Descriptive Statistics: Frequency Distributions and Their Graphs

Descriptive Statistics

Chapter Overview

This chapter introduces the foundational concepts of descriptive statistics, focusing on the organization and graphical representation of data. Key topics include frequency distributions, graphical displays, and measures of central tendency, variation, and position.

• Frequency Distributions and Their Graphs

• More Graphs and Displays

• Measures of Central Tendency

• Measures of Variation

• Measures of Position

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, showing the number of data entries (frequency) in each class.

• Class: A group or interval into which data is divided.

• 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

Each class in a frequency distribution has:

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

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

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

Class Width and Range

The class width is the difference between the lower (or upper) limits of consecutive classes. The range is the difference between the maximum and minimum data entries.

• Class width formula: $\text{Class width} = \text{Lower limit of second class} - \text{Lower limit of first class}$

• Range formula: $\text{Range} = \text{Maximum data value} - \text{Minimum data value}$

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

Constructing a Frequency Distribution

To construct a frequency distribution, follow these steps:

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.

   • Find remaining lower limits by adding the class width to the previous lower limit.

   • Find the upper limit of the first class (ensure classes do not overlap).

   • Find remaining upper class limits.

4. Make a tally mark for each data entry in the appropriate class row.

5. Count the tally marks to find the total frequency for each class.

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: Organizes data into intervals and shows the number of entries in each interval.

• Class Limits: Define the boundaries of each interval.

• Class Width: The size of each interval.

• Range: The spread of the data set.

Applications

• Frequency distributions are used to summarize large data sets and reveal 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: Later sections will cover graphical displays such as histograms, polygons, and ogives, as well as measures of central tendency, variation, and position.