Tabulation

ARITHMETIC & DATA INTERPRETATION HANDBOOK

PAGE 1 — TABULATION: BASIC CONCEPTS

1. What is Tabulation?

Tabulation is the systematic arrangement of collected data in rows and columns so that the information becomes easy to read, compare, analyse and interpret.

Simple Definition:
Tabulation means presenting data in a well-organised table form.

For example, if marks obtained by five students are 45, 60, 72, 55 and 68, these values can be arranged in a table along with the names of the students.

2. Objectives of Tabulation

To present large data in a compact form.
To make comparison easy.
To identify patterns and trends.
To facilitate mathematical calculations.
To simplify data interpretation.
To help in drawing graphs and charts.
To make important information easy to locate.

3. Main Parts of a Statistical Table

Part	Meaning
Table Number	Identification number of the table.
Title	Explains what the table represents.
Headnote	Provides units or other necessary information.
Captions	Headings of columns.
Stubs	Headings of rows.
Body	Main numerical information in the table.
Footnote	Additional explanation or special information.
Source	Origin from which the data were obtained.

4. Example of Simple Tabulation

Student	Marks
Amit	65
Ravi	72
Sita	81
Neha	76

Exam Tip: A good table should have a clear title, proper headings, consistent units and logically arranged data.

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TABULATION • TYPES & FREQUENCY

PAGE 2 — TYPES OF TABULATION

5. Types of Tables

Type	Description
Simple Table	Shows one characteristic or variable.
Two-Way Table	Shows two characteristics simultaneously.
Three-Way Table	Shows three characteristics.
Manifold Table	Shows several characteristics together.
Frequency Table	Shows how often each value or class occurs.

6. Frequency

Frequency is the number of times a particular value or observation occurs.

Total Frequency = Σf

Example 1: Consider the data:

2, 3, 2, 4, 5, 3, 2, 4, 3, 2

The frequency table is:

Value (x)	Frequency (f)
2	4
3	3
4	2
5	1
Total	10

7. Relative Frequency

Relative frequency expresses the frequency of a category as a fraction or proportion of total observations.

Relative Frequency = f / Σf

Percentage Frequency = (f / Σf) × 100

Example 2: If the frequency of a category is 15 and total frequency is 60:

Relative Frequency = 15/60 = 0.25
Percentage Frequency = 0.25 × 100 = 25%

8. Class Interval

When data contains many observations, values are grouped into intervals called class intervals.

Examples:

0–10
10–20
20–30
30–40

Class Width = Upper Class Boundary − Lower Class Boundary

Example 3: For the class interval 20–30:

Class Width = 30 − 20 = 10

9. Class Midpoint

Midpoint (x) = (Lower Limit + Upper Limit) / 2

Example 4: Find the midpoint of 30–40.

x = (30 + 40)/2
= 35

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TABULATION • FORMULAS & DATA ANALYSIS

PAGE 3 — IMPORTANT FORMULAS

10. Arithmetic Mean from a Frequency Table

For a discrete frequency distribution, the arithmetic mean is calculated using:

Mean = Σfx / Σf

Here, x is the observation and f is its frequency.

Example 5:

Suppose:
x = 10, 20, 30
f = 2, 3, 5

``` Σf = 2 + 3 + 5 = 10
Σfx = (10×2) + (20×3) + (30×5)
= 20 + 60 + 150
= 230

Mean = 230/10
= 23 ```

11. Weighted Mean

When different observations have different importance or weights, weighted mean is used.

Weighted Mean = Σwx / Σw

Example 6: Marks are 60, 70 and 80 with weights 2, 3 and 5.

Σwx = 60×2 + 70×3 + 80×5
= 120 + 210 + 400
= 730
Σw = 2 + 3 + 5 = 10

Weighted Mean = 730/10
= 73

12. Cumulative Frequency

Cumulative frequency is obtained by successively adding frequencies.

Less-than CF = f₁ + f₂ + f₃ + ... + fₙ

Example 7:

Frequencies = 4, 6, 8, 5

Less-than cumulative frequencies:
4
4 + 6 = 10
10 + 8 = 18
18 + 5 = 23

13. Percentage from a Table

Percentage = (Part / Total) × 100

Example 8: A table shows 240 students, of whom 72 are girls. Find the percentage of girls.

Percentage = 72/240 × 100
= 30%

14. Ratio from Tabulated Data

Ratio = First Quantity : Second Quantity

Example 9: If a table contains 450 male students and 300 female students:

Ratio = 450 : 300
= 3 : 2
Therefore, Male : Female = 3 : 2.

Shortcut: Always simplify ratios by dividing both terms by their HCF.

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TABULATION • REVISION & PRACTICE

PAGE 4 — SOLVED PRACTICE & ANSWERS

15. Solved Mixed Example

Example 10: The following table shows the number of books read by five students:

Student	Books
A	4
B	6
C	8
D	5
E	7

Find the average number of books.

Total = 4 + 6 + 8 + 5 + 7
= 30

Number of students = 5

Average = 30/5
= 6 books

16. Common Operations on Tabulated Data

Requirement	Method / Formula
Total	Add all observations or frequencies.
Average	Total ÷ Number of observations.
Percentage	Part ÷ Total × 100.
Ratio	Compare two quantities and simplify.
Difference	Larger value − Smaller value.
Increase %	Increase ÷ Original × 100.
Decrease %	Decrease ÷ Original × 100.

17. Practice Questions

The numbers of students in four classes are 40, 45, 50 and 55. Find the total number of students.
A table contains 80 boys and 120 girls. Find the percentage of girls.
The sales of a shop on four days are ₹500, ₹700, ₹600 and ₹800. Find the average daily sales.
A frequency table has frequencies 5, 8, 7 and 10. Find the total frequency.
Find the midpoint of the class interval 40–60.
If 45 out of 180 observations belong to a category, find its percentage frequency.
The values 10, 20 and 30 have frequencies 2, 3 and 5 respectively. Find the arithmetic mean.
Find the ratio of 240 boys to 160 girls.
A table shows sales of ₹20,000 in January and ₹25,000 in February. Find the percentage increase.
The frequencies of four classes are 6, 9, 5 and 10. Find the cumulative frequency after the third class.

18. Answer Key

No.	Answer	No.	Answer
1	190	6	25%
2	60%	7	23
3	₹650	8	3 : 2
4	30	9	25%
5	50	10	20

19. Final Revision

Total Frequency = Σf

Mean = Σfx / Σf

Weighted Mean = Σwx / Σw

Relative Frequency = f/Σf

Percentage = Part/Total × 100

Midpoint = (Lower Limit + Upper Limit)/2

Remember: Tabulation is not merely arranging numbers. A properly constructed table should make the data clear, compact, comparable and easy to analyse.

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