NCERT Solutions for Class 6 Maths Chapter 4: Data Handling and Presentation — Complete Guide

Data is a collection of facts, numbers, or information gathered for a purpose. Data can be:

Qualitative (categorical): Describes qualities or categories that cannot be measured numerically. Examples: favourite colour (red, blue, green), mode of transport (bus, car, bicycle), type of pet (dog, cat, fish).

Quantitative (numerical): Consists of numbers that can be measured or counted. Examples: height ($145$ cm), marks ($87$ out of $100$), number of siblings ($2$).

Raw data is data that has not been organised or processed. For example, a list of $30$ students' favourite fruits written in the order they were asked is raw data. It is hard to draw conclusions from raw data — you need to organise it first.

Tally marks are a quick way to count occurrences. Each occurrence gets one mark ($|$). Every fifth mark crosses the previous four ($\cancel{||||}$), making it easy to count in groups of $5$.

A frequency table lists each category (or value) along with how often it occurs (its frequency).

Example: If students' favourite sports are Cricket, Football, Cricket, Tennis, Cricket, Football, Football, Cricket, Tennis, Cricket, the frequency table is:

The total of all frequencies must equal the total number of data points — this is your verification check.

A pictograph uses pictures or symbols to represent data. Each symbol represents a fixed number of items.

Key elements of a pictograph:
1. Title — tells you what the pictograph is about.
2. Categories — listed in a column or row.
3. Symbols — pictures representing data values.
4. Key (legend) — tells you how many items each symbol represents.

Reading a pictograph: Count the symbols for each category and multiply by the key value.

Drawing a pictograph: Divide each data value by the key value to find how many symbols to draw. If the division is not exact, use a fraction of a symbol (half symbol, quarter symbol).

Choosing the key: The key should be chosen so that the number of symbols is manageable — not too many and not too few. Common choices: $1, 2, 5, 10, 50, 100$ items per symbol.

A bar graph uses rectangular bars of equal width to represent data. The height (or length) of each bar is proportional to the value it represents.

Key elements of a bar graph:
1. Title — describes what the graph shows.
2. Horizontal axis (x-axis) — shows categories.
3. Vertical axis (y-axis) — shows numerical values (with a scale).
4. Bars — rectangles of equal width, with heights proportional to data values.
5. Scale — the numbering on the y-axis (e.g., $1$ unit = $10$ students).
6. Gaps — equal spaces between bars.

Advantages of bar graphs over pictographs:
- More precise (you can read exact values from the scale).
- Easier to compare categories (bar heights are easy to compare visually).
- Can handle larger data values without becoming unwieldy.
- No need for fractional symbols.

The scale of a bar graph determines how the y-axis is numbered. Choosing the right scale is important:

• If data values range from $0$ to $50$, a scale of $1$ unit $= 5$ works well (y-axis: $0, 5, 10, 15, \ldots, 50$).
  - If values range from $0$ to $500$, use $1$ unit $= 50$ or $1$ unit $= 100$.
  - The scale should start at $0$ and go up to at least the largest data value.
  - All bars should fit comfortably in the available space.

Common mistake: Not starting the y-axis at $0$. This can make small differences look large and is considered misleading.

Problem: The following are the favourite colours of $20$ students: Red, Blue, Green, Red, Blue, Red, Green, Blue, Red, Yellow, Blue, Red, Green, Red, Blue, Yellow, Green, Red, Blue, Red.

Organise this data into a frequency table using tally marks.

Solution:

Step 1: Go through the list one by one, making a tally mark for each colour.

Step 2: Count the tally marks for each colour.

Step 3: Verify: $8 + 6 + 4 + 2 = 20$ $\checkmark$

Observations:
- Red is the most popular colour ($8$ students) — this is called the mode.
- Yellow is the least popular ($2$ students).
- More than half the students chose either Red or Blue.

Answer: Frequency table completed. Red is the most popular colour with $8$ votes.

Problem: A survey shows the number of siblings students have:

(a) How many students were surveyed?
(b) What is the most common number of siblings?
(c) How many students have $2$ or more siblings?

Solution:

(a) Total students $= 5 + 12 + 8 + 3 + 2 = 30$.

