# Chapter Main Points and Learning Objectives

Chapter main points

• The most basic way to organizing data is to classify the observations into a frequency distribution—a table that reports the number of observations that fall into each category of the variable being analyzed.
• Constructing a frequency distribution is usually the first step in the statistical analysis of data. To obtain a frequency distribution for nominal and ordinal variables, count and report the number of cases that fall into each category of the variable along with the total number of cases (N). To construct a frequency distribution for interval-ratio variables that have a wide range of values, first combine the scores into a smaller number of groups—known as class intervals—each containing a number of scores.
• Proportions and percentages are relative frequencies. To construct a proportion, divide the frequency (f) in each category by the total number of cases (N). To obtain a percentage, divide the frequency (f) in each category by the total number of cases (N) and multiply by 100.
• Percentage distributions are tables that show the percentage of observations that fall into each category of the variable. Percentage distributions are routinely added to almost any frequency table and are especially important if comparisons between groups are to be considered.
• Cumulative frequency distributions allow us to locate the relative position of a given score in a distribution. They are obtainedby adding to the frequency in each category the frequencies of all the categories below it.
• Cumulative percentage distributions have wider applications than cumulative frequency distributions. A cumulative percentage distribution is constructed by adding to the percentages in each category the percentages of all the categories below it.
• A rate is a number that expresses raw frequencies in relative terms. A rate can be calculated as the number of actual occurrences in a given time period divided by the number of possible occurrences for that period. Rates are often multiplied by some power of 10 to eliminate decimal points and make the number easier to interpret.
• A pie chart shows the differences in frequencies or percentages among categories of nominal or ordinal variable. The categories of the variable are segments of a circle whose pieces add up to 100% of the total frequencies.
• A bar graph shows the differences in frequencies or percentages among categories of a nominal or an ordinal  ariable. The categories are displayed as rectangles of equal width with their height proportional to the frequency or percentage of the category.
• Histograms display the differences in frequencies or percentages among categories of interval-ratio variables. The categories are displayed as contiguous bars with their width proportional to the width of the category and height proportional to the frequency or percentage of that category.
• A line graph shows the differences in frequencies or percentages among categories of an interval-ratio variable. Points representing the frequencies of each category are placed above the midpoint of the category (interval). Adjacent points are then joined by a straight line.
• A time-series chart displays changes in a variable at different points in time. It displays two variables: (1) time, which is labeled across the horizontal axis, and (2) another variable of interest whose values (e.g., frequencies, percentages, or rates) are labeled along the vertical axis.

Learning objectives

1. Construct and analyze frequency, percentage, and cumulative distributions
2. Calculate proportions and percentages
3. Compare and contrast frequency and percentage distributions for nominal, ordinal, and interval-ratio variables
4. Construct and interpret a pie chart, bar graph, histogram, the statistical map, line graph, and time-series chart