Descriptive Statistics

Descriptive Statistics

THE HISTOGRAM

SUMMARY STATISTICS

BIBLIOGRAPHY

Descriptive statistics, which are widely used in empirical research in the social sciences, summarize certain features of the data set in a study. The data set nearly always consists of lists of numbers that describe a population. Descriptive statistics are used to summarize the information in the data using simple measures. Thus, descriptive statistics can help to represent large data sets in a simple manner. However, an incautious use of descriptive statistics can lead to a distorted picture of the data by leaving out potentially important details.

THE HISTOGRAM

Descriptive statistics take as a starting point observations from a population. So suppose we have observed n > 1 draws from a population, and let [ x ₁, …, x_n] ] denote these observations. These observations could, for example, be a survey of income levels in n individual households, in which case x ₁ would be the income level of the first household and so forth. One way of doing this is through the distribution of the data that gives a summary of the frequency of individual observations. The distribution is calculated by grouping the raw observations into categories according to ranges of values. As a simple example, Table 1 reports the distribution of a data set of income levels for 1,654 households in the United Kingdom. The data set has been grouped into five income categories. These categories represent income in U.S. dollars within the following ranges: $0–$700; $701–$1,400; $1,401–$2,100; $2,101–$2,800; and $2,801–$3,500. The second row in Table 1 shows the number of households in each income

Table 1
Distribution of weekly salaries
Weekly salary ($)	0–700	701–1400	1401–2100	2101–2800	2801–3500
SOURCE: UK Family Expenditure Survey, 1995.
Number of households	1160	429	41	17	7
Percentage of households (%)	70.13	25.94	2.48	1.03	0.42

range. The corresponding frequencies are found by dividing each cell with the number of observations; these are given in the third row.

One can also present the frequencies as a graph. This type of graph is normally referred to as a histogram. The frequencies in Table 1 are depicted as a histogram in Figure 1.

SUMMARY STATISTICS

An even more parsimonious representation of the data set can be done through summary statistics. The most typical ones are measures of the center and dispersion of the data. Other standard summary statistics are kurtosis and skewness.

The three most popular measures of the center of the distribution are the mean, median, and mode. The mean,

or average, is calculated by adding up all the observed values and dividing by the number of observations:

The median represents the middle of the set of observations when these are ordered by value. Thus, 50 percent of the observations are smaller and 50 percent are greater than the median. Finally, the mode is calculated as the most frequently occurring value in the data set.

The dispersion of the data set tells how much the observations are spread around the central tendency. Three frequently used measures of this are the variance (and its associated standard deviation), mean deviation, and range. The variance (VAR) is calculated as the sum of squared deviations from the mean, divided by the number of observations:

The standard deviation (SD) is the square-root of the variance: The mean deviation (MD) measures the average absolute deviation from the mean:

The range is calculated as the highest minus the lowest observed value. The range is very sensitive to extremely large or extremely small values, (or outliers), and it may, therefore, not always give an accurate picture of the data.

Skewness is a measure of the degree of asymmetry of the distribution relative to the center. Roughly speaking, a distribution has positive skew if most of the observations are situated to the right of the center, and a negative skew if most of the observations are situated to the left of the center. Skewness is calculated as:

Kurtosis measures the “peakedness” of the distribution. Higher kurtosis means more of the variance is due to infrequent extreme deviations. The kurtosis is calculated as:

SEE ALSO Mean, The; Mode, The; Moment Generating Function; Random Samples; Standard Deviation

BIBLIOGRAPHY

Anderson, David R., Dennis J. Sweeney, and Thomas A. Williams. 2001. Statistics for Business and Economics, 8th ed. Cincinnati, OH: South-Western Thomson Learning.

Freedman, David, Robert Pisani, and Roger Purves. 1998. Statistics, 3rd ed. New York: Norton.

Dennis Kristensen