(b) The highest frequency is $12$, corresponding to $1$ sibling. So having $1$ sibling is most common.

(c) Students with $2$ or more siblings $= 8 + 3 + 2 = 13$.

Answer: (a) $30$ students (b) $1$ sibling (most common) (c) $13$ students.

Problem: The marks obtained by $25$ students in a test (out of $10$) are: $7, 5, 8, 6, 9, 7, 8, 5, 6, 7, 8, 9, 10, 6, 7, 5, 8, 7, 6, 9, 7, 8, 6, 7, 8$.

Prepare a frequency table.

Solution:

Verification: $3 + 5 + 7 + 6 + 3 + 1 = 25$ $\checkmark$

Observations:
- The most common mark is $7$ (frequency $7$) — this is the mode.
- Only $1$ student scored full marks ($10$).
- $16$ students ($7 + 6 + 3$) scored $7$ or above.

Answer: Frequency table completed. Mode $= 7$.

Problem: The heights (in cm) of $20$ students are: $132, 148, 155, 140, 138, 152, 145, 160, 142, 157, 135, 150, 143, 148, 158, 139, 147, 153, 141, 156$.

Organise this data into groups of width $5$ cm.

Solution:

Verification: $1 + 3 + 4 + 4 + 4 + 3 + 1 = 20$ $\checkmark$

Why group data? When data has many different values, listing each one separately makes the table too long. Grouping gives a clearer picture of how the data is distributed.

Answer: Grouped frequency table completed. Most students ($12$ out of $20$) have heights between $140$ and $154$ cm.

Problem: The number of goals scored by a football team in $20$ matches is given:

Find (a) the total goals scored, (b) the mode, (c) the number of matches with fewer than $3$ goals.

Solution:

(a) Total goals $= 0 \times 3 + 1 \times 5 + 2 \times 6 + 3 \times 4 + 4 \times 1 + 5 \times 1$
$= 0 + 5 + 12 + 12 + 4 + 5 = 38$ goals.

(b) The highest frequency is $6$ (for $2$ goals). Mode $= 2$ goals.

(c) Matches with fewer than $3$ goals: $3 + 5 + 6 = 14$ matches.

Answer: (a) $38$ goals (b) Mode $= 2$ (c) $14$ matches.

Problem: Describe how you would collect data on the favourite subjects of students in your class.

Solution:

Step 1: Define the question. "What is your favourite school subject?"

Step 2: List the categories. Maths, Science, English, Social Studies, Hindi, Art/Music, Sports.

Step 3: Collect data. Ask each student and record their answer. Use tally marks on a pre-drawn table.

Step 4: Count and verify. Count tally marks for each subject. Verify that the total equals the number of students asked.

Step 5: Organise into a frequency table.

Step 6: Present. Draw a pictograph or bar graph to present your findings.

Tips for good data collection:
- Ask everyone the same question.
- Each person gives exactly one answer.
- Record responses immediately (do not rely on memory).
- Do not influence the answers ("Don't you think Maths is the best?").

Answer: Steps described for conducting a survey on favourite subjects.

Problem: In a class of $30$ students, $18$ like cricket and $15$ like football. If $8$ like both, how many like neither?

Solution:

Using the inclusion-exclusion principle:

Students who like at least one sport $=$ (Cricket) $+$ (Football) $-$ (Both)
$= 18 + 15 - 8 = 25$

Students who like neither $=$ Total $-$ At least one
$= 30 - 25 = 5$

Answer: $5$ students like neither sport.

Problem: Find the mean of the data:

Solution:

Mean $= \frac{\sum f \times x}{\sum f}$

$\sum f \times x = 2(3) + 4(5) + 6(7) + 8(4) + 10(1)$
$= 6 + 20 + 42 + 32 + 10 = 110$

$\sum f = 3 + 5 + 7 + 4 + 1 = 20$

Mean $= \frac{110}{20} = 5.5$

Answer: The mean is $5.5$.

Problem: A pictograph shows the number of books read by students in a month, where each symbol represents $2$ books.

• Anu: 4 symbols
  - Bina: 3 symbols
  - Charu: 5 symbols
  - Dev: 2 symbols

(a) How many books did each student read? (b) Who read the most? (c) Total books read?

