Introduction to Statistics

Unit I · Lectures 4–5

BDS113 — Introduction to Statistics

Classification, Frequency Distribution & Cumulative Frequency

Organising Raw Data into Meaningful Patterns

Unit I · Lectures 4–5 of 40 · B.Sc. Data Science, 1st Year, Semester II

Introduction to Statistics

Unit I · Lectures 4–5

The Journey — Lectures 4 & 5

🗂️

Lecture 4 — Classification & Frequency Distribution
Types of classification · Key terms (class, tally, frequency) · Grouped vs Ungrouped · Constructing a frequency table

📈

Lecture 5 — Cumulative Frequency Distribution
"Less than" type · "More than" type · Construction · Conversion between types · Ogive introduction

Introduction to Statistics

Unit I · Lecture 4

🤔 Why Organise Raw Data?

📖 The Problem with Raw Data

Raw data (also called unorganised data) is data in its original form — as it was collected. It is often a jumbled mass of numbers with no apparent pattern. Drawing conclusions directly from raw data is like trying to find a specific book in a library where all books are thrown on the floor — impossible without organising them first.

💡 Analogy: A Cupboard

Imagine throwing all your clothes randomly into a cupboard. Finding a specific shirt would take forever. Now imagine organising them: shirts on one shelf, trousers on another, socks in a drawer. Finding anything becomes easy. Classification and frequency distribution do the same thing for data.

💬 Quote: "The goal is to turn data into information, and information into insight." — Carly Fiorina, former CEO of HP. Organisation is the first step in this transformation.

Introduction to Statistics

Unit I · Lecture 4

🗂️ Classification of Data

📖 Definition

Classification is the process of arranging or grouping data into classes (categories) or sub-classes on the basis of some common characteristics. It is the first step in organising raw data into a meaningful form for analysis.

Data can be classified on the basis of four criteria:

1. Chronological (Temporal) Classification

Data arranged according to time — years, months, quarters, days.

Example: Year-wise sales of a company:
2020: ₹50L, 2021: ₹62L, 2022: ₹71L, 2023: ₹85L

2. Geographical (Spatial) Classification

Data arranged according to place — region, state, country, city.

Example: State-wise population:
UP: 20 Cr, Maharashtra: 12 Cr, Bihar: 10 Cr

3. Qualitative Classification

Data arranged according to attributes (qualities) — gender, religion, education.

Example: Gender-wise enrollment:
Male: 250, Female: 310, Other: 5

4. Quantitative Classification

Data arranged according to numerical values — marks, income, height, weight.

Example: Marks-wise student count:
0–20: 5, 20–40: 12, 40–60: 28, 60–80: 35, 80–100: 20

Introduction to Statistics

Unit I · Lecture 4

🎬 Visual: Four Types of Classification

Click each type to explore its definition, basis of classification, and a full example:

📅

Chronological

🌍

Geographical

🏷️

Qualitative

🔢

Quantitative

Introduction to Statistics

Unit I · Lecture 4

📊 What is a Frequency Distribution?

📖 Definition

A frequency distribution is a tabular arrangement of data that shows each class (or value) and the number of observations (frequency) falling into that class. It condenses a large volume of raw data into a compact, meaningful summary.

💡 Analogy: Sorting Marbles by Colour

You have a bag of 100 marbles in different colours. To understand the composition, you sort them: 30 red, 25 blue, 20 green, 15 yellow, 10 white. This sorted table is a frequency distribution — each colour is a class, and the count (30, 25, etc.) is the frequency.

📖 Two Types of Frequency Distribution

1. Ungrouped (Simple) Frequency Distribution: Each individual value is listed separately with its frequency. Best when the data has few distinct values.

2. Grouped Frequency Distribution: Data is divided into class intervals (ranges), and the number of observations in each interval is counted. Best when data has many distinct values spread over a wide range.

Introduction to Statistics

Unit I · Lecture 4

📋 Ungrouped (Simple) Frequency Distribution

📖 When to Use

When the number of distinct values in the data is small (typically fewer than 15–20), we list each value individually and count how many times it appears. This is called an ungrouped or simple frequency distribution.

