Chapter-6: Frequency Distribution and Its Graphical Representations

Frequency distribution is a way to organize data so that the number of observations within each category or interval is displayed. This chapter will cover frequency distribution, cumulative frequency distribution, and their graphical representations, including column diagrams, step diagrams, ogives, histograms, frequency curves of different types, and stemandleaf diagrams. Examples relevant to India will be used for illustration.

 

 

 

 1. Frequency Distribution

 

Definition of Frequency Distribution

 A frequency distribution shows the number of observations (frequency) for each category or interval in a dataset.

 

Characteristics:

 Categories or Intervals: The data is divided into categories (for qualitative data) or intervals (for quantitative data).

 Frequencies: Each category or interval has a corresponding frequency indicating the number of observations in that category or interval.

 

Example:

 A survey of ages of 50 students in a classroom in India can be organized into intervals: 1012 years: 5 students, 1214 years: 15 students, 1416 years: 20 students, 1618 years: 10 students.

 

 

 

 2. Cumulative Frequency Distribution

 

Definition of Cumulative Frequency Distribution

 A cumulative frequency distribution shows the accumulation of frequencies up to each category or interval.

 

Characteristics:

 Cumulative Frequencies: The frequencies are summed as you move from the first interval to the last.

 Running Total: Each cumulative frequency is the sum of the current frequency and all previous frequencies.

 

Example:

 Using the same age data: 1012 years: 5 students (cumulative frequency: 5), 1214 years: 15 students (cumulative frequency: 20), 1416 years: 20 students (cumulative frequency: 40), 1618 years: 10 students (cumulative frequency: 50).

 

 

 

 3. Graphical Representations

 

 

 

3.1 Column Diagram

 

Definition:

 A column diagram, or bar chart, uses vertical bars to represent frequencies for different categories.

 

Characteristics:

 Vertical Bars: Each bar's height represents the frequency of the category.

 Categories: Placed along the horizontal axis.

 

Example:

 A column diagram showing the number of students in different age groups.

 

 

 

3.2 Step Diagram

 

Definition:

 A step diagram shows cumulative frequencies in a steplike fashion.

 

Characteristics:

 Steps: Each step corresponds to a cumulative frequency.

 Horizontal and Vertical Lines: Represent cumulative counts at each interval.

 

Example:

 A step diagram showing the cumulative number of students up to each age interval.

 

 

 

3.3 Ogive

 

Definition:

 An ogive is a line graph that represents the cumulative frequency distribution.

 

Characteristics:

 Cumulative Frequencies: Plotted on the vertical axis.

 Continuous Line: Connects points corresponding to cumulative frequencies.

 

Example:

 An ogive showing the cumulative number of students in each age interval.

 

 

 

3.4 Histogram

 

Definition:

 A histogram uses adjacent bars to show the frequency distribution of continuous data.

 

Characteristics:

 Adjacent Bars: Bars touch each other to show continuous data intervals.

 Interval Widths: Represent ranges of data.

 

Example:

 A histogram showing the distribution of ages among students.

 

 

 

3.5 Frequency Curve

 

Definition:

 A frequency curve is a smooth curve that represents the frequency distribution.

 

Types:

 Symmetrical: Bellshaped curve (normal distribution).

 Skewed: Curve that leans to one side (positively or negatively skewed).

 

Example:

 A frequency curve showing the distribution of exam scores in a school.

 

 

 

3.6 StemandLeaf Diagram

 

Definition:

 A stemandleaf diagram organizes data to show its shape and distribution.

 

Characteristics:

 Stems: Represent leading digits.

 Leaves: Represent trailing digits.

 

Example:

 A stemandleaf diagram showing test scores: scores of 65, 67, 68, 70, 72, 75.

 

 

 

 Illustrations with Examples

 

1. Frequency Distribution Example:

    The number of cars sold by different brands in India in a month: Maruti (25), Hyundai (15), Tata (10), Mahindra (8), Honda (7).

 

2. Cumulative Frequency Distribution Example:

    Cumulative car sales: Maruti (25), Hyundai (40), Tata (50), Mahindra (58), Honda (65).

 

3. Column Diagram Example:

    The number of tourists visiting different Indian states.

 

4. Step Diagram Example:

    Cumulative enrollment in different courses over a year.

 

5. Ogive Example:

    Cumulative rainfall over months in a year in a particular region of India.

 

6. Histogram Example:

    The distribution of household incomes in an Indian village.

 

7. Frequency Curve Example:

    The distribution of marks in a national exam in India.

 

8. StemandLeaf Diagram Example:

    Ages of participants in a survey: 23, 25, 26, 28, 30, 31, 32, 33, 34, 41, 43, 45, 47, 49.

 

 

 

 References

 

1. "Research Methodology: Methods and Techniques" by C.R. Kothari.

2. "Statistics for Management" by Richard I. Levin and David S. Rubin.

3. Government of India Census Reports.

4. "Data Visualization: A Practical Introduction" by Kieran Healy.

5. "The Visual Display of Quantitative Information" by Edward R. Tufte.

6. National Sample Survey Office (NSSO) reports.

7. "Elementary Statistics: A Step by Step Approach" by Allan G. Bluman.