What is data description in research?

Introduction

Almost all research investigations involve studying a sample of individuals randomly selected from a population with the goal of applying what is learned from the sample to all the individuals who constitute the population. A critical part of this enterprise entails organizing, summarizing, and characterizing the data collected from the sample in meaningful ways. To accomplish this aim, researchers use statistical procedures and graphing techniques. Among these techniques are frequency distributions, measures of central tendency, and measures of variability. In addition, the numbers that constitute research data have different meanings. This is reflected in the scales of measurement to which numbers adhere.

Scales of Measurement

Not all numbers are created equal. Different numbers have different meanings and thus have different characteristics. For example, the number 24 on the back of a baseball player’s jersey does not indicate that the player is twice as good as another player who wears the number 12. On the other hand, $24 does indicate twice as many dollars as $12. To differentiate these characteristics, one must understand the scale of measurement to which numbers adhere.

There are four scales of measurement. In ascending order, they are nominal, ordinal, interval, and ratio. Each scale has all the characteristics of the preceding scale plus one additional unique characteristic. Numbers that adhere to the nominal scale simply represent different categories or groups, such as the numbers 1 or 2 to indicate the gender of a research subject. The ordinal scale has the characteristic of different categories but also reflects relative magnitude or degree of measurement, such as ranking photographs from 1 to 5 based on their aesthetic qualities. Both features of separate categories and relative magnitude are reflected in the next scale, the interval scale, with the added characteristic that the distances between successive numbers on the scale are of equal interval. Temperatures on either the Celsius or Fahrenheit scale would be examples of an interval scale of measurement. Finally, numbers that adhere to the ratio scale of measurement reflect the three characteristics of the interval scale along with an absolute zero point, with a value of 0 representing the absence of the measurement. The variables, for example, of time, height, or body weight all would adhere to the ratio scale of measurement. In a research context, knowing the scale of measurement to which numbers adhere will have an impact on the type of statistical procedure used to analyze the data.

Organizing Data

At the completion of any research study, the data collected need to be organized and summarized in ways that allow the researcher to identify trends or other interesting consistencies in the results. One of the techniques for organizing and summarizing data, especially large sets of data, is the frequency distribution. Frequency distributions allow the researcher to tabulate the frequencies associated with specific response categories and also allow for the data to be summarized and characterized in a more manageable fashion. The organized frequency data are then presented in table form, with the response categories organized in ascending or descending order. Organizing the results in such a manner will facilitate making interpretations from and conclusions about the data.

Generally speaking, there are two types of frequency distribution: simple frequency distribution and grouped frequency distribution. These two types of frequency distribution are constructed identically, with one exception. The simple frequency distribution entails categorizing frequencies for each and every possible response category or score (symbolized as X), while grouped frequency distributions combine specific categories or specific scores into groups called class intervals. Grouping frequencies into class intervals has the advantage of making data sets with wide-ranging categories or scores easier to manage and thus easier to summarize. However, doing so does come with a price. By grouping categories or scores together, the researcher loses some specificity with regard to the number of frequencies associated with particular categories or scores.

Much information can be gleaned from a frequency distribution table. Apart from
listing the scores or response categories and their frequencies (symbolized as
f), frequency distributions often contain columns indicating the
percentage of the total frequency each particular frequency represents (symbolized as %
f), the cumulative frequency counts (symbolized as cum
f) and their associated percents (symbolized as % cum
f), the products of each pair of f ×
X terms (symbolized as fX), and the products of each
pair of f × X
2 terms (symbolized
fX
2). Each of these two latter columns, along with the
frequency column, is summed (indicated by the capital Greek letter Σ). These sums are
then used in calculating the values of the mean and the standard
deviation.

An example of the use of a frequency distribution can be seen in the case of a researcher interested in determining the frequencies with which heights (in inches) present in a sample of fifty subjects. (The number of subjects in a research study is indicated by N.) Each subject’s height measurement might be presented in the simple frequency distribution in figure 1.

Organizing the data in this manner allows the researcher to make sense of the
data by identifying the most frequent (65 inches) and least frequent (59 inches and 71
inches) height, along with the percentage of the total associated with each height
category, and by recording the cumulative frequencies and their associated percentages.
Moreover, by examining the values in the “f” column, the manner in
which the heights are distributed over the various categories can be easily ascertained.
In this distribution, for example, the greatest number of frequencies are associated
with height categories that lie toward the middle range of scores, while very few
frequencies are associated with heights that lie at either the upper or lower end of the
range of height scores. This type of distribution is called a bell-shaped curve or a
normal
distribution and is often a characteristic of psychological and
behavioral data.

Measures of Central Tendency

Because research data represent large sets of numbers, it is desirable, in fact necessary, to summarize these data to facilitate making sense of them. In an attempt to accomplish this goal, researchers calculate summary statistics that provide one value whose purpose is to reflect the general characteristics of the data. The most frequently used summary statistics are called measures of central tendency, and they include the mean or arithmetic average, the median or middle point of the distribution, and the mode or most frequently encountered score in the distribution.