📝 Example: Number of Children per Family

Raw data from 20 families:
2, 1, 3, 2, 0, 1, 2, 3, 1, 0, 2, 1, 2, 3, 1, 0, 2, 1, 2, 1

📊 Solution: Simple Frequency Table

No. of Children (x)	Tally Marks	Frequency (f)
0	\|\|\|	3
1	\|\|\|\| \|\|	7
2	\|\|\|\| \|\|\|\| \|	7
3	\|\|\|	3
Total		Σf = 20

💡 Key Point: The sum of all frequencies must equal the total number of observations (Σf = N). This is a quick check for correctness.

Introduction to Statistics

Unit I · Lecture 4

📖 Key Terms of Grouped Frequency Distribution

1. Class (Class Interval)

A class is a group or range into which data is divided. E.g., 0–10, 10–20, 20–30 are three classes. Each class has a lower limit and an upper limit.

2. Class Limits

Each class has two boundaries:
• Lower Class Limit (LCL): The smallest value in the class. E.g., in 20–30, LCL = 20.
• Upper Class Limit (UCL): The largest value in the class. E.g., in 20–30, UCL = 30.

3. Class Width (Size / Length)

Class width = Upper Limit − Lower Limit.
In the class 20–30: width = 30 − 20 = 10.
All classes should ideally have equal width.

4. Class Mark (Midpoint)

Class Mark = (Lower Limit + Upper Limit) / 2.
In the class 20–30: midpoint = (20+30)/2 = 25.
The class mark represents the entire class in calculations.

5. Frequency (f)

The frequency of a class is the number of observations that fall within that class interval. It is counted using tally marks (groups of 5: ||||).

6. Tally Marks

A method of counting using vertical strokes in groups of five. The fifth mark crosses the previous four (||||). This makes counting easy: each group of 5 = 5 observations.

Introduction to Statistics

Unit I · Lecture 4

🎬 Visual: Anatomy of a Class Interval

Click "Reveal" to label each part of a class interval step by step:

Introduction to Statistics

Unit I · Lecture 4

📝 Grouped Frequency Distribution: Worked Example

📝 Problem

The marks (out of 100) of 30 students in a Statistics test are:
12, 35, 42, 56, 78, 23, 45, 67, 89, 15, 28, 51, 73, 95, 38, 62, 47, 84, 19, 33, 58, 71, 26, 44, 69, 91, 37, 53, 76, 60

Task: Construct a grouped frequency distribution with class width of 10.

📝 Step-by-Step Solution

Step 1: Find the range = Maximum value − Minimum value = 95 − 12 = 83.
Step 2: Decide the class width = 10 (given).
Step 3: Determine number of classes ≈ Range / Class width = 83 / 10 ≈ 9 classes.
Step 4: Start from the lowest value (or a convenient round number below it). Start at 10.
Step 5: Form classes: 10–20, 20–30, 30–40, 40–50, 50–60, 60–70, 70–80, 80–90, 90–100.
Step 6: Count observations in each class using tally marks.

Class Interval	Tally Marks	Frequency (f)	Class Mark (x)
10 – 20	\|\|\| \|\|	3	15
20 – 30	\|\|\| \|	4	25
30 – 40	\|\|\|\| \|	4	35
40 – 50	\|\|\|\|	4	45
50 – 60	\|\|\|\|	4	55
60 – 70	\|\|\|\| \|\|	5	65
70 – 80	\|\|\|\|	3	75
80 – 90	\|\|\|	2	85
90 – 100	\|\|	1	95
Total		Σf = 30

Introduction to Statistics

Unit I · Lecture 4

⚖️ Exclusive vs Inclusive Methods

There are two ways to form class intervals. The choice depends on whether the upper limit of one class is the same as the lower limit of the next class.

1. Exclusive Method (Continuous Classes)

The upper limit of one class is the lower limit of the next class. The upper limit is excluded from that class — it belongs to the next class. Classes are continuous (no gap between them).

