Frequency Distributions and Graphical Representation in Statistics

Frequency Distributions

Definition and Purpose

A frequency distribution is a systematic arrangement of raw data into a table, showing the number of occurrences (frequency) for each distinct value or range of values. This organization helps in understanding the pattern and spread of data.

Simple Frequency Table Example

Age of Actor	Frequency
27 – 32	1
33 – 38	10
39 – 44	6
45 – 50	4

Grouped Frequency Distribution Example

Class limit	Class boundary	Midpoint	Frequency	Cumulative frequency
27 – 32	26.5 – 32.5	29.5	1	1
33 – 38	32.5 – 38.5	35.5	10	11
39 – 44	38.5 – 44.5	41.5	6	17
45 – 50	44.5 – 50.5	47.5	4	21

Components of a Frequency Table

• Class width: The length of a class interval.

• Class limit: The endpoints of a class interval.

• Class boundary: The true limits between classes, without gaps.

• Midpoint: The average of the lower and upper class boundaries. $\text{Midpoint} = \frac{\text{Lower boundary} + \text{Upper boundary}}{2}$

• Frequency: The number of data points in each class.

• Cumulative frequency: The running total of frequencies up to a certain class.

• Sample size: The total frequency (sum of all frequencies).

Types of Frequency Distribution

Categorical vs. Grouped Frequency Distributions

• Categorical frequency distribution: Used for qualitative (non-numeric) data. No midpoint is calculated.

• Grouped frequency distribution: Used for quantitative (numeric) data, organized into intervals (classes).

Constructing Frequency Distributions

Steps for Categorical Frequency Distribution

1. Collect categorical data (e.g., blood type, car type).

2. Count the frequency of each category.

3. Tabulate the results.

Steps for Grouped Frequency Distribution

1. Rearrange data from lowest to highest.

2. Compute the class width: $\text{Class width} = \frac{\text{Range}}{\text{Number of classes}}$ If the result is a decimal, round up to the next whole number.

3. Set up the frequency distribution table.

4. Use the smallest data value to start the first class limit.

5. Calculate class boundaries: $\text{Lower boundary} = \text{Lower class limit} - 0.5$ $\text{Upper boundary} = \text{Upper class limit} + 0.5$

6. Calculate midpoints and cumulative frequencies for each class.

Graphical Representation of Frequency Distributions

Histogram

A histogram is a vertical bar graph representing the frequency of data within each class interval. The x-axis shows class boundaries, and the y-axis shows frequencies.

Frequency Polygon

A frequency polygon is a line graph that connects points plotted at the midpoints of each class interval, with the y-axis representing frequency.

Ogive (Cumulative Frequency Graph)

An ogive is a graph that displays cumulative frequencies using points plotted at the class boundaries on the x-axis.

Dot Plot

A dot plot is a simple graph where each data value is represented by a dot above its corresponding value on the number line.

Example: Frequency Table for Graphs

Class limits	Frequency	Class boundary	Midpoint	Cumulative frequency
1 – 5	21	0.5 – 5.5	3	21
6 – 10	25	5.5 – 10.5	8	46
11 – 15	15	10.5 – 15.5	13	61
16 – 20	8	15.5 – 20.5	18	69
21 – 25	8	20.5 – 25.5	23	69
26 – 30	6	25.5 – 30.5	28	75

Shape of Distributions

Types of Distribution Shapes

• Uniform: Frequencies are evenly spread across all values.

• Bell-shaped (Normal): Highest frequency occurs in the middle, with frequencies tapering off symmetrically to both sides.

• Skewed right: The right tail (higher values) is longer than the left tail.

• Skewed left: The left tail (lower values) is longer than the right tail.

Example: Dot Plot Construction

Given data: 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7 Each value is represented by a dot above its number on the number line.

Summary Table: Frequency Distribution Components

Component	Definition
Class limit	Endpoints of a class interval
Class boundary	True limits between classes
Midpoint	Average of class boundaries
Frequency	Number of data points in a class
Cumulative frequency	Running total of frequencies
Class width	Length of a class interval

Additional info: These notes expand on the brief points in the original file, providing definitions, formulas, and examples for clarity and completeness.