Data Handling and Presentation
Why This Matters
Look around you. Numbers and facts are everywhere.
How many students in your class like cricket? How many like football? Which day did the most books get borrowed from the school library? Which sweet should the teacher buy more of for a party?
To answer questions like these, you first need facts. A collection of such facts is called data. But here is the problem. A long list of facts is hard to read. Imagine a list of 40 students’ favourite games, all jumbled up. You cannot tell at a glance which game wins.
So we do two things. First, we organise the data neatly. Second, we turn it into a picture — a pictograph or a bar graph. A picture lets you see the answer in one quick look.
This skill is useful for life, not just for the exam. News on TV, cricket scores, weather charts, prices in the market — they all use these same pictures. Once you learn to read and draw them, you can understand the world a little better.
The Big Idea
Data is just a collection of facts or numbers. On its own, a pile of facts is messy and hard to read. So we organise it (using tally marks and a frequency table) and then show it as a picture (a pictograph or a bar graph). A good picture answers questions in one glance: which is the most, which is the least, and how the groups compare. The trick that makes big numbers fit on a small page is a key (in a pictograph) or a scale (in a bar graph) — one symbol or one unit stands for many.
Let’s Break It Down
What is data?
Suppose you ask everyone in your class, “What is your favourite game?” You write down each answer. That list of answers is data.
Data means any collection of facts, numbers, or observations that tells us something.
Here are some everyday examples of data:
- The favourite colour of each student in your class.
- The number of runs a batter scored in each match.
- The daily temperature for one week.
- The shoe size of each child in a group.
So data is simply information that you have collected. The first step is always to collect it — by asking, counting, or measuring.
But raw data is messy. Imagine your friend reads out 30 favourite games one after another: “Kabaddi, Hockey, Cricket, Hockey, Football, Cricket, Hockey…” Could you tell which game is the most popular just by listening? Very hard! Your brain cannot hold all of it.
That is exactly why we need the next step: organising the data.
Your teacher wants to know 'What is the capital of India?' Does she need to collect data from the class to answer this?
No. She does not need to collect data. The capital of India is a known fact (New Delhi) — it is the same for everyone, so there is nothing to count or survey. You collect data only when the answer varies from person to person or thing to thing (like favourite games), or when you do not already know it.
Organising data — tally marks and frequency tables
Let us make that messy list neat. Suppose a teacher asks each student which sweet they like best, and the choices are jalebi, gulab jamun, gujiya, barfi and rasgulla.
As each student says their choice, the teacher puts a small mark next to that sweet. This mark is called a tally mark.
A tally mark is just one stroke, like this: |. One stroke means one count. So if four students choose barfi, you draw four strokes: | | | |.
But there is a clever rule. When you reach the fifth count, you do not draw a fifth straight line. Instead, you draw a line across the first four. So a group of five looks like four lines with a slash through them. Why do this? Because our eyes can count in fives much faster than counting many single lines. A group of five is easy to spot.
The number of times something happens is called its frequency. For example, if 6 students chose jalebi, then the frequency of jalebi is 6.
A table that shows each choice, its tally marks, and its frequency (count) is called a frequency table.
The picture below shows a complete frequency table for the sweets. Look at how the tally marks are bundled into fives, and how each row ends with the count.
To read a tally count, just add up the bundles. Each full bundle is 5. Then add the leftover single lines.
Let us practise reading tally marks with a worked example.
A frequency table shows that gulab jamun has one full bundle of five tally marks and then four more single lines. How many students chose gulab jamun?
- First, count the full bundle. One bundle of five tally marks = 5 students.
- Now count the leftover single lines. There are 4 single lines = 4 students.
- Add them together: 5 + 4 = 9. So 9 students chose gulab jamun.
Sometimes data comes as plain numbers, not categories — like shoe sizes: 4, 5, 3, 4, 3, 5, 5, 4, 6, and so on. A neat way to organise numbers is to put them in ascending order (smallest to largest). Let’s refresh what that means.
Pictographs
A neat table is good. But a picture is even better, because you can compare amounts in one glance.
A pictograph shows data using small pictures or symbols. Each picture stands for a fixed number of things.
If you have only a few items, you can let one picture stand for one thing. For example, one small drawing of a student = 1 student.
But what if the numbers are big — say 250 kites, or 50 children? Drawing 250 tiny kites would take forever and would not fit on the page! So we use a key.
A key (also called a scale) tells you how many real things one picture stands for. For example, the key might say: one picture = 5 students. Then a row with 4 pictures means 4 × 5 = 20 students.
The picture below shows a pictograph of how students travel to school. Look at the key in the top corner first — it is the most important part.
So reading a pictograph has two simple steps. Step 1: read the key. Step 2: count the pictures in a row and multiply by the key value.
But there is a small puzzle. What if the key is “1 picture = 10 children” and you need to show 25 children? You cannot draw 2 and a half pictures… or can you? Yes! We draw a half picture to mean half of the key value.
