Descriptive Statistics

This page describes graphical and pictorial methods of descriptive statistics and the three most common measures of descriptive statistics (central tendency, dispersion, and association).

Descriptive statistics can be useful for two purposes: 1) to provide basic information about variables in a dataset and 2) to highlight potential relationships between variables. The three most common descriptive statistics can be displayed graphically or pictorially and are measures of:

• Graphical/Pictorial Methods

• Measures of Central Tendency

• Measures of Dispersion

• Measures of Association

There are several graphical and pictorial methods that enhance researchers' understanding of individual variables and the relationships between variables. Graphical and pictorial methods provide a visual representation of the data. Some of these methods include:

• Histograms

• Scatter plots

• Geographical Information Systems (GIS)

• Sociograms

Histograms

• Visually represent the frequencies with which values of variables occur

• Each value of a variable is displayed along the bottom of a histogram, and a bar is drawn for each value

• The height of the bar corresponds to the frequency with which that value occurs

Scatter plots

• Display the relationship between two quantitative or numeric variables by plotting one variable against the value of another variable

• For example, one axis of a scatter plot could represent height and the other could represent weight. Each person in the data would receive one data point on the scatter plot that corresponds to his or her height and weight

Geographic Information Systems (GIS)

• A GIS is a computer system capable of capturing, storing, analyzing, and displaying geographically referenced information; that is, data identified according to location

• Using a GIS program, a researcher can create a map to represent data relationships visually

Sociograms

• Display networks of relationships among variables, enabling researchers to identify the nature of relationships that would otherwise be too complex to conceptualize

Glossary terms related to graphical and pictorial methods:

Measures of central tendency are the most basic and, often, the most informative description of a population's characteristics. They describe the "average" member of the population of interest. There are three measures of central tendency:

Mean -- the sum of a variable's values divided by the total number of values
Median -- the middle value of a variable
Mode -- the value that occurs most often

Example:
The incomes of five randomly selected people in the United States are \$10,000, \$10,000, \$45,000, \$60,000, and \$1,000,000.

Mean Income = (10,000 + 10,000 + 45,000 + 60,000 + 1,000,000) / 5 = \$225,000
Median Income = \$45,000
Modal Income = \$10,000

The mean is the most commonly used measure of central tendency. Medians are generally used when a few values are extremely different from the rest of the values (this is called a skewed distribution). For example, the median income is often the best measure of the average income because, while most individuals earn between \$0 and \$200,000, a handful of individuals earn millions.

Glossary terms related to measures of central tendency:

Measures of dispersion provide information about the spread of a variable's values. There are four key measures of dispersion:

• Range

• Variance

• Standard Deviation

• Skew

Range is simply the difference between the smallest and largest values in the data. The interquartile range is the difference between the values at the 75th percentile and the 25th percentile of the data.

Variance is the most commonly used measure of dispersion. It is calculated by taking the average of the squared differences between each value and the mean.

Standard deviation, another commonly used statistic, is the square root of the variance.

Skew is a measure of whether some values of a variable are extremely different from the majority of the values. For example, income is skewed because most people make between \$0 and \$200,000, but a handful of people earn millions. A variable is positively skewed if the extreme values are higher than the majority of values. A variable is negatively skewed if the extreme values are lower than the majority of values.

Example:
The incomes of five randomly selected people in the United States are \$10,000, \$10,000, \$45,000, \$60,000, and \$1,000,000:

Range = 1,000,000 - 10,000 = 990,000
Variance = [(10,000 - 225,000)2 + (10,000 - 225,000)2 + (45,000 - 225,000)2 + (60,000 - 225,000)2 + (1,000,000 - 225,000)2] / 5 = 150,540,000,000
Standard Deviation = Square Root (150,540,000,000) = 387,995
Skew = Income is positively skewed

Glossary terms related to measures of dispersion:

Measures of association indicate whether two variables are related. Two measures are commonly used:

• Chi-square

• Correlation

Chi-Square

• As a measure of association between variables, chi-square tests are used on nominal data (i.e., data that are put into classes: e.g., gender [male, female] and type of job [unskilled, semi-skilled, skilled]) to determine whether they are associated*

• A chi-square is called significant if there is an association between two variables, and nonsignificant if there is not an association

To test for associations, a chi-square is calculated in the following way: Suppose a researcher wants to know whether there is a relationship between gender and two types of jobs, construction worker and administrative assistant. To perform a chi-square test, the researcher counts up the number of female administrative assistants, the number of female construction workers, the number of male administrative assistants, and the number of male construction workers in the data. These counts are compared with the number that would be expected in each category if there were no association between job type and gender (this expected count is based on statistical calculations). If there is a large difference between the observed values and the expected values, the chi-square test is significant, which indicates there is an association between the two variables.

*The chi-square test can also be used as a measure of goodness of fit, to test if data from a sample come from a population with a specific distribution, as an alternative to Anderson-Darling and Kolmogorov-Smirnov goodness-of-fit tests. As such, the chi square test is not restricted to nominal data; with non-binned data, however, the results depend on how the bins or classes are created and the size of the sample

Correlation

• A correlation coefficient is used to measure the strength of the relationship between numeric variables (e.g., weight and height)

• The most common correlation coefficient is Pearson's r, which can range from -1 to +1.

• If the coefficient is between 0 and 1, as one variable increases, the other also increases. This is called a positive correlation. For example, height and weight are positively correlated because taller people usually weigh more

• If the correlation coefficient is between -1 and 0, as one variable increases the other decreases. This is called a negative correlation. For example, age and hours slept per night are negatively correlated because older people usually sleep fewer hours per night