📝 Example

Class	Includes
0 – 10	0 ≤ x < 10 (10 excluded)
10 – 20	10 ≤ x < 20 (20 excluded)
20 – 30	20 ≤ x < 30 (30 excluded)

A value of 10 falls in the class 10–20, NOT in 0–10.

💡 Note: The last class may include its upper limit. E.g., the class 90–100 includes both 90 and 100 when marks are out of 100.

2. Inclusive Method (Discontinuous Classes)

Both the lower limit and upper limit are included in the class. There is a gap between the upper limit of one class and the lower limit of the next. Classes are discontinuous.

📝 Example

Class	Includes
0 – 9	0, 1, 2, ... , 9 (both included)
10 – 19	10, 11, ... , 19 (both included)
20 – 29	20, 21, ... , 29 (both included)

A value of 10 falls in the class 10–19.

💡 Conversion: To convert inclusive to exclusive, subtract 0.5 from the lower limit and add 0.5 to the upper limit. E.g., 10–19 becomes 9.5–19.5.

Introduction to Statistics

Unit I · Lecture 4

🎬 Visual: Exclusive vs Inclusive — Number Line

Click each method to see how a value of 25 is classified:

Introduction to Statistics

Unit I · Lecture 4

⚠️ Open-End Classes

📖 What are Open-End Classes?

An open-end class is a class that has no lower limit or no upper limit (or both). This happens when extreme values exist in the data that do not fit neatly into regular intervals. Open-end classes are written with the word "Below" or "Above" (or symbols like "<" or ">").

📝 Example: Monthly Income of Workers

Monthly Income (₹)	Frequency
Below 5,000	8
5,000 – 10,000	15
10,000 – 20,000	22
20,000 – 50,000	30
50,000 and above	5
Total	80

📖 Identifying Open-End Classes

"Below 5,000" — No lower limit → open at the bottom.

"50,000 and above" — No upper limit → open at the top.

The classes 5,000–10,000, 10,000–20,000, and 20,000–50,000 have both limits → closed classes.

💡 Limitation: Open-end classes make it impossible to calculate the class mark (midpoint) precisely. This creates difficulties in calculating measures like mean. They should be used only when necessary.

Introduction to Statistics

Unit I · Lecture 4

✅ Rules for Constructing a Frequency Distribution

Number of classes: Typically between 5 and 15. Too few classes hide detail; too many defeat the purpose of grouping. A common rule: number of classes ≈ √N (where N is total observations) or use Sturges' formula: k = 1 + 3.322 × log₁₀(N).
Equal class width: All classes should ideally have the same width. Unequal widths are acceptable only when data is sparse in some ranges.
Exhaustive: Every observation must fall into exactly one class. No data point should be left out, and no data point should appear in two classes.
Mutually exclusive: Classes must not overlap. In the exclusive method, the boundary value belongs to the next class. In the inclusive method, there is a clear gap between classes.
Start from a round number: Begin classes from a convenient number (e.g., 0, 10, 100) rather than the exact minimum value. This makes the table easier to read.
Verification: The sum of all frequencies (Σf) must equal the total number of observations (N). If Σf ≠ N, there is an error.
Avoid too many open-end classes: Use open-end classes only for the extreme ends of the distribution when extreme values exist. Prefer closed classes wherever possible.

Introduction to Statistics

Unit I · Lecture 5

— Lecture 5 —

Cumulative Frequency Distribution

"Less Than" Type · "More Than" Type · Ogives · Applications

Introduction to Statistics

Unit I · Lecture 5

📈 What is Cumulative Frequency?

📖 Definition

A cumulative frequency is the running total of frequencies. Instead of showing how many observations fall in each class, it shows how many observations fall below (or above) a certain value. Cumulative frequency distribution is obtained by successively adding the frequencies.

💡 Analogy: Climbing Stairs

Imagine climbing stairs. At each step, you have climbed a certain number of steps so far. If Step 1 = 3 steps, Step 2 = 5 more, Step 3 = 2 more, then cumulatively you have climbed: 3, 8 (3+5), 10 (3+5+2). Cumulative frequency works the same way — it keeps a running total.