The picture below shows how a half picture works.
Now let’s draw and read a pictograph together in a worked example.
A kite seller sold these kites: Rani 300, Chaman 250, Rukhsana 100. Using the key '1 kite picture = 100 kites', how many pictures should each person's row have? And who bought the most?
- First read the key: 1 picture stands for 100 kites. So to find the number of pictures, divide each amount by 100.
- Rani: 300 ÷ 100 = 3 pictures. Chaman: 250 ÷ 100 = 2 and a half pictures (because 250 is two full hundreds plus a half hundred). Rukhsana: 100 ÷ 100 = 1 picture.
- The most kites is the largest number, which is Rani’s 300. So Rani’s row has the most pictures (3 full pictures), and Rani bought the most kites.
Bar graphs
A pictograph is friendly and fun. But drawing lots of little pictures is slow. And when numbers are odd (like 27 or 33), half-pictures get clumsy.
So for bigger or trickier data, we use a bar graph.
A bar graph shows data using bars (rectangles). The taller (or longer) the bar, the bigger the number. All bars have the same width and equal gaps between them, so it is fair to compare them. Only their heights change.
A bar graph has two lines, called axes (say “ax-eez”):
- The horizontal line (going across, at the bottom) lists the categories — like the names of the sweets.
- The vertical line (going up, on the left) shows the numbers — like the number of students.
Just like a pictograph needs a key, a bar graph needs a scale. The scale tells you how much one unit of height means. If the scale is “1 unit = 1 student”, then a bar 6 units tall means 6 students. If numbers are big, you might choose “1 unit = 10 crores” so the bars fit on the page.
The picture below shows the favourite-sweets data (the same data from our frequency table) drawn as a bar graph. Notice the axis labels and the scale written at the top.
Reading a bar graph is easy. To find a value, look at the top of a bar and read across to the number line on the left. To compare, just see which bar is taller.
Let’s read this bar graph together.
Using the favourite-sweets bar graph in Figure 4.4, answer: (a) Which sweet is the most popular? (b) How many more students chose Gujiya than Barfi?
- The most popular sweet has the tallest bar. The tallest bar is Gujiya, reaching 13. So Gujiya is the most popular.
- Read the two bars. Gujiya reaches 13 students. Barfi reaches 3 students.
- ”How many more” means we subtract: 13 − 3 = 10. So 10 more students chose Gujiya than Barfi.
How do you draw a bar graph from a table? Follow these four simple steps:
- Draw the two axes (one across, one going up).
- Choose a scale that makes the tallest number fit nicely on your page.
- Write the category names along the bottom, with equal gaps.
- For each category, draw a bar as tall as its number.
You have now met three ways to show the same data: a table, a pictograph, and a bar graph. The picture below puts all three side by side for the same little set of facts, so you can see they all say the same thing.
When should you pick a pictograph and when a bar graph? The table below compares them.
| Feature | Pictograph | Bar graph |
|---|---|---|
| Shows data using | Small pictures/symbols | Bars (rectangles) |
| Big number trick | A key (1 picture = many) | A scale (1 unit = many) |
| Best for | Small, fun, eye-catching data | Larger data; exact, neat reading |
| Odd numbers like 27 | Awkward (needs part-pictures) | Easy (bar just stops there) |
| Reading a value | Count pictures × key | Read bar top against the number line |
A bar graph has no scale written on it. The bars are 4 units, 6 units and 2 units tall. Can you say how many students each bar stands for?
No, you cannot. Without a scale you do not know what one unit means. One unit could be 1 student, or 10 students, or 100 students. The heights tell you which bar is bigger, but not the actual numbers. This is exactly why every bar graph (and every pictograph) must show its scale or key.
Common Mistakes
When reading a pictograph, I just count the pictures and that's the answer. So 4 pictures means 4.
In your earlier classes, a pictograph often used one picture for one thing, so counting pictures gave the answer directly. That habit sticks.
Check the key first. Each picture may stand for many things. If the key says 1 picture = 5 students, then 4 pictures means 4 × 5 = 20 students, not 4. Always multiply the picture count by the key value.
To draw a group of five tally marks, I draw five straight lines: | | | | |.
It feels natural — five counts, so five strokes. One stroke per count seems the simplest, most honest way.
The fifth mark must cross the first four. So five is four upright lines with a slash through them, not five upright lines. This bundling is the whole point — it lets you count quickly in fives.
In a bar graph, a taller bar always means the category came earlier or is more important.
We read top to bottom and left to right, so 'taller' or 'first' can feel like 'better' or 'more important'.
A taller bar only means a bigger number (a higher count or value). Order and importance have nothing to do with bar height — the bar simply shows how much.
I can make the bars in a bar graph any width I like, as long as the heights are correct.
Since only the height carries the number, the width feels like it doesn't matter, so any width seems fine.
All bars must have the same width and equal gaps. If widths differ, a fatter bar looks 'bigger' even with the same height, and the comparison becomes unfair. Keep widths equal so only heights do the talking.
