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Data Handling and Presentation

NCERT Class 6 · Mathematics Based on NCERT Class 6 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter introduces students to the concept of data — any collection of facts, numbers, measures, observations, or descriptions that convey information. It explains why we collect and organise data, and how to present it visually using tally marks, frequency tables, pictographs, and bar graphs. The big idea is that data, when properly organised and presented, helps us quickly understand, compare, and make inferences about the world around us, such as finding the most popular game in class or the busiest traffic hour.
Key Concepts (10)
Data
Any collection of facts, numbers, measures, observations or other descriptions of things that convey information about those things is called data.
e.g. A list of favourite colours collected from classmates is data.
Tally mark
A mark '|' used to count items; when the count reaches 5, a line is drawn through the previous four marks to make a group of five: ||||.
e.g. To count 7 students who chose rasgulla, we write |||| ||.
Frequency
The number of times a particular value or observation occurs in a data set.
e.g. In the sweet preference table, the frequency of students who chose jalebi is 6.
Pictograph
A visual representation of data using pictures or symbols, where each picture or symbol represents a fixed number of items (the scale).
e.g. A pictograph showing modes of travel where one bus symbol represents 1 student.
Scale (in pictograph)
The key that tells what each symbol or picture represents in a pictograph.
e.g. In a pictograph of tractors, if one tractor symbol = 1 tractor, the scale is '= 1 Tractor'.
Bar graph
A visual representation of data using bars of uniform width, where the length or height of each bar represents the frequency or quantity for that category.
e.g. A bar graph showing the number of students absent in each class, with bars of different heights.
Scale (in bar graph)
The conversion factor that tells how many units of the quantity are represented by one unit length of a bar.
e.g. In a bar graph of population, 1 unit length = 10 crores.
Column graph
A bar graph with vertical bars (columns).
e.g. A bar graph showing the heights of mountains with vertical bars is called a column graph.
Infographic
A data visualisation that is beautified with extensive artistic and visual imagery to communicate information more clearly and quickly in a visually pleasing way.
e.g. A picture of mountain-shaped triangles instead of rectangular bars to represent mountain heights.
Frequency distribution table
A table that organises data by listing each category or value alongside its frequency (often using tally marks).
e.g. The table of sweet preferences with columns for sweet name, tally marks, and number of students.
Formulas & Equations (4)
Frequency = Number of times an observation occurs
Reads as: Frequency equals the number of times an observation occurs
Variables: No variables; it is a definition.
Use when: When counting how many times a particular value appears in a data set.
⚠ Students sometimes confuse frequency with the total number of observations.
Total number of items = Sum of frequencies of all categories
Reads as: Total number of items equals the sum of the frequencies of all categories
Variables: No variables; it is a relationship.
Use when: When finding the total number of observations from a frequency table.
⚠ Students forget to add frequencies for all categories, especially if some categories have zero frequency.
Scale: 1 unit length = k units of the quantity (e.g., 1 unit length = 10 crores)
Reads as: One unit length on the graph represents k units of the quantity being measured
Variables: k = the number of units represented by one unit length on the graph
Use when: When drawing a bar graph or pictograph to decide how tall or long each bar or symbol should be.
⚠ Students forget to start the scale from zero on the vertical axis, leading to misleading graphs.
Height of bar (in units) = Frequency ÷ Scale factor
Reads as: Height of bar in units equals frequency divided by the scale factor
Variables: Frequency = the actual number; Scale factor = number of units represented by 1 unit length
Use when: When calculating how many units tall to draw a bar in a bar graph.
⚠ Students forget to divide by the scale factor and draw bars with heights equal to the raw frequency, making the graph too large.
How It Works
Collecting and Organising Data
  1. Step 1: Identify the question or inquiry (e.g., 'What is the most popular game in our class?').
  2. Step 2: Collect data by asking each person or observing each item (e.g., ask each student their favourite game).
  3. Step 3: Record the raw data as a list.
  4. Step 4: Organise the data using tally marks in a frequency distribution table.
  5. Step 5: Analyse the table to answer the original question (e.g., find the category with the highest frequency).
💡 The raw list is data, but it is not useful until it is organised. Always use tally marks to count frequencies accurately.
Drawing a Pictograph
  1. Step 1: Choose a symbol or picture to represent the items.
  2. Step 2: Decide on a scale (e.g., 1 symbol = 5 students).
  3. Step 3: For each category, draw the appropriate number of symbols (including half symbols if needed).
  4. Step 4: Add a key or scale to show what each symbol represents.
  5. Step 5: Label the categories and give the pictograph a title.
💡 If the frequency is not an exact multiple of the scale, you may need to use a fraction of a symbol (e.g., half a symbol for 5 when scale is 10).
Drawing a Bar Graph
  1. Step 1: Draw a horizontal line (x-axis) and a vertical line (y-axis).
  2. Step 2: On the x-axis, write the names of the categories with equal spacing.
