Descriptive Statistics: Frequency Distributions and Their Graphs

Descriptive Statistics

Chapter Outline

• 2.1 Frequency Distributions and Their Graphs

• 2.2 More Graphs and Displays

• 2.3 Measures of Central Tendency

• 2.4 Measures of Variation

• 2.5 Measures of Position

Frequency Distributions and Their Graphs

Frequency Distribution

A frequency distribution is a table that organizes data into classes or intervals, showing the number of data entries in each class. This is fundamental for summarizing large data sets and identifying patterns.

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

• 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 consecutive classes.

• Range: The difference between the maximum and minimum data entries.

Example Table: Frequency Distribution

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

Constructing a Frequency Distribution

1. Decide on the number of classes: Typically between 5 and 20 for meaningful patterns.

2. Find the class width:

   • Determine the range: $\text{Range} = \text{Maximum} - \text{Minimum}$

   • Divide the range by the number of classes: $\text{Class width} = \frac{\text{Range}}{\text{Number of classes}}$

   • 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 subsequent lower limits.

   • Upper limit of a class is one less than the lower limit of the next class.

4. Make tally marks for each data entry in the appropriate class.

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

Example: Out-of-Pocket Prescription Medicine Expenses

Given a data set of 30 U.S. adults' expenses, construct a frequency distribution with 7 classes.

• Number of classes = 7

• Class width calculation: $\frac{\text{Range}}{7} = 35.71$ (rounded up to 36)

• First lower limit: 155; subsequent limits found by adding class width.

Lower limit	Upper limit
155	190
191	226
227	262
263	298
299	334
335	370
371	406

Midpoints, Relative Frequency, and Cumulative Frequency

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

• Relative Frequency: The proportion of data in a class. $\text{Relative frequency} = \frac{f}{n}$

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

Example Table: Expanded Frequency Distribution

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

Graphs of Frequency Distributions

Frequency Histogram

• A bar graph representing the frequency distribution.

• Horizontal axis: quantitative data values.

• Vertical axis: frequencies of the classes.

• Bars must touch, indicating continuous data.

Class Boundaries

• Class boundaries are the numbers that separate classes without forming gaps.

• Used to ensure bars in histograms touch.

• Example: For classes 155–190 and 191–226, the boundary is halfway between 190 and 191, i.e., 190.5.

Frequency Polygon

• A line graph that emphasizes the continuous change in frequencies.

• Uses class midpoints or boundaries for the horizontal axis.

Relative Frequency Histogram

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

Cumulative Frequency Graph (Ogive)

• A line graph displaying cumulative frequency at each upper class boundary.

• Shows how many data values are below a particular value.

Example: Patterns in Data

• The most common range for expenses is $299 to $334.

• About half of the expenses are less than $299.

Technology in Statistics

• Histograms, frequency polygons, and ogives can be constructed using calculators (e.g., TI-84 Plus) or software (e.g., Minitab, Excel, StatCrunch).

Summary Table: Frequency Distribution for Out-of-Pocket Prescription Medicine Expenses

Class	Class Boundaries	Frequency, f
155–190	154.5–190.5	3
191–226	190.5–226.5	2
227–262	226.5–262.5	5
263–298	262.5–298.5	6
299–334	298.5–334.5	7
335–370	334.5–370.5	4
371–406	370.5–406.5	3

Key Formulas

• Class Width: $\text{Class width} = \frac{\text{Range}}{\text{Number of classes}}$

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

• Relative Frequency: $\text{Relative frequency} = \frac{f}{n}$

Example Application

Frequency distributions and their graphical representations are essential for summarizing and interpreting data in statistics. They help identify patterns, central tendencies, and variability in data sets, which are foundational for further statistical analysis.

Additional info: These notes cover Section 2.1 of a college-level statistics textbook, focusing on frequency distributions and their graphical representations. Later sections (2.2–2.5) would cover additional descriptive statistics topics.