Descriptive Statistics: Frequency Distributions and Their Graphs

Descriptive Statistics

Overview

Descriptive statistics involves methods for organizing, displaying, and describing data using tables, graphs, and summary measures. This section focuses on frequency distributions and their graphical representations, which are foundational tools in statistics for summarizing large data sets.

Frequency Distributions and Their Graphs

Frequency Distribution

A frequency distribution is a table that shows classes or intervals of data with a count of the number of data entries in each class. It helps to organize raw data into a more interpretable format.

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

• Frequency (f): The number of data values in each class.

Example Table:

Class	Frequency (f)
180–199	7
200–219	8
220–239	6
240–259	4
260–279	2

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 upper and lower class boundaries. Calculated as:

$\text{Class Width} = \text{Upper Class Limit} - \text{Lower Class Limit} + 1$

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

Steps for Constructing a Frequency Distribution

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

2. Find the range of the data.

3. Divide the range by the number of classes and round up to get the class width.

4. Choose the lower limit of the first class.

5. List the lower limits for all classes.

6. Find the upper limits for all classes.

7. Tally the data into classes and count the frequencies.

Example: Constructing a Frequency Distribution

Given a set of data on pocket expenses, the steps above are followed to create a frequency distribution table. For instance, if the data range is 155 and the number of classes is 7:

$\text{Class Width} = \frac{155}{7} \approx 22.14$ (rounded up to 23)

Midpoint of a Class

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

$\text{Midpoint} = \frac{\text{Lower Class Limit} + \text{Upper Class Limit}}{2}$

Relative Frequency

Relative frequency is the proportion or percentage of data values that fall in a particular class:

$\text{Relative Frequency} = \frac{f}{n}$

where $f$ is the frequency of the class and $n$ is the total number of data values.

Cumulative Frequency

Cumulative frequency is the sum of the frequencies for that class and all previous classes.

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

Class	Frequency	Midpoint	Relative Frequency	Cumulative Frequency
180–199	7	189.5	0.14	7
200–219	8	209.5	0.16	15
220–239	6	229.5	0.12	21
240–259	4	249.5	0.08	25
260–279	2	269.5	0.04	27

Graphs of Frequency Distributions

Frequency Histogram

A frequency histogram is a bar graph that represents the frequency distribution. The horizontal axis is quantitative and measures the data values, while the vertical axis measures the frequencies of the classes. Bars are drawn for each class, and consecutive bars must touch.

Class Boundaries

Class boundaries are the values that separate classes without forming gaps between them. They are used when drawing histograms to ensure bars touch.

Frequency Polygon

A frequency polygon is a line graph that emphasizes the continuous change in frequencies. It is constructed by plotting points at the midpoints of each class and connecting them with straight lines.

Relative Frequency Histogram

A relative frequency histogram is similar to a frequency histogram, but the vertical axis measures relative frequencies instead of absolute frequencies.

Cumulative Frequency Graph (Ogive)

An ogive is a line graph that displays the cumulative frequency of each class at its upper boundary. The cumulative frequencies are marked on the vertical axis, and the upper class boundaries are marked on the horizontal axis.

Steps for Constructing an Ogive

1. Construct a frequency distribution table with cumulative frequencies.

2. Specify the horizontal (upper class boundaries) and vertical (cumulative frequencies) scales.

3. Plot points representing the upper class boundaries and their corresponding cumulative frequencies.

4. Connect the points in order from left to right.

5. The graph should start at the lower boundary of the first class (cumulative frequency = 0) and end at the upper boundary of the last class (cumulative frequency = sample size).

Summary Table: Types of Frequency Distribution Graphs

Graph Type	Vertical Axis	Horizontal Axis	Purpose
Frequency Histogram	Frequency	Class Boundaries or Midpoints	Shows frequency of data in each class
Frequency Polygon	Frequency	Class Midpoints	Shows continuous change in frequency
Relative Frequency Histogram	Relative Frequency	Class Boundaries or Midpoints	Shows proportion of data in each class
Ogive	Cumulative Frequency	Upper Class Boundaries	Shows cumulative totals

Applications

• Frequency distributions and their graphs are used to summarize large data sets, identify patterns, and facilitate further statistical analysis.

• Histograms and polygons help visualize the distribution and spread of data.

• Ogives are useful for determining percentiles and medians.

Additional info: Some steps and definitions have been expanded for clarity and completeness based on standard statistics curriculum.