The calculations for the mean, median, and mode are quite simple. Adding all the scores together and dividing by the number of scores in the distribution obtains the mean. In the simple frequency distribution above, the sum of all the scores is indicated by ΣfX, while the number of scores is reflected in the Σf term. Thus, the mean for this set of data is 3,224 ÷ 50, or 64.48 inches. The median or middle point of this distribution of fifty scores lies somewhere between the twenty-fifth and twenty-sixth scores. Since this point in the distribution does not have an actual score associated with it, the convention is to estimate the value of this score by averaging the twenty-fifth and twenty-sixth scores. It just so happens that the twenty-fifth and twenty-sixth scores in the above distribution both have values of 64 inches, therefore making the median (64 + 64) ÷ 2, or 64 inches. Lastly, since the mode is the most frequently exhibited score in the distribution, its identification in the above simple frequency distribution is obtained by looking down the “f” column and determining the largest frequency and its associated height score. By doing so, the mode for this distribution is determined to be 65 inches.

The measures of central tendency are not only useful for using a single value to characterize large data sets but also helpful in identifying the shape of the distribution. For example, it is known that distributions whose mean, median, and mode values are all the same (or similar) most likely are normal distributions. A distribution whose mean value is larger than its median value is most likely to be positively skewed. Positively skewed distributions are those that have the majority of their frequencies toward the lower end of the range of scores. In contrast, when the median value of a distribution exceeds the mean value, then the majority of scores fall at the upper end of the range of scores and the distribution is described as being negatively skewed.

Of the three measures of central tendency, the mean is the most frequently used. However, the use of this statistic will depend on the shape of the distribution and the scale of measurement to which the scores adhere. The mean should be used when working with either interval or ratio data and also when the distribution is approximately normal and does not contain many excessively extreme scores at either end of the distribution. The last criterion is important because the presence of extreme scores in the distribution can severely distort the value of the mean. For this reason, government statistics that summarize income or house prices, for example, typically report median values. When the mean is inappropriate to use, the median is the statistic of choice, as long as the data adhere to either the ordinal, interval, or ratio scale. The mode can be used with any scale of measurement and is typically the statistic of choice with nominal data.

Measures of Variability

Although measures of central tendency are useful statistics, they reflect only one aspect of the data. Another important feature of the data is the amount of spread or dispersion that exists among the scores. This dimension is reflected in another class of statistics called measures of variability.

The most straightforward measure of variability is the range. (It should be noted that the range is not used with nominal data.) The range sample reflects how far apart two extreme scores in the distribution are from each other. In its simplest form, the range is calculated by subtracting the least value from the greatest value. In the height data in Figure 1, the range would equal 14 inches (that is, 72 inches minus 58 inches). A variant of the range is the interquartile range, and its calculation entails taking a difference between the scores that lie at the twenty-fifth and seventy-fifth percentiles. Again, using the height data, it can be seen in the “cum f” column that the scores that lie at the twenty-fifth and seventy-fifth percentiles are 66 and 63, respectively; thus the interquatile range would equal 3 inches. Another variant of the range is called the semi-interquartile range, and its calculation is the interquartile range divided by 2. For the height data, the semi-interquartile range would be 1.5 inches. The range, or one of its variants (typically the semi-interquartile range), is used as the measure of variability when the median is used as the measure of central tendency.

The utility of the ranges as measures of variability is quite limited, since their calculations involve using only two scores from the distribution. The variance and standard deviation, on the other hand, use all the scores in their calculations and thus are better measures, but they do require that the data fit either the interval or ratio scale of measurement. The variance is obtained by applying the following formula to the data: [ΣX2 − (ΣX)2 ÷ N)] ÷ N. Thus, the variance for the height data discussed above would be equal to [208,210 − (3,2242 ÷ 50)] ÷ 50 or 6.53 inches. (Note that a mathematically equivalent formula is [Σ(X − mean)2]÷ N]). The standard deviation is simply the square root of the variance, and its value would be 2.56 inches. Of these two measures of variability, the standard deviation is the one almost always used, and it is reported when the appropriate measure of central tendency is the mean.

The variance and standard deviation are important in a number of ways. First, the variance represents a measurement that reflects the average squared dispersion between each score and the mean of the distribution. (Note that mathematically, squaring the difference between the score and the mean when calculating the variance or standard deviation is necessary to avoid always obtaining a quotient of 0.) Second, the variance can be interpreted as an estimate of the margin of error when using the mean to predict the value of a randomly selected individual’s score from the population. For example, based on the sample of fifty subjects presented in the frequency distribution above, the height of a randomly selected person from the population would be 64.48 ± 6.53 inches. Another example of a variance measure would be the margin of error that accompanies the results of most public opinion polls. Finally, the standard deviation is used in calculating a standardized score, also known as a z-score. A standardized score is equal to the squared difference between the score and the mean divided by the standard deviation (that is, z = (X − mean)2 ÷ standard deviation). Standardized scores are helpful in comparing the relative performance of scores that come from different populations and samples and are also used in determining various proportions of the population associated with different regions of the normal distributions. For example, 68.26 percent of the scores in a normal distribution will fall within ±1 z-score unit (or ±1 standard deviation unit) from the mean, while 95.44 percent will fall within ±2 z-score units (±2 standard deviation units).