💡 Key Insight: A regular frequency table tells you "how many students scored between 40–50." A cumulative frequency table tells you "how many students scored below 50." The second question is often more useful in practice.

Introduction to Statistics

Unit I · Lecture 5

🔀 Two Types of Cumulative Frequency

1. "Less Than" Type (Ascending / Upward)

Cumulative frequencies are calculated from the top of the table, adding frequencies downward. It answers: "How many observations are less than the upper limit of each class?"

📝 How to Compute

Start with the frequency of the first class.
Add the frequency of the second class to the first cumulative frequency.
Continue adding until the last class.

Formula: CF(i) = f(1) + f(2) + ... + f(i)

2. "More Than" Type (Descending / Downward)

Cumulative frequencies are calculated from the bottom of the table, adding frequencies upward. It answers: "How many observations are more than or equal to the lower limit of each class?"

📝 How to Compute

Start with the frequency of the last class.
Add the frequency of the second-last class to the last cumulative frequency.
Continue adding upward until the first class.

Formula: CF(i) = f(i) + f(i+1) + ... + f(last)

Introduction to Statistics

Unit I · Lecture 5

📝 Worked Example: Cumulative Frequency Distribution

📝 Given Data

Marks obtained by 40 students in a Statistics exam:

Class Interval (Marks)	Frequency (f)
0 – 10	2
10 – 20	5
20 – 30	8
30 – 40	12
40 – 50	7
50 – 60	4
60 – 70	2
Total	40

📝 Solution: Both Types of Cumulative Frequency

Marks	f	"Less Than" CF (Cumulate ↓)	"More Than" CF (Cumulate ↑)
0 – 10	2	2	40
10 – 20	5	2 + 5 = 7	40 − 2 = 38
20 – 30	8	7 + 8 = 15	38 − 5 = 33
30 – 40	12	15 + 12 = 27	33 − 8 = 25
40 – 50	7	27 + 7 = 34	25 − 12 = 13
50 – 60	4	34 + 4 = 38	13 − 7 = 6
60 – 70	2	38 + 2 = 40	6 − 4 = 2

💡 Verification: The last "Less Than" CF must equal N (40). The first "More Than" CF must also equal N (40). The last "More Than" CF equals the frequency of the last class (2). If these checks fail, recheck your addition.

Introduction to Statistics

Unit I · Lecture 5

🎬 Visual: Build Cumulative Frequency Step by Step

Click "Less Than" or "More Than" to watch the cumulative frequencies being calculated row by row:

Introduction to Statistics

Unit I · Lecture 5

📊 "Less Than" Cumulative Frequency Table

📖 How to Read

In a "Less Than" CF table, the class boundaries replace class intervals. The cumulative frequency against a value x tells us: "How many observations are less than x."

📝 From Our Example

Marks (Less Than)	Cumulative Frequency
Less than 10	2
Less than 20	7
Less than 30	15
Less than 40	27
Less than 50	34
Less than 60	38
Less than 70	40

💡 How to Interpret

• "Less than 30": CF = 15
→ 15 students scored below 30 marks.

• "Less than 50": CF = 34
→ 34 students scored below 50 marks.

• "Less than 70": CF = 40
→ All 40 students scored below 70 marks (since max is out of 70).

💡 Note: In "Less Than" type, the first CF equals the first frequency, and the last CF equals N (total observations).

Introduction to Statistics

Unit I · Lecture 5

📊 "More Than" Cumulative Frequency Table

📖 How to Read

In a "More Than" CF table, the lower class limits are used. The cumulative frequency against a value x tells us: "How many observations are greater than or equal to x."

📝 From Our Example

Marks (More Than)	Cumulative Frequency
More than or equal to 0	40
More than or equal to 10	38
More than or equal to 20	33
More than or equal to 30	25
More than or equal to 40	13
More than or equal to 50	6
More than or equal to 60	2

💡 How to Interpret

• "More than 0": CF = 40
→ All 40 students scored 0 or above (trivially true).

