Data description in research

• TYPE OF PSYCHOLOGY: Psychological methodologies

Data description refers to how the results from research studies are organized, summarized, and characterized statistically.

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 facilitates 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 regarding 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 (cum f) and their associated percents (% cum f), the products of each pair of f × x terms (fx), and the products of each pair of f × x² terms (fx²). 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 a simple frequency distribution table. Organizing the data in this manner allows the researcher to make sense of the data by identifying the most frequent and least frequent heights, 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 a typical height distribution, the greatest number of frequencies are usually 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 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 center, or 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 simple. Adding all the scores together and dividing by the number of scores in the distribution obtains the mean. The median or middle point of a distribution of fifty scores would lie 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. Lastly, since the mode is the most frequently exhibited score in the distribution, it can be identified in a simple frequency distribution by merely looking down the “f” column and determining the largest frequency and its associated score.

The measures of central tendency are useful not only for using a single value to characterize large data sets, but also for helping identify the shape of the distribution. For example, it is known that distributions whose mean, median, and mode values are all the same or similar are most likely 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 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 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 highest value. A variant of the range is the interquartile range, the calculation of which entails taking a difference between the scores that lie at the twenty-fifth and seventy-fifth percentiles. Another variant is called the semi-interquartile range, which is the interquartile range divided by 2. 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 range as measures of variability is quite limited, since its calculation involves 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. A mathematically equivalent formula is [Σ(x − mean)²] ÷ N]. The standard deviation is simply the square root of the variance. Of these two measures of variability, the standard deviation is almost always the one 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. (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. One 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. They 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 will often 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, a bar graph, or a 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.

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.

On the other hand, data derived from an experiment involving dependent and independent variables are usually plotted using a line graph. A line graph typically plots the mean of the dependent variable as a function of the independent variables being 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.

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 known 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 these assumptions to be 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.

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