Data Analysis

ARITHMETIC & DATA INTERPRETATION HANDBOOK

PAGE 1 — DATA ANALYSIS: BASIC CONCEPTS

1. What is Data Analysis?

Data Analysis is the process of collecting, organising, examining, comparing and interpreting numerical or categorical information to obtain meaningful conclusions.

Simple Definition:
Data analysis means converting raw data into useful information by using calculations, comparisons, ratios, percentages, averages and other statistical methods.

Data may be presented through tables, bar graphs, line graphs, pie charts, schedules or statements. In competitive examinations, questions generally require calculations based on the information given.

2. Types of Data

Type	Meaning	Example
Qualitative	Describes qualities or categories.	Gender, colour, category
Quantitative	Expressed numerically.	Income, marks, population
Discrete	Countable numerical values.	Number of students
Continuous	Can take values within an interval.	Height, weight, temperature

3. Basic Operations in Data Analysis

Finding totals.
Finding averages.
Finding percentages.
Finding ratios.
Finding differences.
Finding percentage increase or decrease.
Comparing two or more data sets.
Finding missing values.
Finding maximum and minimum values.

4. Total and Average

Total = Sum of all observations

Average = Total / Number of observations

Example 1: The sales of a shop for five days are ₹400, ₹500, ₹600, ₹700 and ₹800. Find the total and average.

Total = 400 + 500 + 600 + 700 + 800
= ₹3,000

Average = 3000/5
= ₹600

5. Percentage

Percentage = (Part / Total) × 100

Example 2: A school has 800 students, of whom 320 are girls. Find the percentage of girls.

Percentage = 320/800 × 100
= 40%

Exam Tip: Before calculating, identify whether the question asks for a total, difference, ratio, percentage, average or growth rate.

DATA ANALYSIS • IMPORTANT FORMULAS

PAGE 2 — CORE FORMULAS

6. Ratio Analysis

A ratio compares two quantities of the same type.

Ratio = First Quantity : Second Quantity

Example 3: If a company's sales in two years are ₹60 lakh and ₹45 lakh:

Ratio = 60 : 45
= 4 : 3

7. Difference

Difference = Larger Value − Smaller Value

Example 4: Production in 2025 was 7,500 units and in 2026 it was 8,700 units.

Difference = 8700 − 7500
= 1,200 units

8. Percentage Increase

Percentage Increase = (Increase / Original Value) × 100

Example 5: Production increases from 500 units to 650 units.

Increase = 650 − 500 = 150
Percentage Increase = 150/500 × 100
= 30%

9. Percentage Decrease

Percentage Decrease = (Decrease / Original Value) × 100

Example 6: Sales decrease from ₹80,000 to ₹68,000.

Decrease = 80,000 − 68,000
= ₹12,000

Percentage Decrease = 12,000/80,000 × 100
= 15%

10. Growth Rate

Growth Rate = [(New Value − Old Value) / Old Value] × 100

Example 7: Population increases from 2,00,000 to 2,30,000.

Growth Rate = 30,000/2,00,000 × 100
= 15%

11. Weighted Average

When observations have different weights, use weighted average.

Weighted Average = Σwx / Σw

Here x represents the value and w represents its weight.

Example 8: Values 20, 30 and 40 have weights 2, 3 and 5.

Σwx = 20×2 + 30×3 + 40×5
= 40 + 90 + 200
= 330

Σw = 2 + 3 + 5 = 10
Weighted Average = 330/10
= 33

DATA ANALYSIS • TABLES & COMPARISON

PAGE 3 — DATA INTERPRETATION FORMULAS

12. Frequency Analysis

Frequency tells us how many times a particular value occurs.

Total Frequency = Σf

Mean = Σfx / Σf

Example 9: Consider:

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

Σf = 10
Σfx = 10×2 + 20×3 + 30×5
= 230

Mean = 230/10
= 23

13. Percentage Share

Percentage Share = (Individual Value / Total Value) × 100

Example 10: A company's total expenditure is ₹10 lakh. The expenditure on salaries is ₹2.5 lakh.