  3. Step 3: Choose a suitable scale for the y-axis (e.g., 1 unit length = 10 runs).
  4. Step 4: Mark the scale on the y-axis starting from zero.
  5. Step 5: For each category, draw a bar of uniform width with height equal to (frequency ÷ scale factor) in units.
  6. Step 6: Keep uniform gaps between bars.
  7. Step 7: Label the axes and give the graph a title.
💡 The scale must start from zero. Bars must be of equal width and have equal spaces between them.
Interpreting a Bar Graph
  1. Step 1: Read the title to understand what data is shown.
  2. Step 2: Look at the labels on the x-axis to identify the categories.
  3. Step 3: Look at the scale on the y-axis to know what each unit length represents.
  4. Step 4: Compare the heights of the bars to find the highest (maximum) and lowest (minimum) values.
  5. Step 5: Use the scale to read the exact value for each bar.
  6. Step 6: Answer questions by comparing or adding values from different bars.
💡 Always check the scale before reading values. A bar that is 5 units tall may represent 50 if the scale is 1 unit = 10.
Exam Traps — Don't Make These Mistakes
⚠ Confusing tally marks with the final frequency
✗ Students write the tally marks themselves as the answer, e.g., '||||' instead of '4'.
→ Tally marks are a counting tool, not the final answer. The examiner expects the numerical frequency.
✓ Write the number, e.g., '4 students' or 'frequency = 4'.
⚠ Forgetting the scale in a pictograph or bar graph
✗ Students draw a pictograph or bar graph without including the key or scale.
→ Without a scale, the reader cannot know what each symbol or unit length represents, making the graph meaningless.
✓ Always include a clear key or scale, e.g., '= 10 children' or '1 unit length = 5 students'.
⚠ Drawing bars with heights equal to raw frequency when scale is not 1:1
✗ For a bar graph with scale 1 unit = 10 runs, a student draws a bar of height 80 for a score of 80 runs.
→ The bar height must be in units of the scale, not the raw number. 80 runs should be drawn as 8 units (80 ÷ 10).
✓ Calculate the bar height: frequency ÷ scale factor. For 80 runs with scale 1 unit = 10 runs, draw a bar of height 8 units.
⚠ Misreading a half symbol in a pictograph
✗ Students count a half symbol as 1 full unit.
→ A half symbol represents half of the scale value. If 1 symbol = 10 children, half a symbol = 5 children.
✓ Count the half symbol as half of the scale value. For scale = 10, half symbol = 5.
⚠ Not starting the vertical axis from zero
✗ Students sometimes start the y-axis from a number greater than zero to make differences look bigger.
→ Bar graphs must start from zero to accurately represent proportions. Starting from a non-zero value is misleading.
✓ Always mark zero at the bottom of the vertical axis and scale up from there.
⚠ Confusing 'data collection needed' vs 'not needed'
✗ Students say data collection is needed for questions like 'When did India get independence?'
→ That is a factual question with a known answer; no new data needs to be collected from people.
✓ Data collection is needed only when the answer is not already known and must be gathered from observations or surveys.
Compare & Contrast
Method of visual representation
Pictograph
Bar graph
A pictograph uses pictures or symbols to represent data, while a bar graph uses rectangular bars of uniform width. Pictographs can be more visually appealing but are harder to draw when data is large or not exact multiples of the scale; bar graphs are more precise and easier to draw for large data sets.
Role in data organisation
Tally mark
Frequency
A tally mark is a single stroke used to count items (a tool), while frequency is the final numerical count of how many times an observation occurs (the result). Tally marks are grouped in fives to make counting easier; frequency is the total number of tally marks for a category.
Exam Focus
Marks pattern: 2-3 MCQs (1 mark each), 1 Short Answer (2-3 marks) on drawing or interpreting a pictograph or bar graph, 1 Long Answer (4-5 marks) on collecting, organising, and presenting data in a table and graph.
Most asked: Drawing and interpreting bar graphs · Drawing and interpreting pictographs · Creating frequency distribution tables using tally marks · Choosing the correct scale for a graph · Identifying the most/least frequent category from a graph or table
Don't skip: Draw a bar graph from a given frequency table and answer questions based on it.; Given a pictograph, find the total number of items or compare two categories.; Complete a frequency table using tally marks from raw data.
One-Line Revision (read aloud before the exam)
  1. Data is any collection of facts, numbers, measures, observations, or descriptions that convey information.
  2. Tally marks are used to count frequencies: each group of five is written as ||||.
  3. Frequency is the number of times a particular observation occurs in a data set.
  4. A pictograph uses pictures or symbols to represent data, with a key showing what each symbol means.
  5. A bar graph uses bars of uniform width and equal spacing to represent data visually.
  6. The scale of a bar graph or pictograph must be clearly stated and must start from zero.
  7. To find the most popular item, look for the category with the highest frequency.
  8. Arranging data in ascending order makes it easier to find the smallest and largest values.
  9. Data collection is needed when the answer is not already known and must be gathered from observations or surveys.
  10. A frequency distribution table organises data into categories with their frequencies.
  11. When drawing a bar graph, the height of the bar = frequency ÷ scale factor.
  12. Infographics are visually appealing but can be misleading if not drawn accurately.
  13. A column graph is a bar graph with vertical bars.
  14. The total number of items in a data set is the sum of all frequencies.

