Descriptive Statistics: Frequency Distributions and Their Graphs

Chapter 2: 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

Section 2.1: Frequency Distributions and Their Graphs

Objectives

• How to construct a frequency distribution, including 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 shows classes or intervals of data with a count of the number of entries in each class. The frequency, denoted as f, of a class is the number of data entries in the class.

Class	Frequency, f
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 number that can belong to the class.

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

• Class width: The distance between lower (or upper) limits of consecutive classes. For example, class width = 6 – 1 = 5.

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

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 limits of subsequent classes.

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

   • Continue for 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 f for each class.

Example: Constructing a Frequency Distribution

Given a data set of out-of-pocket prescription medicine expenses for 30 adults, construct a frequency distribution with 7 classes.

• Number of classes = 7

• Class width = (max - min)/7 ≈ 35.71, round up to 36

• First lower limit = 155 (minimum value), next lower limit = 155 + 36 = 191, etc.

• First upper limit = 190 (one less than next lower limit), next upper limit = 190 + 36 = 226, etc.

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

Midpoint, Relative Frequency, and Cumulative Frequency

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

• Relative frequency: $\text{Relative frequency} = \frac{f}{n}$, where $n$ is the sample size.

• Cumulative frequency: The sum of the frequency for that class and all previous classes. The cumulative frequency of the last class equals the sample size $n$.

Example Table: Frequency Distribution with Midpoints and Frequencies

Class	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. The horizontal scale is quantitative, and the vertical scale measures frequencies. Bars must touch.

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

• Frequency Polygon: A line graph that emphasizes the continuous change in frequencies. Plots frequency at class midpoints or boundaries and connects the points.

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

Example: Frequency Histogram and Polygon

For the prescription expense data, both histograms and polygons can be constructed using either class midpoints or class boundaries. Patterns such as the most common expense range and the distribution's shape can be observed.

Key Formulas

• Class width: $\text{Class width} = \frac{\text{Range}}{\text{Number of classes}}$ (rounded up)

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

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

Example: Interpreting Patterns

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

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

Summary

• Frequency distributions organize data into classes and show how many data points fall into each class.

• Histograms, polygons, and relative frequency graphs visually represent the distribution of data.

• Key calculations include class width, midpoints, relative and cumulative frequencies.