Salary Share = 2.5/10 × 100
= 25%

14. Comparison of Two Values

Percentage Difference = |A − B| / Reference Value × 100

The denominator depends on the wording of the question. If the question asks percentage increase from A to B, A is the base.

15. Finding a Missing Value

Missing-value questions can often be solved using the known total, average or percentage.

Missing Value = Required Total − Sum of Known Values

Example 11: The average of five numbers is 42. Four numbers are 35, 40, 45 and 50. Find the fifth number.

Total = 42 × 5 = 210
Sum of known values = 35 + 40 + 45 + 50 = 170
Missing value = 210 − 170
= 40

16. Ratio-to-Total Method

If two quantities are in the ratio a:b and their total is T:

First Quantity = a/(a+b) × T

Second Quantity = b/(a+b) × T

Example 12: Boys and girls are in the ratio 3:2. If total students are 500, find the number of boys and girls.

Boys = 3/5 × 500 = 300
Girls = 2/5 × 500 = 200

17. Data Comparison Table

Quantity	Year 1	Year 2	Difference
Production	5,000	6,000	1,000
Sales	4,000	5,200	1,200
Profit	800	1,000	200

Shortcut: When comparing several rows, calculate the difference first and then calculate the percentage only if required. This reduces calculation errors.

DATA ANALYSIS • SHORTCUTS & PRACTICE

PAGE 4 — REVISION & PRACTICE SET

18. Important Formula Chart

Concept	Formula
Total	Σx
Average	Σx/n
Frequency Total	Σf
Frequency Mean	Σfx/Σf
Weighted Mean	Σwx/Σw
Percentage	Part/Total × 100
Percentage Increase	Increase/Original × 100
Percentage Decrease	Decrease/Original × 100
Growth Rate	(New−Old)/Old × 100
Ratio	First : Second
Ratio Share	a/(a+b) × Total
Relative Frequency	f/Σf

19. Common Mistakes to Avoid

1. Using the new value instead of the original value as the denominator for percentage increase.

2. Confusing percentage with percentage points.

3. Adding percentages without checking whether they have the same base.

4. Using a ratio without simplifying it.

5. Ignoring units such as ₹, lakh, crore, tonnes or kilometres.

20. Practice Questions

The sales of a company for three years are ₹20 lakh, ₹25 lakh and ₹30 lakh. Find the total sales.
The marks of five students are 60, 70, 80, 90 and 100. Find the average.
A town has 50,000 people, of whom 32,500 are adults. Find the percentage of adults.
Production rises from 4,000 units to 5,000 units. Find the percentage increase.
Sales fall from ₹90,000 to ₹72,000. Find the percentage decrease.
The ratio of male to female employees is 5:3. If there are 640 employees, find the number of males and females.
The values 10, 20 and 30 have frequencies 3, 4 and 3. Find the arithmetic mean.
Total expenditure is ₹12 lakh and expenditure on salaries is ₹3 lakh. Find the salary percentage.
The average of six numbers is 35. Five numbers are 20, 30, 35, 40 and 45. Find the sixth number.
A company's output increases from 80,000 units to 92,000 units. Find the growth rate.

21. Answer Key

No.	Answer	No.	Answer
1	₹75 lakh	6	400 males, 240 females
2	80	7	20
3	65%	8	25%
4	25%	9	40
5	20%	10	15%

22. Final Revision

Data Analysis = Organise → Calculate → Compare → Interpret

Golden Rule: In every data-analysis problem, first identify the base value. For percentage increase or decrease, the original value is normally the base. For percentage share, the total is the base.

Fast Method: Read the table carefully → identify required values → write the correct formula → substitute numbers → simplify → check the unit and reasonableness of the answer.