Descriptive Statistics: Frequency Distributions and Their Graphs

Chapter 2: Descriptive Statistics

Chapter Outline

• Frequency Distributions and Their Graphs

• More Graphs and Displays

• Measures of Central Tendency

• Measures of Variation

• Measures of Position

Section 2.1: Frequency Distributions and Their Graphs

Section Objectives

• How to construct a frequency distribution including class limits, midpoints, relative frequencies, cumulative frequencies, and boundaries.

• How to construct frequency histograms, frequency polygons, relative frequency histograms, and ogives.

Frequency Distribution

A frequency distribution is a table that organizes data into classes or intervals and shows the number of entries (frequency) in each class.

• Class: A range of data values grouped together.

• Frequency (f): The number of data entries in a class.

Class	Frequency
1 – 5	5
6 – 10	8
11 – 15	6
16 – 20	8
21 – 25	5
26 – 30	4

Class Limits

• Lower class limit: The smallest value that can belong to a class.

• Upper class limit: The largest value that can belong to a class.

Class	Lower Class Limit	Upper Class Limit
1 – 5	1	5
6 – 10	6	10
11 – 15	11	15
16 – 20	16	20
21 – 25	21	25
26 – 30	26	30

Class Width and Range

• Class width: The difference between the lower (or upper) limits of two consecutive classes. Formula: $\text{Class width} = \text{Lower limit of next class} - \text{Lower limit of current class}$

• Range: The difference between the maximum and minimum data entries. Formula: $\text{Range} = \text{Maximum data entry} - \text{Minimum data entry}$

Example: For the classes above, class width = 6 - 1 = 5.

Constructing a Frequency Distribution

1. Decide on the number of classes (usually between 5 and 20 for clarity).

2. Find the class width:

   • Determine the range of the data.

   • Divide the range by the number of classes.

   • Round up to the next convenient number.

3. Find the class limits:

   • Use the minimum data entry as the lower limit of the first class.

   • Add the class width to get the lower limit of the next class.

   • Find the upper limit of the first class (one less than the lower limit of the next class).

   • Continue for remaining classes.

Example: If the minimum value is 1 and the class width is 5, the classes would be 1–5, 6–10, etc.

Key Terms and Concepts

• Frequency Distribution Table: Organizes data into classes and shows the frequency for each class.

• Class Limits: Define the boundaries of each class.

• Class Width: Determines the size of each class interval.

• Range: Measures the spread of the data.

Applications

• Frequency distributions are used to summarize large data sets and identify patterns.

• They are foundational for constructing histograms and other graphical representations in statistics.

Example Table: Frequency Distribution

Class	Frequency
1 – 5	5
6 – 10	8
11 – 15	6
16 – 20	8
21 – 25	5
26 – 30	4

Additional info: These notes are based on textbook slides for "Elementary Statistics" by Ron Larson, Pearson Education, and cover foundational concepts in descriptive statistics relevant for college-level statistics courses.