Graphs

It has been said that a picture is worth a thousand words. This is also true when it comes to research data. Researchers often will present their data or summary statistics calculated from their data in graphic form. There are several ways to do this. Frequency data are often displayed via a frequency polygon, bar graph, or histogram. All three of these types of graphs plot the frequency data as a function of the score categories on a set of X,Y axes, as is illustrated in figures 2, 3, and 4.

Apart from the obvious differences in their look, these three graphs differ in another way. Frequencies associated with either the nominal or ordinal scale of measurement should be plotted using a bar graph, while data that reflect some quantifiable measurement (that is, quantitative data) can be plotted using either a frequency polygon or a histogram. It is the convention to use a frequency polygon, rather than a histogram, when there is a large range of scores to be plotted on the X-axis. Another important feature of frequency polygons is that the left- and right-hand sides of the curve are anchored to the X-axis. This is accomplished by starting the X-axis off with the score below the lowest score in the data set and ending the X-axis with the score above the highest score. There are no frequencies associated with these two X values, thus the curve is anchored to the X-axis.

Data derived from an experiment, on the other hand, are usually
plotted using a line graph. A line graph typically plots the mean of some measure
(called the dependent
variable) as a function of the variables (called the independent variables) studied in the experiment. Also included in a line graph
are T-bars that extend from the mean. The T-bars represent each plotted mean’s measure
of variability, often expressed in terms of ±1 standard deviation unit. Line graphs do,
however, require that the data be derived from either an interval or ratio scale of
measurement. Experimental data that are either ordinal or nominal in nature should be
plotted using a bar graph.

The example of a line graph provided in figure 5(see page 540) presents fictitious data for illustration purposes. The relationship expressed in this graph is the effect of number of alcoholic drinks (one independent variable) on the number of words recalled (the dependent variable) in both male and female subjects (a second independent variable). The mean values for each of the eight groups of subjects are plotted along with their standard deviations, represented by the T-bars that extend upward and downward from each point in the graph. In this set of fictitious data, it can seen that, on average, female subjects were more adversely affected by either one or two alcoholic drinks, but showed the same degree of memory impairment as male subjects when three alcoholic drinks were administered.

Inferential Statistics

All the methods thus far described are descriptive in nature; that is, they
simply describe, summarize, and illustrate the trends that exist in the data. Many other
statistical procedures exist that enable researchers to make inferences about the
population at large based on sample data. These procedures are referred to as
inferential
statistics. Essentially, the goal of all
inferential statistics is to establish, within a specific probability of
certainty, whether groups of subjects performed differently and whether these
differences are attributable to the effects of the independent variable being studied in
the investigation.

Inferential statistical tests fall into two broad categories, depending on whether two groups or more than two groups of subjects were studied. Further, each broad category is subdivided into two subcategories called parametric and nonparametric tests. Parametric tests are used when certain assumptions about the population from which the subjects were selected can be safely made (that is, the population is normally distributed and the population variances of the groups are similar), while nonparametric tests do not require that these assumptions are met. Analyzing research data with inferential statistical tests is a critical component of the scientific process that enables researchers to identify cause-and-effect relationships in nature.

Bibliography

Bruning, James L., and B. L.
Kintz. Computational Handbook of Statistics. 4th ed. New York:
Longman, 1997. Print.

Hanneman, Robert, Augustine J. Kposowa, and Mark
Riddle. Basic Statistics for Social Research. San Francisco:
Jossey-Bass, 2013. Print.

Heiman, Gary W. Basic
Statistics for the Behavioral Sciences. 5th ed. Boston: Houghton, 2006.
Print.

Keppel, Geoffrey.
Design and Analysis: A Researcher’s Handbook. 5th ed. Englewood
Cliffs: Prentice, 2007. Print.

Larson, Ron, and Elizabeth Farber.
Elementary Statistics: Picturing the World. Boston: Pearson, 2012.
Print.

Siegel, Sidney.
Nonparametric Statistics for the Behavioral Sciences. 2d ed. New
York: McGraw, 1988. Print.

Spatz, Chris. Basic
Statistics: Tales of Distributions. 9th ed. Belmont: Wadsworth, 2008.
Print.

Wheelan, Charles J. Naked Statistics:
Stripping the Dread from the Data. New York: Norton, 2013.
Print.