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, measures of central tendency, measures of variation, and measures of 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. This helps to summarize large data sets and identify patterns.

• Class: A group or interval into which data values are grouped.

• 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:

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

For example, $6 - 1 = 5$

• Range formula:

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

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.

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

Example: Frequency Distribution Construction

Suppose you have a data set of cell phone screen times (in minutes) for U.S. adults. To construct a frequency distribution with seven classes:

• Find the minimum and maximum values.

• Calculate the range and class width.

• Determine class limits and tally frequencies.

Application: Frequency distributions are used to summarize and visualize data, making it easier to identify trends and patterns, such as the most common range of values or the distribution of data across intervals.

Key Terms Summary

• Frequency Distribution: Table organizing data into intervals/classes.

• Class Limits: Boundaries for each class interval.

• Class Width: Difference between consecutive class limits.

• Range: Spread of the data set.

Additional info:

• Frequency distributions are foundational for constructing histograms, polygons, and other graphical displays in statistics.

• Choosing appropriate class width and number of classes is essential for meaningful data representation.