Solution:

Since each symbol $= 2$ books:

(a)
- Anu: $4 \times 2 = 8$ books
- Bina: $3 \times 2 = 6$ books
- Charu: $5 \times 2 = 10$ books
- Dev: $2 \times 2 = 4$ books

(b) Charu read the most ($10$ books).

(c) Total $= 8 + 6 + 10 + 4 = 28$ books.

Answer: (a) Anu: $8$, Bina: $6$, Charu: $10$, Dev: $4$. (b) Charu. (c) $28$ books.

Problem: A pictograph shows the number of cars sold by a dealer each month. Each symbol represents $10$ cars.

• January: 3 symbols
  - February: 2.5 symbols
  - March: 4 symbols
  - April: 3.5 symbols

How many cars were sold each month? What is the total?

Solution:

• January: $3 \times 10 = 30$ cars
  - February: $2.5 \times 10 = 25$ cars
  - March: $4 \times 10 = 40$ cars
  - April: $3.5 \times 10 = 35$ cars

Total $= 30 + 25 + 40 + 35 = 130$ cars.

Note: The half symbol ($0.5$) represents $5$ cars (half of the key value $10$).

Answer: January: $30$, February: $25$, March: $40$, April: $35$. Total: $130$ cars.

Problem: Draw a pictograph for the following data using a key of $1$ symbol $= 5$ students.

Solution:

Divide each value by $5$ to find the number of symbols:
- Apple: $20 \div 5 = 4$ symbols
- Banana: $15 \div 5 = 3$ symbols
- Mango: $30 \div 5 = 6$ symbols
- Orange: $10 \div 5 = 2$ symbols
- Grapes: $25 \div 5 = 5$ symbols

Favourite Fruits of Students (Key: each symbol $= 5$ students)

At a glance, you can see Mango is the most popular and Orange is the least popular.

Answer: Pictograph drawn with key: $1$ symbol $= 5$ students.

Problem: The number of trees planted by four classes is: Class A: $42$, Class B: $36$, Class C: $54$, Class D: $48$. Choose an appropriate key and draw a pictograph.

Solution:

Step 1: Look at the data values: $42, 36, 54, 48$. All are multiples of $6$.

Step 2: Choose key: $1$ symbol $= 6$ trees.

Step 3: Calculate symbols:
- Class A: $42 \div 6 = 7$ symbols
- Class B: $36 \div 6 = 6$ symbols
- Class C: $54 \div 6 = 9$ symbols
- Class D: $48 \div 6 = 8$ symbols

Alternative key: $1$ symbol $= 12$ trees would give $3.5, 3, 4.5, 4$ symbols — this works too, using half symbols.

Answer: Key of $6$ trees per symbol works well, giving $7, 6, 9, 8$ symbols respectively.

Problem: Two pictographs show the production of two factories. Factory A produces $200$ units (shown with $4$ symbols) and Factory B produces $350$ units (shown with $7$ symbols). What is the key? How many more units does Factory B produce?

Solution:

Factory A: $4$ symbols $= 200$ units, so each symbol $= \frac{200}{4} = 50$ units.

Verify with Factory B: $7 \times 50 = 350$ units. $\checkmark$

Difference: $350 - 200 = 150$ units (or $3$ symbols).

Answer: Key: $1$ symbol $= 50$ units. Factory B produces $150$ units more.

Problem: Why might a pictograph not be suitable for the data: City A population $= 2{,}350{,}000$, City B $= 1{,}780{,}000$, City C $= 3{,}120{,}000$?

Solution:

With a key of $1$ symbol $= 100{,}000$ people:
- City A: $23.5$ symbols
- City B: $17.8$ symbols
- City C: $31.2$ symbols

This requires drawing many symbols AND using fractional symbols that are hard to draw precisely.

A bar graph would be much more suitable for this data because:
1. You can use a scale on the y-axis to represent large numbers precisely.
2. There is no need for fractional symbols.
3. The exact values can be read from the scale.

Answer: A pictograph is impractical for large numbers with non-round values. A bar graph is more appropriate.