• "More than 30": CF = 25
→ 25 students scored 30 or more.

• "More than 60": CF = 2
→ Only 2 students scored 60 or more.

💡 Note: In "More Than" type, the first CF equals N (total observations), and the last CF equals the frequency of the last class.

Introduction to Statistics

Unit I · Lecture 5

📈 What is an Ogive? (Introduction)

📖 Definition

An ogive (pronounced "oh-jive") is a graph of the cumulative frequency distribution. It is a smooth curve drawn by plotting the upper class boundaries (for "Less Than" type) or lower class boundaries (for "More Than" type) on the X-axis and the corresponding cumulative frequencies on the Y-axis, then joining the points.

Introduction to Statistics

Unit I · Lecture 5

📐 Finding the Median from an Ogive

📖 The Median — Quick Reminder

The median is the middle value of a dataset when arranged in order. For N observations, the median is the value of the (N/2)th observation. It divides the data into two equal halves — 50% below and 50% above.

📝 Steps to Find Median from "Less Than" Ogive

Step 1: Calculate N/2. For our example: N = 40, so N/2 = 20.

Step 2: Locate the value 20 on the Y-axis (cumulative frequency axis).

Step 3: Draw a horizontal line from Y = 20 to meet the ogive curve.

Step 4: From the point of intersection, draw a vertical line down to the X-axis.

Step 5: The value where the vertical line meets the X-axis is the median.

💡 Our Example

N/2 = 20. Looking at our ogive, the value 20 on the CF axis falls between the CF values 15 (at marks = 30) and 27 (at marks = 40).

By interpolation (or reading from the ogive), the median ≈ 33.3 marks.

Interpretation: 50% of students scored below 33.3 marks, and 50% scored above.

💡 Similarly: The "More Than" ogive can also be used. Both ogives intersect at the median point (the point where "Less Than" and "More Than" curves cross).

Introduction to Statistics

Unit I · Lecture 5

🎬 Visual: Find the Median from the Ogive

Click "Find Median" to see the construction lines drawn on the ogive:

Introduction to Statistics

Unit I · Lecture 5

🔗 Relationship: Frequency ↔ Cumulative Frequency

📖 Key Conversion Formulas

Frequency → Cumulative Frequency:
• "Less Than" CF at class i = f₁ + f₂ + ... + fᵢ (add downward)
• "More Than" CF at class i = fᵢ + fᵢ₊₁ + ... + fₙ (add upward)

Cumulative Frequency → Frequency:
• f(i) = CF(i) − CF(i−1) for "Less Than" type (subtract consecutive CF values)
• f(i) = CF(i) − CF(i+1) for "More Than" type (subtract consecutive CF values)

📝 Example: Recover Frequency from "Less Than" CF

Less Than	CF	f = CF(i) − CF(i−1)
Less than 10	2	2 − 0 = 2
Less than 20	7	7 − 2 = 5
Less than 30	15	15 − 7 = 8
Less than 40	27	27 − 15 = 12
Less than 50	34	34 − 27 = 7
Less than 60	38	38 − 34 = 4
Less than 70	40	40 − 38 = 2

