Whether you intend to employ qualitative methods or quantitative methods or mixed methods to investigate the research questions for your dissertation, you will encounter statistics in several contexts. The obvious application of statistics that you may encounter is the need to use statistics for data analysis in your quantitative or mixed methods study. Perhaps you are not planning to employ quantitative methods - what is the value of this course for qualitative researchers? If your study involves a group of participants, you may find it helpful and informative to describe your participants' characteristics. To help your reader understand your participants' context, you might also find it useful to describe the characteristics of the setting for your study. These examples represent the producer role of using statistics.

Even if you are not intending to include numerical data in your study, chances are very good that other researchers working in your topic have employed statistical techniques, which you need to understand and evaluate. So, instead of a producer role, everyone who conducts or reads research studies plays a critical consumer role. When you review previous research (Chapter 2 of your dissertation), if that research involves statistics, you must be able to understand how the researcher applied the statistical techniques and what the results mean. Furthermore, you need to be able to challenge the quantitative approach to research in meaningful ways. This course aims to build your skills in filling both roles.

The mathematics of statistics has its roots in probability theory - the branch of mathematics that deals with chance and uncertainty. Explaining the characteristics of games of chance through mathematics was one of the early goals of probability theory. Essentially understanding games of chance involves predicting outcomes (e.g., heads or tails; blackjack; a royal flush; a roll of seven). How might understanding phenomena and predicting outcomes be linked? They are linked through a process of reasoning called statistical inference, which uses a known state of nature as the basis for predictions about the future or about a wider context. To read more about the general background of statistics, you might visit: http://en.wikipedia.org/wiki/Statistics.

Scan through the list in the preceding paragraph once more. In addition to the categories of physical, social, and school data, can you discern other differences between these variables? For example, would you describe the process of assigning of numbers to height to be similar to the process of assigning numbers to ethnic classifications? Hopefully, you find these two processes to differ in a fundamental way - namely, heights are measured with a measuring device like a tape measure and assigned a length whereas ethnicities are based on ancestry and any number assigned to a specific ethnicity is completely arbitrary. For example, you could assign a 1 to Asian or a 5 to Asian and, as long as no other group is assigned a 1 or 5, either would be an adequate quantification of ethnicity.

Let's describe these differences more formally. Variables are measured at different levels - called levels of measurement. There are four levels: nominal, ordinal, interval, and ratio. Before explaining each one, you might wonder why these measurement levels matter. The reason is simple - the level of measurement determines which statistical techniques are appropriate to use. The assignment of numbers to values of the variable is often called coding.

Nominal - the numbers assigned to the values of the variable are completely arbitrary - they are just labels that are consistently applied. For example, gender is a variable that typically has two values, male and female. You can assign 1 to represent male and 2 to represent female or vice versa or 5 to represent female and 3 to represent male. As long as all males are assigned the same number and all females are assigned the same number (and the two numbers are different), the coding of gender is appropriate. Variables measured at the nominal level are also called qualitative or categorical variables. When there are just two values, as is the case with gender, the variable is called dichotomous. Dichotomous variables are quite common in research because they represent the division of a sample into two groups.

Ordinal - the numbers assigned to the values of the variable indicate order but not the actual size. The prototype to remember is rankings. For example, runners who finish a race are labeled first, second, and third (these are called ordinal numbers, by the way). The place in which they finish does not indicate their actual time though. In fact, the difference between first place and second place may be 2 seconds, while the difference between second and third may be 10 seconds. In more formal terms, the intervals between consecutive values on an ordinal scale are not equal. Think of ranking your students by some ability - the top five students may be very close in ability levels and then there may be a substantial decrease between the fifth place student and the sixth place student.

Interval - the numbers assigned to the values of the variable indicate measured amounts. The intervals between these numbers are equal. For example, temperature is measured on an interval scale - the interval between 40 degrees and 50 degrees is the same as the interval between 70 degrees and 80 degrees. Most educational variables are treated as if they were measured on interval scales.

Ratio - the numbers assigned to the values of the variable meet the properties of an interval scale and include a "true" zero, which makes a comparison of values meaningful. A true zero is the complete lack of the measured quantity. If we counted the change in students' pockets, students without any change would be assigned a 0. Reporting that Juan had twice the change that Nina had would make sense. On the other hand, if we measured math ability using a math test, a student who scored 0 on the test can't be said to lack all math ability. Furthermore, reporting that Beatrix, who scored a 90, is twice as able, mathematically, as Que, who scored a 45, isn't meaningful.

Return to the previous list of variables and try your hand at classifying them according to these four levels of measurement.

- Descriptive statistics summarize data by reporting a number that represents the entire set of data. For example, the mean (average) score represents a summary of all of the scores on a test.
- Descriptive statistics can also be used to organize a set of data. For example, a table of age ranges and tallies (frequencies) of participants within those age ranges helps to organize the observed values of the age variable.
- Descriptive statistics allow researchers to illustrate entire sets of data so that overall patterns can be seen. For example, a pie chart showing ethnic classifications can describe these characteristics for a group of students. Likewise, a graph that compares reading levels and hearing abilities can illustrate how these two variables are related.

- Inferential statistics are used to compare groups on one or more variables. For example, reading ability of girls might be compared to that of boys for a subgroup of students in order to make general statements about the comparable abilities of girls and boys.
- Inferential statistics can also be used to compare two variables within a group of people. For example, the relationship between nutritional habits and school achievement levels might be studied for a subgroup of students so that general statements might be made about how these two variables could be linked.
- Similar to comparing variables, inferential statistics can be used to generate mathematical models that help to predict particular outcomes. For example, an admission officer might use historical data about high school performance and subsequent success in college to help inform an admissions decision.