Chapter 2 Frequency Distributions and Percentiles

Data Suppose that you are working on a study of social development. Of particular interest is the age at which aggressive tendencies first appear in children. You begin data collection (measuring the aggressiveness variable) by asking the teacher of a preschool class to rate the aggressiveness of the 20 children in the class using the scale:

Frequency distribution A frequency distribution is a tabulation of the number of occurrences of each score value.

Grouped Frequency Distribution

Grouped Frequency Distribution II

Grouped Frequency Distribution III - Rules 1.Compute the range of your scores by subtracting the lowest score from the highest score (Range = High Score – Low Score). 2.Divide the range by 8 and 15. Find a convenient number in between those two values. That will be your class interval. This is also known as your “bin width.” 3.Beginning with your starting value, construct intervals of increasing value. 4.Count the number (frequency) of scores in each interval.

Relative Frequency Histogram

Relative Frequency Histogram II

Characteristics of Distributions A symmetric distribution can be divided into two halves that are mirror images of each other. A positively skewed distribution has score values with low frequencies that trail off toward the positive numbers (to the right). A negatively skewed distribution has score values with low frequencies that trail off toward the negative numbers (to the left). A unimodal distribution has one peak. A bimodal distribution has two peaks. A multimodal distribution has more than two peaks.

Central Tendency & Variability The central tendency of a distribution is the score value near the center of the distribution. It is a typical or representative score value. Variability is the degree to which the measurements in a distribution differ from one another.

Why is the distribution for the boys more variable than that for the girls? Does this represent a basic difference between the sexes, or is it due to socialization? Should these data be used to argue for social change? Of course, these particular questions are not statistical questions, and statistical analyses will not provide answers to them. The point is that statistical analysis can transform the data into a form that helps you to ask good questions. But it is up to you to ask the questions.

Percentiles We have been working with whole distributions and paying little attention to individual measurements within a distribution. At times, however, the individual measurements are of great importance. Consider this example. One of your professors has just handed back your exam. On the top, in bold red ink, is the number 32. At first you are worried. But when you see the 25 on your neighbor’s paper, you begin to feel a little better. Still, is 32 a good score? Percentile ranks provide just such an index of goodness by giving the relative standing of a score—the score’s location in the distribution relative to the other measurements inthe distribution. More formally: The percentile rank of a score value is the percent of measurements in the distributionbelow that score value.

Nearest Rank Method

Example Consider an ordered population of 10 data values {3, 6, 7, 8, 8, 10, 13, 15, 16, 20}. What are the 25th, 50th, 75th and 100th percentiles of this list using the Nearest Rank method?

Linear Interpolation Method

End of Chapter 2…