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

Visit the following websites for more information:

Glossary terms related to graphical and pictorial methods:

GIS

Histogram

Scatter Plot

Sociogram

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.

Visit the following websites for more information:

Glossary terms related to measures of central tendency:

Average

Central Tendency

Confidence Interval

Mean

Median

Mode

Moving Average

Point Estimate

Univariate Analysis

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 75^{th} percentile and the 25^{th} 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

Visit the following websites for more information:

- Descriptive Statistics
- Survey Research Tools
- Variance and Standard Deviation
- Summarizing and Presenting Data
- Skewness
- Skewness Simulation

Glossary terms related to measures of dispersion:

Confidence Interval

Distribution

Kurtosis

Point Estimate

Quartiles

Range

Skewness

Standard Deviation

Univariate Analysis

Variance

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

Visit the following websites for more information:

- Chi-Square Procedures for the Analysis of Categorical Frequency Data
- Chi-square Analysis
- Correlation

Glossary terms related to measures of association:

Association

Chi Square

Correlation

Correlation Coefficient

Measures of Association

Pearson's Correlational Coefficient

Product Moment Correlation Coefficient