Problem: A bar graph shows the marks scored by five students. The y-axis scale goes from $0$ to $100$ in steps of $10$. The bars show: Asha ($70$), Bala ($85$), Chitra ($60$), Deepak ($90$), Esha ($75$). Answer the following:

(a) Who scored the highest? (b) Who scored the lowest? (c) What is the difference between the highest and lowest marks? (d) How many students scored above $70$?

Solution:

(a) Deepak scored the highest: $90$ marks.

(b) Chitra scored the lowest: $60$ marks.

(c) Difference $= 90 - 60 = 30$ marks.

(d) Students scoring above $70$: Bala ($85$), Deepak ($90$), Esha ($75$) $= 3$ students.

Note: Asha scored exactly $70$, which is NOT above $70$.

Answer: (a) Deepak ($90$) (b) Chitra ($60$) (c) $30$ marks (d) $3$ students.

Problem: In a bar graph, the y-axis shows values $0, 20, 40, 60, 80, 100$. A bar reaches up to the line between $60$ and $80$. What value does this bar represent?

Solution:

The line between $60$ and $80$ is at $\frac{60 + 80}{2} = 70$.

So the bar represents $70$.

Tip: When a bar falls between two gridlines, estimate the value by looking at where it falls. If it is exactly halfway, take the average. If it is closer to one gridline, adjust accordingly.

Answer: The bar represents $70$.

Problem: A bar graph shows the number of students in five different clubs:

(a) Which club has the most members? (b) How many more members does Sports have than Art? (c) What is the total membership across all clubs? (d) What fraction of total members are in the Science club?

Solution:

(a) Sports club has the most members: $55$.

(b) Difference $= 55 - 25 = 30$ more members.

(c) Total $= 45 + 30 + 55 + 25 + 35 = 190$ members.

(d) Fraction in Science $= \frac{45}{190} = \frac{9}{38}$.

Answer: (a) Sports ($55$) (b) $30$ more (c) $190$ (d) $\frac{9}{38}$.

Problem: A bar graph shows a city's monthly rainfall (in mm) for January through June: $10, 15, 30, 55, 80, 120$. Describe the trend.

Solution:

The rainfall increases every month:
- Jan to Feb: $+5$ mm
- Feb to Mar: $+15$ mm
- Mar to Apr: $+25$ mm
- Apr to May: $+25$ mm
- May to Jun: $+40$ mm

Trend: The rainfall shows a steadily increasing trend from January to June, with the rate of increase also growing. This is consistent with the approach of the monsoon season.

The total rainfall over six months: $10 + 15 + 30 + 55 + 80 + 120 = 310$ mm.

June alone accounts for $\frac{120}{310} \approx 38.7\%$ of the total.

Answer: Rainfall increases steadily, with June having the highest rainfall ($120$ mm).

Problem: A horizontal bar graph shows the production of wheat (in thousands of tonnes) by five states. The x-axis goes from $0$ to $300$ in steps of $50$. State A's bar extends to $250$, State B to $175$, State C to $200$, State D to $125$, State E to $300$.

(a) Which state produces the most wheat? (b) What is the combined production of States B and D? (c) How much more does State E produce than State A?

Solution:

(a) State E produces the most: $300$ thousand tonnes.

(b) Combined $= 175 + 125 = 300$ thousand tonnes (same as State E alone!).

(c) Difference $= 300 - 250 = 50$ thousand tonnes.

Answer: (a) State E ($300$ thousand tonnes) (b) $300$ thousand tonnes (c) $50$ thousand tonnes.

Problem: A bar graph shows Company A's sales as $100$ units and Company B's sales as $110$ units. But the y-axis starts at $95$ instead of $0$. Why is this misleading?

Solution:

When the y-axis starts at $95$:
- Company A's bar height is $100 - 95 = 5$ units tall.
- Company B's bar height is $110 - 95 = 15$ units tall.

Company B's bar appears three times as tall as Company A's, even though its sales are only $10\%$ higher ($110$ vs. $100$).

If the y-axis started at $0$:
- Company A's bar would be $100$ units tall.
- Company B's bar would be $110$ units tall.

Company B's bar would be only $10\%$ taller — a much more accurate visual representation.

