interest.

To make meaningful inferences, descriptive statistics must be used properly, and that begins with understanding when to use each quantitative descriptive technique. Common descriptive statistics in multi-method studies are the three measures of central tendency: mean (x −, M), median, and mode. The three measures of central tendency provide a set of values that describe the typical score in a distribution of scores. The measures of central tendency are calculated from [Page 210]continuous data (e.g., test scores) and not categorical scores (e.g., gender identification). For example, in educational research a common variable is an achievement score such as reading comprehension, and in gambling research a common variable is the speed of play on a slot machine.

Figure 1 Histogram of Reading Achievement Scores

The mean, or average value, for the reading comprehension example is the summation of all scores divided by the number of test scores. The median is the middle score of all the ordered achievement scores. The mode is the most common, or highest frequency, achievement score. There can be more than one mode in a data set.

Measures of variability (i.e., score dispersion) are typically reported for continuous data and include the range, variance, and standard deviation of the scores. The range for a set of scores, or the distribution data, is calculated by subtracting the largest score from the smallest score. From the example of students' reading comprehension scores, this is the highest test score minus the lowest test score. The standard deviation (SD) is the average distance that scores are from the mean. The more dispersed the values, the larger the standard deviation. In a normally distributed data set (i.e., looks like a bell curve), 68% of the values will be within 1 standard deviation above or below the mean. The standard deviation is more commonly provided because it is easily interpreted, whereas the variance simply indicates that variability in the observed scores exists. If the variance, and therefore the standard deviation, is zero, then all of the scores are the same. A fourth, albeit less commonly provided, dispersion descriptor is the interquartile range. The interquartile range is the distance between the 25th and 75th percentiles and indicates where the middle 50% of the values are located.

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