Descriptive Statistics: Frequency Distributions and Their Graphs

Descriptive Statistics

Chapter Overview

This chapter introduces the foundational concepts of descriptive statistics, focusing on the organization, summarization, and graphical representation of data. The main topics include frequency distributions, various types of graphs, 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 2.1 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 range of values into which data are grouped.

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

Example Table:

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

Class Limits and Class Width

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

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

Steps to Construct a Frequency Distribution

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

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 each class (one less than the lower limit of the next class).

4. Tally the data into the appropriate classes.

5. Count the tallies to determine the frequency for each class.

Example: Constructing a Frequency Distribution

• Given a data set of cell phone screen times for 30 adults, construct a frequency distribution with 7 classes.

• Calculate the class width:

  • Range = 405 (max) - 155 (min) = 250

  • Class width = $\frac{250}{7} \approx 35.7$ (round up to 36)

• First lower class limit: 155

• Subsequent lower class limits: Add 36 to each previous lower limit.

• First upper class limit: 190 (one less than next lower limit)

• Continue for all classes.

Midpoints, Relative Frequency, and Cumulative Frequency

• Midpoint of a class: $\text{Midpoint} = \frac{\text{Lower class limit} + \text{Upper class limit}}{2}$

• Relative Frequency: $\text{Relative Frequency} = \frac{\text{Class frequency}}{\text{Sample size } n}$

• Cumulative Frequency: The sum of the frequency for that class and all previous classes.

Example Table: Frequency Distribution with Midpoints, Relative and Cumulative Frequencies

Class	Midpoint	Frequency	Relative Frequency	Cumulative Frequency
155–190	172.5	3	0.10	3
191–226	208.5	2	0.07	5
227–262	244.5	5	0.17	10
263–298	280.5	6	0.20	16
299–334	316.5	7	0.23	23
335–370	352.5	4	0.13	27
371–406	388.5	3	0.10	30

Patterns in Data

• The most common range for screen time is 299 to 334 minutes.

• About half of the times are less than 299 minutes.

Graphs of Frequency Distributions

Frequency Histogram

A frequency histogram is a bar graph representing the frequency distribution. The horizontal axis is quantitative (data values), and the vertical axis shows frequencies. Bars are adjacent (no gaps).

• Class Boundaries: Values that separate classes without forming gaps. Used so bars touch in the histogram.

• To find class boundaries, add/subtract half the gap between upper and lower class limits.

Frequency Polygon

A frequency polygon is a line graph that emphasizes the continuous change in frequencies. Points are plotted at class midpoints and connected by straight lines.

• Start and end the graph at the horizontal axis, one class width before the first midpoint and after the last midpoint.

Relative Frequency Histogram

Similar to a frequency histogram, but the vertical axis shows relative frequencies instead of actual frequencies.

Cumulative Frequency Graph (Ogive)

An ogive is a line graph displaying cumulative frequencies at each upper class boundary.

• Horizontal axis: upper class boundaries

• Vertical axis: cumulative frequencies

• Start at the lower boundary of the first class (cumulative frequency = 0)

• End at the upper boundary of the last class (cumulative frequency = sample size)

Example Table: Ogive Construction

Upper Class Boundary	Frequency	Cumulative Frequency
190.5	3	3
226.5	2	5
262.5	5	10
298.5	6	16
334.5	7	23
370.5	4	27
406.5	3	30

Technology in Constructing Histograms

• Statistical calculators (e.g., TI-84 Plus), Minitab, Excel, and StatCrunch can be used to construct histograms and other graphs.

• Data is entered into lists, and graphing functions are used to display histograms and analyze frequencies.

Summary Table: Types of Graphs for Frequency Distributions

Graph Type	Main Purpose	Key Features
Frequency Histogram	Show frequency of data in intervals	Bars touch, quantitative axis
Frequency Polygon	Show trends in frequencies	Line graph, uses midpoints
Relative Frequency Histogram	Show proportion of data in intervals	Vertical axis is relative frequency
Ogive	Show cumulative totals	Line graph, uses upper class boundaries

Key Terms

• Frequency Distribution

• Class Limits

• Class Width

• Range

• Midpoint

• Relative Frequency

• Cumulative Frequency

• Histogram

• Frequency Polygon

• Ogive

Example Application: Frequency distributions and their graphs are widely used in summarizing survey data, analyzing test scores, and visualizing patterns in large data sets.