Introduction to Statistics

Unit I · Lectures 4–5

🔑 Complete Summary: Lectures 4–5

Classification is the process of arranging data into groups based on common characteristics. Four types: Chronological (time), Geographical (place), Qualitative (attributes), Quantitative (numerical).
A frequency distribution is a table showing each class/value and the number of observations (frequency) in it. It can be ungrouped (few distinct values) or grouped (many values, divided into class intervals).
Key terms: Class (range), Class limits (LCL & UCL), Class width (UCL − LCL), Class mark ((LCL + UCL)/2), Frequency (count), Tally marks (counting tool).
Exclusive method: Upper limit of one class = lower limit of next class. The upper limit is excluded. Classes are continuous. (E.g., 0–10, 10–20)
Inclusive method: Both limits are included in each class. There is a gap between consecutive classes. (E.g., 0–9, 10–19). To convert to exclusive: subtract 0.5 from LCL, add 0.5 to UCL.
Open-end classes lack a lower or upper limit (e.g., "Below 5,000" or "50,000 and above"). They make midpoint calculations impossible — use only when necessary.
Cumulative frequency is the running total of frequencies. "Less Than" type adds frequencies downward (tells how many observations fall below a value). "More Than" type adds frequencies upward (tells how many observations are above or equal to a value).
An ogive is the graph of a cumulative frequency distribution. The "Less Than" ogive is an S-shaped curve rising from left to right. The "More Than" ogive falls from left to right.
The median can be found from the ogive: locate N/2 on the Y-axis, draw a horizontal line to the curve, then drop a vertical line to the X-axis. The intersection gives the median.
Frequency can be recovered from cumulative frequency by taking the difference of consecutive CF values: f(i) = CF(i) − CF(i−1).

Introduction to Statistics

Unit I · Lectures 4–5

🧠 Quick Quiz — Lectures 4–5

Q1: In the exclusive method, the class interval 20–30 includes which values?

A) 20 and 30 (both included)

B) 20 included, 30 excluded

C) 20 excluded, 30 included

D) Both 20 and 30 excluded

Q2: The class mark of the interval 40–60 is:

A) 40

B) 60

C) 50

D) 20

Q3: Classification of data by time is called:

A) Geographical

B) Chronological

C) Qualitative

D) Quantitative

Q4: The "Less Than" cumulative frequency for the last class always equals:

A) 0

B) The frequency of the last class

C) N (total number of observations)

D) The class width

Q5: The inclusive class 15–19 is converted to exclusive form as:

A) 15–20

B) 14.5–19.5

C) 16–19

D) 15.5–18.5

Q6: An ogive is a graph of:

A) Frequency distribution

B) Cumulative frequency distribution

C) Class marks vs frequencies

D) Tally marks

Q7: "Below 10" in a frequency distribution is an example of:

A) A closed class

B) An open-end class

C) An exclusive class

D) A class mark

Q8: If the "Less Than" CF values are 5, 12, 20, 28, 30, the frequencies are:

A) 5, 12, 20, 28, 30

B) 30, 25, 18, 10, 2

C) 5, 7, 8, 8, 2

D) 5, 17, 37, 65, 95

Introduction to Statistics

Unit I · Lectures 4–5

📝 Homework & Practice

📝 Practice Tasks

1. The following data gives the marks (out of 50) of 25 students:
   22, 35, 18, 41, 29, 33, 15, 27, 38, 45, 12, 31, 25, 40, 36, 20, 28, 34, 42, 19, 26, 37, 23, 30, 44

   (a) Arrange the data in ascending order.
   (b) Construct a grouped frequency distribution with class width of 5 using the exclusive method.
   (c) Identify all class marks and verify that the class width is uniform.

2. Convert the following inclusive classes to exclusive classes:
   (a) 10 – 14    (b) 20 – 29    (c) 100 – 199    (d) 0 – 4

3. Given the frequency distribution below, construct both "Less Than" and "More Than" cumulative frequency tables:

Class	5–10	10–15	15–20	20–25	25–30
f	3	7	10	6	4

4. The cumulative frequencies ("Less Than" type) of a distribution are: 4, 10, 18, 25, 28. Recover the original frequency distribution.

5. Draw a "Less Than" ogive for the data in Question 3 and use it to estimate the median.

6. BONUS: Explain why the sum of all frequencies must equal N. What would it mean if Σf ≠ N?

💡 Tip: Practice constructing frequency tables by hand — counting with tally marks builds your intuition. Then verify using the Σf = N check. This skill is fundamental to every topic that follows.

Introduction to Statistics

Unit I · Lectures 4–5

Thank You!

Questions? Let's discuss!

Upcoming in Unit I

Lecture 6–7: Representation of Data — Tabular, Graphical & Diagrammatic
Lecture 8: Review & Problem-Solving Session

Unit I · Lectures 4–5 · BDS113 Introduction to Statistics · B.Sc. Data Science, Semester II