Main Points

Statistics versus Parameters

• A parameter is a characteristic of a population. It is a numerical or graphic way to summarize data obtained from the population.
• A statistic, on the other hand, is a characteristic of a sample. It is a numerical or graphic way to summarize data obtained from a sample.

Types of Numerical Data

• There are two fundamental types of numerical data a researcher can collect. Categorical data are data obtained by determining the frequency of occurrences in each of several categories. Quantitative data are data obtained by determining placement on a scale that indicates amount or degree.

Techniques for Summarizing Quantitative Data

• A frequency distribution is a two-column listing, from high to low, of all the scores along with their frequencies. In a grouped frequency distribution, the scores have been grouped into equal intervals.
• A frequency polygon is a graphic display of a frequency distribution. It is a graphic way to summarize quantitative data for one variable.
• A graphic distribution of scores in which only a few individuals receive high scores is called a positively skewed polygon; one in which only a few individuals receive low scores is called a negatively skewed polygon.
• The normal distribution is a theoretical distribution that is symmetrical, and in which a large proportion of the scores are concentrated in the middle of the distribution.
• A distribution curve is a smoothed out frequency polygon.
• The distribution curve of a normal distribution is called a normal curve. It is bell-shaped, and its mean, median, and mode are identical.
• There are several measures of central tendency (averages) that are used to summarize quantitative data. The two most common are the mean and the median.
• The mean of a distribution is determined by adding up all of the scores and dividing this sum by the total number of scores.
• The median of a distribution marks the point above and below which half of the scores in the distribution lie.
• The mode is the most frequent score in a distribution.
• The term "variability," as used in research, refers to the extent to which the scores on a quantitative variable in a distribution are spread out.
• The most common measure of variability used in educational research is the standard deviation.
• The range, another measure of variability, represents the difference between the highest and lowest scores in a distribution.
• A five-number summary of a distribution reports the lowest score, the first quartile, the median, the third quartile, and the highest score.
• Five-number summaries of distributions are often portrayed graphically by the use of boxplots.

Standard Scores and the Normal Curve

• Standard scores use a common scale to indicate how an individual compares to other individuals in a group. This simplest form of standard score is a z score. A z score expresses how far a raw score is from the mean in standard deviation units.
• The major advantage of standard scores is that they provide a better basis for comparing performance on different measures than do raw scores.
• The term "probability," as used in research, refers to a prediction of how often a particular event will occur. Probabilities are usually expressed in decimal form.

Correlation

• A correlation coefficient is a numerical index expressing the degree of relationship that exists between two quantitative variables. The one most commonly used in educational research is the Pearson r.
• A scatterplot is a graphic way to describe a relationship between two quantitative variables.

Techniques for Summarizing Categorical Data

• There are a variety of graphic techniques researchers use to summarize categorical data, including frequency tables, bar graphs, and pie charts.
• A crossbreak table is a graphic way to report a relationship between two or more categorical variables.