basic concepts

The researchers conducting this study were interested in showing that the CSR approach was more effective than the traditional approach used by ele-mentary schools with LEP students. The population for this study may be de-fined as all students in U.S. elementary schools that enroll LEP students.

However, it is almost impossible for researchers to collect data from an entire population, so Klingner and Vaughn conducted the study using a sample of 37 fifth-grade students in an elementary school. To make statements concern-ing the population, the researchers needed to use a valid procedure to general-ize the findings from the sample to the population.

A numeric summary obtained from a sample is called a statistic. Assuming that 72% of the 37 students were LEP students, the number 72% is a statistic because it was obtained from the sample. A numeric summary that concerns a population is called a parameter. Using the above example, the percentage of LEP students in U.S. elementary schools is a parameter. In most cases, param-eters are unknown and need to be estimated from statistics.

One assessment in this study was a vocabulary definition test in which the students were required to define a set of English words. Each child took the test before and after a CSR session. The researchers were interested in examining

whether the average of the posttest was higher than that of the pretest. This de-sign had two variables: pretest and posttest. A variable is defined as the prop-erty of an event that may have different values. Taking the variable pretest as an example, the 37 students may have different scores on the pretest due to their different knowledge of English vocabulary. Variables can be either numeric or nonnumeric. In this study, both pre- and posttest are numeric variables, so that arithmetic calculations can be carried out on these variables. For nonnumeric variables, such as gender, L1, and learning strategies used by second language (L2) students, frequency tables can be used to present the number of raw oc-currences and the percentages of the ococ-currences. For example, if the variable LEP indicates the status of students in the sample with yes indicating a LEP stu-dent and no indicating a non-LEP stustu-dent, the variable LEP is a nonnumeric variable. The occurrences and percentages of the LEP students can be obtained from the sample.

In general, variability exists among individuals and within an individual over time. Descriptive statistics summarize and present such variability. Im-portant summary statistics include mean, median, mode, variance, and stan-dard deviation. Mean, median, and mode measure the central tendency of a variable, while variance and standard deviation summarize how much the data are spread out. The concepts and calculations for mean, median, mode, vari-ance, and standard deviation are illustrated using the following numerical ex-ample. Suppose that we replicated Klingner and Vaughn’s (2000) study and obtained a sample of 10 students with scores on the pretest and posttest as pre-sented in Table 2.1.

The mean is the average from the scores. The formula calculating the sam-ple mean is X ⫽ 兺 X兾N, where X represents the original scores and N is the sample size. For the pretest in our example, X ⫽ (18 ⫹ 34 ⫹ 19 ⫹ 22 ⫹ 15 ⫹ 28⫹ 10 ⫹ 21 ⫹ 16 ⫹ 20)兾10 ⫽ 20.3. Likewise, we can calculate the mean for the posttest as 25.

The median is the score that lies in the middle if the data are arranged in an increasing or decreasing order. For example, we may arrange the data for the pretest in an increasing order: 10, 15, 16, 18, 19, 20, 21, 22, 28 and 34. The middle score lies between 19 and 20. Therefore, the median for the pretest is

table 2.1

Simulated Scores on the Vocabulary Definition Test for the 10 Students

Student 1 2 3 4 5 6 7 8 9 10

Pretest 18 34 19 22 15 28 10 21 16 20

Posttest 23 31 19 28 21 36 20 24 26 28

(19⫹ 20)兾2 ⫽ 19.5. Using same method, we find that the median for the posttest is (24⫹ 26)兾2 ⫽ 25. When sample size N is an odd number, the me-dian is the actual score in the middle.

The mode is the score with the highest frequency in the data. In the posttest data, we have two scores equaling 28 and all other scores appear only once. The score 28 has the highest frequency, and therefore the mode for the posttest is 28. In the pretest, all scores appear only once. Since no score has the highest frequency in the pretest, no mode exists.

The formula for obtaining the variance of a variable is兺 (X ⫺ X )²兾(N ⫺ 1).

For the pretest, X is 20.3. We can calculate the variance for the sample: ((18⫺ 20.3)²⫹ (34 ⫺ 20.3)²⫹ … ⫹ (20 ⫺ 20.3)²)兾(10 ⫺ 1) ⫽ 45.57. The stan-dard deviation is the square root of the variance. For the pretest, the stanstan-dard deviation is 45 57. ⫽ 6.75. In published literature, the standard deviation is often reported along with the mean to indicate how much the data deviated from the central point. As an exercise, the reader may verify that the variance and standard deviation for the posttest are 28.27 and 5.32, respectively.

The statistical procedures introduced in this chapter belong to the area of inferential statistics. Inferential statistics allow researchers to generalize their findings from a sample to a population through the process of hypothesis test-ing. For example, if CSR is found to be more effective than the traditional classroom approach for the 37 students in the sample, can the researchers con-fidently infer that CSR is more effective for classes with LEP students in all U.S. elementary schools? Inferential statistics play an important role in this decision-making process.

Researchers may design a study to examine the effectiveness of CSR be-tween students in classes using CSR and students in traditional classes. To start with, they should randomly select two classes in the same elementary school and implement CSR in one of the classes. To show the effectiveness of CSR, the two selected classes should be as similar as possible at the outset of the study. For example, it is better for the two classes to be chosen from the same grade, to be similar in class size, and to have similar percentages of LEP students. The logic here is that since the students in the two classes are similar at the beginning of the experiment, if the students in the CSR class score higher on the vocabulary test at the end of the semester than the students in the traditional class, it can be assumed that the CSR method is relatively more effective than the traditional method.

An independent variable is the factor that the experimenter can manipulate or arrange. In our example, the instructional method (CSR vs. the traditional method) is the independent variable because the experimenter has arranged the two teaching methods for comparison. On the other hand, the values of a

dependent variable cannot be arranged by the experimenter but only obtained from the participants. In our example, the scores on the vocabulary test for the students are the values of the dependent variable.