Descriptive Statistics: Frequency Distributions and Their Graphs

Descriptive Statistics

Section 2.1: Frequency Distributions and Their Graphs

Descriptive statistics involves methods for organizing, displaying, and describing data using tables, graphs, and summary measures. This section focuses on organizing raw data into meaningful frequency distributions and visualizing them with various types of graphs.

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

Objectives

• Construct frequency distributions, including limits, midpoints, relative frequencies, cumulative frequencies, and boundaries.

• Construct frequency histograms, relative frequency histograms, frequency polygons, and ogives.

Frequency Distributions

Definition and Purpose

• 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 (f) of a class is the number of data entries in that class (also called "count").

Example Frequency Distribution Table

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

Class Limits and 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 (or Bin Width): The distance between lower (or upper) limits of consecutive classes.

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

Formula for Class Width:

$\text{Class width} = \frac{\text{Range}}{\#\text{classes}} = \frac{\text{max} - \text{min}}{\#\text{classes}}$

Always round up the class width to the next whole number.

Steps to Construct a Frequency Distribution

1. Determine the minimum and maximum values in the data set.

2. Decide on the number of classes (typically between 5 and 20).

3. Calculate the class width using the formula above.

4. List the lower class limit of the first class (usually the minimum value), then add the class width to get subsequent lower limits.

5. Find the upper class limits (one less than the next lower class limit).

6. Tally the data into the appropriate classes and count the frequencies.

Example: Cell Phone Screen Times

• Data: 30 values ranging from 155 to 405 minutes.

• Number of classes: 7

• Class width: $\frac{405-155}{7} = 35.7$ (rounded up to 36)

Class	Frequency, f
155–190	3
191–226	2
227–262	5
263–298	6
299–334	7
335–370	4
371–406	3

Midpoints

• The midpoint of a class is the average of the lower and upper class limits.

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

Example: For the class 155–190, midpoint = $(155 + 190)/2 = 172.5$

Relative Frequency

• The relative frequency of a class is the proportion or percentage of the data that falls in that class.

$\text{Relative frequency} = \frac{\text{class frequency}}{\text{sample size}} = \frac{f}{n}$

Cumulative Frequency

• The cumulative frequency for a class is the sum of the frequencies for that class and all previous classes.

• The cumulative frequency of the last class equals the sample size $n$.

Example Table: Frequency, Midpoint, Relative, and Cumulative Frequency

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

Class Boundaries

• Class boundaries are the numbers that separate classes without forming gaps between them. They are used so that bars in a histogram touch.

• To find boundaries, take half the gap between the upper limit of one class and the lower limit of the next class.

• Example: For classes 155–190 and 191–226, the gap is 1 (191–190), so boundaries are 154.5–190.5 and 190.5–226.5.

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

Graphing Quantitative Data

Types of Graphs

• Quantitative Data: Histograms, Frequency Polygon, Dot Plots, Stem-and-Leaf Plots, Pareto/Bar Plot, Box Plots

• Qualitative Data: Pie (Circle) Chart, Bar Graph

Histograms

• A frequency histogram is a bar graph representing the frequency distribution.

• The horizontal axis is quantitative (data values), and the vertical axis is frequency.

• Bars must touch (no gaps).

Relative Frequency Histograms

• Same shape and horizontal scale as the frequency histogram, but the vertical axis shows relative frequencies instead of counts.

Frequency Polygon

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

• Plotted using class midpoints (or boundaries) and frequencies.

Ogive (Cumulative Frequency Graph)

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

• Helps visualize how many data values are below a particular value.

Applications of Ogives

• Business: Cumulative sales, revenue targets, inventory levels.

• Quality Control: Cumulative defect rates, product performance.

• Environmental Science: Cumulative rainfall, pollution, temperature distributions.

Summary Table: Frequency Distribution for Cell Phone Screen Times

Class	Class boundaries	Frequency, f	Midpoint	Relative frequency	Cumulative frequency
155–190	154.5–190.5	3	172.5	0.10	3
191–226	190.5–226.5	2	208.5	0.07	5
227–262	226.5–262.5	5	244.5	0.17	10
263–298	262.5–298.5	6	280.5	0.20	16
299–334	298.5–334.5	7	316.5	0.23	23
335–370	334.5–370.5	4	352.5	0.13	27
371–406	370.5–406.5	3	388.5	0.10	30

Example Interpretation: From the relative frequency histogram, 20% of adults have screen times between 262.5 and 298.5 minutes.

Additional info: These notes cover the foundational concepts for organizing and visualizing quantitative data, which are essential for further statistical analysis.