3.4 Histograms

Definitions The frequency of a class is the number of observations in the class. The relative frequency of a class is the proportion of the observations in the class. Sum of Relative Frequencies For a numerical variable, the sum of the relative frequencies of all the classes is equal to 1.

Example: Constructing Histograms Use the table on page 174 to help construct a frequency histogram for the test score distribution.

Solution Because the numbers 30, 40, 50, …, and 100 are the lower class limits, we write these numbers as well as 110 equally spaced on the horizontal axis. Because the largest frequency is 10, we write the numbers 0, 2, 4, 6, 8, and 10 equally spaced on the vertical axis. Next, we write “Points” and “Frequency” on the appropriate axes. Then we draw rectangles whose widths are 10 (the class widths) and whose heights are the class frequencies. (see next slide)

Solution

Area of Bars of a Density Histogram The following statements are true for a density histogram. The area of each bar is equal to the relative frequency of the bar’s class. The total area of the bars is equal to 1.

Example: Using a Density Histogram to Find Proportions The average prices of 2014 Major League Baseball® (MLB) tickets at the stadiums are described by the density histogram below. Find the proportion of stadiums whose average price of 2014 MLB tickets is 1. between $30 and $39.99, inclusive. 2. less than $20. 3. at least $20.

Solution 1. between $30 and $39.99, inclusive. The proportion is equal to the total area of the green rectangles, which is 0.10 + 0.07 = 0.17.

Solution 2. less than $20 All average ticket prices less than $20 are between $15 and $20. The proportion for this price range is equal to the area of the orange bar, which is 0.13.

Solution 3. at least $20 The proportion of stadiums whose average price is at least $20 is equal to the sum of the areas of the blue bars and green bars. So, the proportion is 0.33 + 0.27 + 0.10 + 0.07 + 0.03 + 0.07 = 0.87.

Unimodal, bimodal, and multimodal distributions A distribution is unimodal if it has one mound, bimodal if it has two mounds, and multimodal if it has more than two mounds.

Skewed-left, skewed-right, and symmetric distributions If the left tail of a unimodal distribution is longer than the right tail, then the distribution is skewed left. If the right tail of a unimodal distribution is longer than the left tail, then the distribution is skewed right. If the left tail of a distribution is roughly the mirror image of the right tail, the distribution is symmetric.

Order of Determining the Four Characteristics of a Distribution with a Numerical Variable We often determine the four characteristics of a distribution with a numerical variable in the following order: 1. Identify all outliers. a. For outliers that stem from errors in measurement or recording, correct the errors, if possible. If the errors cannot be corrected, remove the outliers. b. For other outliers, determine whether they should be analyzed in a separate study.

Order of Determining the Four Characteristics of a Distribution with a Numerical Variable 2. Determine the shape. If the distribution is bimodal or multimodal, determine whether subgroups of the data should be analyzed separately. 3. Measure and interpret the center. 4. Describe the spread.

Model A model is a mathematical description of an authentic situation. We say the description models the situation.