MCQs — 10 Questions with Answers

Q1. What is data?

  • A. Any collection of facts, numbers, measures and observations ✓
  • B. Only numbers written on paper
  • C. Questions you ask your teacher
  • D. Pictures drawn in a book

Answer: A — Data is any collection of facts, numbers, measures, and observations that give us information about things.

Q2. When should you collect data?

  • A. When you want to know something that only people can tell you ✓
  • B. When you already know the answer from learning
  • C. When you want to draw pictures
  • D. Only on Mondays

Answer: A — You collect data when you need new information from people or observations, not when you already know the answer.

Q3. What does one tally mark (|) represent?

  • A. 5 counts
  • B. 1 count ✓
  • C. 10 counts
  • D. 3 counts

Answer: B — One tally mark represents 1 count, and when you reach 5, you draw a line through the previous four marks to show a group of 5.

Q4. If a table shows: Jalebi - 6 students, Gulab jamun - 9 students, Gujiya - 12 students, which sweet was chosen by the most students?

  • A. Jalebi
  • B. Gulab jamun
  • C. Gujiya ✓
  • D. Cannot determine

Answer: C — By comparing the frequency numbers (6, 9, 12), Gujiya has the highest number, so it was chosen by the most students.

Q5. Shoe sizes in a class arranged in order are: 3, 3, 4, 4, 4, 5, 5, 5, 5, 6. What is the frequency of size 5?

  • A. 2
  • B. 3
  • C. 4 ✓
  • D. 5

Answer: C — Counting how many times '5' appears in the arranged list: 5, 5, 5, 5 = frequency of 4 students wear size 5.

Q6. What does the frequency of a choice mean?

  • A. The colour of the data
  • B. How many times that choice was selected ✓
  • C. The name of the person
  • D. The date of collection

Answer: B — Frequency tells you how many people chose a particular option or how many times something appears in your data.

Q7. You have a list: Cricket, Hockey, Cricket, Football, Hockey, Cricket. If you arrange this data by frequency, which sport appears most often?

  • A. Football (1 time)
  • B. Hockey (2 times)
  • C. Cricket (3 times) ✓
  • D. All appear equally

Answer: C — Counting: Cricket appears 3 times, Hockey 2 times, Football 1 time, so Cricket is most frequent.

Q8. In a pictograph, one car symbol = 2 students. If there are 5 car symbols in a row, how many students does that represent?

  • A. 5 students
  • B. 10 students ✓
  • C. 7 students
  • D. 2 students

Answer: B — Since one symbol = 2 students, 5 symbols = 5 × 2 = 10 students; the key tells you what each symbol means.

Q9. Which of the following questions requires you to collect data?

  • A. What is the capital of India?
  • B. What is the favourite colour among your classmates? ✓
  • C. When did India get independence?
  • D. What is 5 + 5?

Answer: B — Only the favourite colour question needs data collection by asking classmates; the others are factual questions with fixed answers.

Q10. If shoe sizes are arranged as: 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, what is the smallest shoe size?

  • A. 6
  • B. 5
  • C. 4
  • D. 3 ✓

Answer: D — When data is arranged in ascending order (smallest to largest), the smallest shoe size is the first number: 3.

Flashcards

What is data?