Lesson: Always check if the y-axis starts at $0$. A truncated axis can make small differences look dramatic.

Answer: Starting the y-axis at $95$ instead of $0$ exaggerates the visual difference between the two companies.

Problem: Draw a bar graph for the following data:

Solution:

Step 1: Choose the scale. Data ranges from $10$ to $40$. A scale of $1$ unit $= 5$ students works well. Y-axis: $0, 5, 10, 15, 20, 25, 30, 35, 40$.

Step 2: Draw the axes. Horizontal axis for modes of transport. Vertical axis for number of students.

Step 3: Label the axes. X-axis: "Mode of Transport." Y-axis: "Number of Students." Title: "Modes of Transport Used by Students."

Step 4: Draw the bars. Each bar has equal width (say, $1$ cm) with equal gaps (say, $0.5$ cm).
- Bus: bar height $= 40$ (reaches the $40$ line)
- Car: bar height $= 25$
- Bicycle: bar height $= 15$
- Walk: bar height $= 30$
- Auto: bar height $= 10$

Step 5: Verify. Check that each bar height matches the data value. Ensure all bars have the same width and equal spacing.

Answer: Bar graph drawn with $5$ bars, scale of $5$ students per unit.

Problem: Draw a bar graph for the population of five cities (in thousands): Delhi ($200$), Mumbai ($180$), Bangalore ($120$), Chennai ($90$), Kolkata ($150$).

Solution:

Data ranges from $90$ to $200$ (in thousands). Choose scale: $1$ unit $= 20$ thousand.

Y-axis labels: $0, 20, 40, 60, 80, 100, 120, 140, 160, 180, 200$.

Bar heights:
- Delhi: $200$ (top of scale)
- Mumbai: $180$
- Bangalore: $120$
- Chennai: $90$ (between $80$ and $100$ gridlines)
- Kolkata: $150$ (between $140$ and $160$)

Tip for Chennai: Since $90$ falls between gridlines, carefully estimate its position (halfway between $80$ and $100$).

Answer: Bar graph drawn with scale of $20$ thousand per unit.

Problem: The marks of $5$ students in Maths and Science are:

Draw a double bar graph and compare.

Solution:

A double bar graph shows two bars side by side for each category, using different colours or patterns.

Scale: $1$ unit $= 10$ marks. Y-axis: $0, 10, 20, \ldots, 100$.

For each student, draw two bars:
- Maths bar (e.g., blue): heights $80, 65, 90, 75, 85$
- Science bar (e.g., orange): heights $70, 75, 85, 80, 90$

Observations:
- Anu and Chitra scored higher in Maths than Science.
- Bala, Dev, and Esha scored higher in Science than Maths.
- Chitra has the highest Maths score ($90$).
- Esha has the highest Science score ($90$).

Include a legend showing which colour represents Maths and which represents Science.

Answer: Double bar graph drawn comparing Maths and Science marks.

Problem: A class survey on favourite sports gave: Cricket ($15$), Football ($12$), Basketball ($8$), Tennis ($5$), Badminton ($10$). Draw a bar graph and answer: What percentage of students chose Cricket?

Solution:

Total students $= 15 + 12 + 8 + 5 + 10 = 50$.

Scale: $1$ unit $= 2$ students. Y-axis: $0, 2, 4, 6, 8, 10, 12, 14, 16$.

Percentage choosing Cricket $= \frac{15}{50} \times 100 = 30\%$.

Answer: $30\%$ of students chose Cricket. Bar graph drawn with appropriate scale.

Problem: A bar graph shows monthly sales. January: $50$, February: $45$, March: $60$, April: $55$, May: $70$, June: $65$. In which months did sales increase compared to the previous month?

Solution:

Compare each month to the previous:
- Feb ($45$) vs. Jan ($50$): Decrease ($-5$)
- Mar ($60$) vs. Feb ($45$): Increase ($+15$)
- Apr ($55$) vs. Mar ($60$): Decrease ($-5$)
- May ($70$) vs. Apr ($55$): Increase ($+15$)
- Jun ($65$) vs. May ($70$): Decrease ($-5$)

Answer: Sales increased in March and May.