IIT Madras BS Descriptive Statistics

Certainly! Below is a detailed explanation of the key topics in the IIT Madras BS Descriptive Statistics PDF, presented with examples, questions, and step-by-step solutions in a clear and structured layout[1].

1. Introduction to Statistics

Statistics is the science of collecting, organizing, analyzing, interpreting, and presenting data.

Key Concepts

Population: All elements of interest (e.g., all houses in Tamil Nadu).
Sample: A subset of the population (e.g., 1000 houses from Tamil Nadu).
Descriptive Statistics: Summarizing and describing data.
Inferential Statistics: Drawing conclusions about a population from sample data.

Example:
A teacher wants to know the average marks of all students in a school. She collects a sample of students and calculates their average. If she uses this to estimate the school average, she is using inferential statistics.

2. Data Types and Organization

2.1 Unstructured vs. Structured Data

Unstructured Data: Not organized (e.g., social media posts).
Structured Data: Organized (e.g., a table of student records).

Example Table:

Name	Gender	Date of Birth	Marks	Board
Anjali	F	17 Feb, 2003	484	State
Pradeep	M	3 June, 2002	514	ICSE

2.2 Classification of Data

Categorical Data: Labels (e.g., Gender, Board).
Numerical Data: Numbers (e.g., Marks).

Scales of Measurement:

Nominal: Categories (e.g., Gender).
Ordinal: Ordered categories (e.g., ratings: poor, good, excellent).
Interval: Numerical, no true zero (e.g., temperature in °C).
Ratio: Numerical, true zero (e.g., height, weight).

3. Describing Categorical Data

3.1 Frequency Distribution

Counts of each category.

Example:
Data: A, A, B, C, A, D, A, B, D, C

Category	Frequency
A	4
B	2
C	2
D	2

3.2 Relative Frequency

Frequency divided by total observations.

Category	Frequency	Relative Frequency
A	4	0.4
B	2	0.2
C	2	0.2
D	2	0.2

3.3 Charts

Pie Chart: Shows proportions.
Bar Chart: Shows counts.
Pareto Chart: Bar chart sorted by frequency.

Pie Chart Example:
A: 40%, B: 20%, C: 20%, D: 20%

4. Describing Numerical Data

4.1 Types of Variables

Discrete: Countable (e.g., number of people).
Continuous: Measurable (e.g., height, weight).

4.2 Organizing Data

Frequency Table: For discrete data.
Class Intervals: For continuous data.

Example:
Marks obtained by 50 students:

Class Interval	Frequency
30-40	3
40-50	6
50-60	18
60-70	17
70-80	4
80-90	2

4.3 Stem-and-Leaf Diagram

Shows individual data points.

Example:
Ages: 15, 22, 29, 36, 31, 23, 45, 10, 25, 28, 48

Stem	Leaf
1	0 5
2	2 3 5 8 9
3	1 6
4	5 8

5. Measures of Central Tendency and Dispersion

5.1 Mean

Sum of values divided by number of values.

Example:
Data: 2, 12, 5, 7, 6, 7, 3
Mean = (2+12+5+7+6+7+3)/7 = 6

5.2 Median

Middle value in ordered data.

Example:
Ordered data: 2, 3, 5, 6, 7, 7, 12
Median = 6 (4th value)

5.3 Mode

Most frequent value.

Example:
Data: 2, 12, 5, 7, 6, 7, 3
Mode = 7 (appears twice)

5.4 Range

Difference between maximum and minimum.

Example:
Data: 1, 2, 3, 4, 5
Range = 5 - 1 = 4

5.5 Variance and Standard Deviation

Measure of spread.

Example:
Data: 68, 79, 38, 68, 35, 70, 61, 47, 58, 66
Mean = (68+79+38+68+35+70+61+47+58+66)/10 = 59

Variance = Σ(xi - mean)² / n = 1898 / 10 = 189.8
Standard Deviation = √189.8 ≈ 13.78

6. Percentiles and Quartiles

6.1 Percentiles

Value below which a given percentage of data falls.

Example:
Data: 35, 38, 47, 58, 61, 66, 68, 68, 70, 79
25th percentile:
n = 10, p = 0.25
np = 2.5 → 3rd value = 47

6.2 Quartiles

Q1: 25th percentile
Q2: 50th percentile (median)
Q3: 75th percentile

Example:
From above, Q1 = 47, Q2 = (61+66)/2 = 63.5, Q3 = 68

7. Association Between Variables

7.1 Categorical Variables

Contingency Table: Shows counts for combinations of categories.
Stacked Bar Chart: Shows counts for each category.

Example:
Gender vs. Smartphone Ownership

Gender	No	Yes	Total
Female	10	34	44
Male	14	42	56

8. Numerical Variables

8.1 Scatter Plot

Shows relationship between two numerical variables.

Example:
Age vs. Height

Age (years)	Height (cm)
1	75
2	85
3	94
4	101
5	108

9. Counting Principles

9.1 Addition Rule

If you can do A in n₁ ways or B in n₂ ways, total = n₁ + n₂.

Example:
Choose a shirt (4 options) or a pant (3 options) → 4 + 3 = 7 ways

9.2 Multiplication Rule

If you can do A and B, total = n₁ × n₂.

Example:
Choose a shirt and a pant → 4 × 3 = 12 ways

10. Factorial, Permutation, and Combination

10.1 Factorial

n! = n × (n-1) × … × 1

Example:
5! = 5 × 4 × 3 × 2 × 1 = 120

10.2 Permutation

Arrangement of r objects from n:
nPr = n! / (n-r)!

Example:
Arrange 3 books from 5: 5P3 = 60

10.3 Combination

Selection of r objects from n:
nCr = n! / (r!(n-r)!)

Example:
Choose 2 students from 4: 4C2 = 6

11. Example Questions and Solutions

Q1: Find the mean, median, and mode of: 10, 20, 30, 40, 50, 60, 70, 80

Solution:

Mean: (10+20+30+40+50+60+70+80)/8 = 45
Median: (40+50)/2 = 45
Mode: No mode (all unique)

Q2: A class has 60 students. In how many ways can a captain and vice-captain be chosen?

Solution:
Order matters → Permutation
60P2 = 60 × 59 = 3540

Q3: From a pack of 52 cards, how many ways to choose 4 cards of the same suit?