Data is any collection of facts, numbers, measures, observations or descriptions of things that give us information about those things.

How do you collect data from your class?

Ask each student the question, write down their answers, and make a list of all the responses.

What is a tally mark in data handling?

A tally mark is a small line (|) used to count, and when you reach 5, you draw a line through the previous four (||||) to make a group of 5.

What does frequency mean?

Frequency is the number of times something appears or how many people chose a particular option in your data.

How should you arrange data to find patterns?

Arrange data in ascending order (smallest to largest) or descending order (largest to smallest) to easily spot which items appear most or least often.

What is a pictograph?

A pictograph is a way to show data using pictures or symbols instead of writing numbers.

What must a pictograph always have?

A pictograph must have a key that tells you what each symbol or picture means (for example, one car symbol = 1 student).

Why is organising data useful?

Organising data helps you find answers quickly, spot which items are most or least popular, and make smart decisions based on patterns.

How do tally marks help count faster?

Tally marks help you count as you go, and grouping them in fives (||||) makes it easy to add up the total at the end.

What question should you ask before collecting data?

Ask yourself: 'Do I need to collect new information, or do I already know the answer from what I have learned?'

Important Board Questions

Define data in one sentence. [1 mark]

Data is a collection of facts, numbers, measures, observations or descriptions. Give a simple example like favourite games or shoe sizes.

Complete the tally mark table: If 8 students chose 'Hockey' and you want to show this using tally marks, write how many complete groups of 5 and how many extra marks you need. [2 marks]

One group of tally marks = 4 marks + 1 line through = 5 total. Divide 8 by 5 to find complete groups and remainder.

Your teacher asked 20 students about their favourite fruits. You collected: 8 like Mangoes, 7 like Bananas, 5 like Apples. Arrange these in ascending order of frequency and explain why arranging data helps you make decisions. [3 marks]

Ascending order means smallest to largest frequency. Explain that it helps you see which fruit is most/least popular so the teacher knows what to buy more of.

A pictograph shows modes of travel with the key: one bus symbol = 3 students. If 4 symbols are shown for 'School Bus', 2 symbols for 'Cycle', and 3 symbols for 'Walking', (a) How many students travel by school bus? (b) Which mode is most popular? (c) Draw a simple pictograph showing this data. [5 marks]

Multiply: 4 symbols × 3 = 12 students; 2 × 3 = 6 students; 3 × 3 = 9 students. Compare numbers to find most popular. Draw rows with symbols and label them clearly with the key.

True or False: (a) Frequency means the colour of the data. (b) Ascending order means arranging from largest to smallest. Give one reason for each answer. [2 marks]

(a) False — frequency means how many times something appears. (b) False — ascending is small to large; descending is large to small.

Assertion-Reason Questions for Data Handling and Presentation

Q1. Assertion (A):

Data is any collection of facts, numbers, or observations that gives us information.

Reason (R):

When we ask students about their favourite colour and write down the answers, we are collecting data.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both statements are true, and the reason correctly explains what data is through a clear example. Collecting information is the definition of data collection.

Q2. Assertion (A):

Tally marks help us organize and count data quickly.

Reason (R):

Tally marks are pictures that look like objects in the data.

Answer: C — A is true but R is false — The assertion is true—tally marks do help count quickly—but the reason is false. Tally marks are simple lines (| or ||||), not pictures of objects. That would be a pictograph.

Q3. Assertion (A):

Arranging data in ascending order means arranging from smallest to largest numbers.

Reason (R):

This helps us easily find the smallest value, largest value, and compare different numbers in the data.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both are true and the reason correctly explains WHY ascending order is useful. It makes it easy to see patterns and find minimum and maximum values quickly.

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Case-Based Questions (CBQ) for Data Handling and Presentation

Case Study:

Ravi's mother asked him to find out which fruit the students in his class like the most. Ravi asked all 30 students and recorded their answers using tally marks. He found: Mango (|||| ||||), Apple (||||), Banana (|||| ||), Orange (|||). Now Ravi wants to decide which fruit to buy for the class party.

Q1. How many students chose Mango as their favourite fruit?

  • A. 10 students ✓
  • B. 9 students
  • C. 5 students
  • D. 3 students

Answer: A

Q2. Which fruit should Ravi buy the most for the party?

  • A. Apple
  • B. Banana
  • C. Mango ✓
  • D. Orange

Answer: C

Q3. How many students in total chose Apple or Orange?

  • A. 7 students
  • B. 8 students ✓
  • C. 9 students
  • D. 10 students

Answer: B

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