Quick Check
A pictograph uses the key: one picture = 10 books. A row has 3 full pictures and 1 half picture. How many books is that?
What is the frequency of a category?
In a bar graph, what makes one bar taller than another?
Practice Problems
Easy
A frequency table for favourite fruits shows: Apple 5, Banana 8, Mango 3. (a) Which fruit is most liked? (b) How many students were asked in total?
(a) The most liked fruit has the highest count. The highest is Banana with 8. So Banana is most liked.
(b) The total number of students is the sum of all the counts: 5 + 8 + 3 = 16 students.
In a pictograph the key is: one picture = 5 students. The 'Cricket' row has 6 full pictures. How many students like cricket?
Each picture stands for 5 students. There are 6 pictures.
So the number of students = 6 × 5 = 30 students like cricket.
Medium
A bar graph shows students absent in each class. Class I = 3, Class II = 5, Class III = 4, Class IV = 2, Class V = 0. (a) Which class had full attendance (nobody absent)? (b) How many students were absent altogether?
(a) Full attendance means zero students absent. Class V has a count of 0, so Class V had full attendance.
(b) Add up all the absences: 3 + 5 + 4 + 2 + 0 = 14 students were absent altogether.
A pictograph uses the key: one picture = 100 kites. Make a plan for these sales — Rani 300, Jasmeet 450, Rukhsana 100. How many pictures (including any half pictures) should each row have?
Divide each amount by the key value of 100 to find the number of pictures.
Rani: 300 ÷ 100 = 3 pictures.
Jasmeet: 450 ÷ 100 = 4 and a half pictures (450 is four full hundreds plus a half hundred, which we show as a half picture).
Rukhsana: 100 ÷ 100 = 1 picture.
Challenge
Traffic police counted vehicles each hour. A bar graph (scale: 1 unit = 100 vehicles) shows: 6–7 a.m. → 1.5 units, 7–8 a.m. → 12 units, 8–9 a.m. → 10 units, 9–10 a.m. → 8 units. (a) How many vehicles passed during 7–8 a.m.? (b) How many vehicles passed in the two hours from 8 a.m. to 10 a.m. together?
First use the scale: 1 unit = 100 vehicles. So to turn units into vehicles, multiply each height by 100.
(a) 7–8 a.m. is 12 units tall. So 12 × 100 = 1200 vehicles passed during 7–8 a.m.
(b) The two hours are 8–9 a.m. (10 units) and 9–10 a.m. (8 units). 8–9 a.m. = 10 × 100 = 1000 vehicles. 9–10 a.m. = 8 × 100 = 800 vehicles. Add them: 1000 + 800 = 1800 vehicles passed from 8 a.m. to 10 a.m.
Look at Figure 4.4 (favourite sweets). The total number of students surveyed is the sum of all five counts. (a) Find the total. (b) The shopkeeper has 30 sweets in total to give out, one per student. Will there be enough?
(a) Read all five bar heights from Figure 4.4: Jalebi 6, Gulab jamun 9, Gujiya 13, Barfi 3, Rasgulla 7. Add them: 6 + 9 + 13 + 3 + 7 = 38 students in total.
(b) There are 38 students but only 30 sweets. Since 30 is less than 38, there will not be enough. The shopkeeper is short by 38 − 30 = 8 sweets.
Summary
- Data is any collection of facts, numbers, or observations. You collect it by asking, counting, or measuring.
- Raw data is messy, so we organise it. Tally marks record counts (one stroke = one count); a group of five is four lines with a fifth line crossing them.
- A frequency table lists each category, its tally marks, and its count. The frequency is how many times a category appears.
- A pictograph shows data using pictures. A key tells you how many things one picture stands for. To read it: count the pictures and multiply by the key. A half picture means half the key value.
- A bar graph shows data using bars of equal width. The height shows the number — taller means bigger. The categories sit along the bottom; the numbers run up the left.
- Every pictograph needs a key and every bar graph needs a scale, so big numbers fit on a small page. Without it, you cannot read the real values.
- The same data can be shown as a table, a pictograph, or a bar graph — pick the table for exact counts and a graph for comparing at a glance.
What’s Next
You can now collect, organise, and picture data like a pro. Next, in Chapter 5 — Prime Time, you will go inside the numbers themselves. You will meet factors, multiples, and the special “prime” numbers that are the building blocks of every other number. The careful, step-by-step thinking you used here to read graphs will help you spot the patterns hidden in numbers.
Frequently Asked Questions
What is data and what is a frequency table in class 6 maths?
Data is a collection of facts or numbers you gather -- like the favourite colours of everyone in your class. A frequency table is an organised list that shows how many times each item appears. It makes it easy to see which item is most or least common.
How do tally marks work in data handling?
Tally marks are a quick way to count. You draw one vertical line for each count. After four lines, you draw a diagonal line across them to make a bundle of 5. This makes it easy to count large amounts quickly, because you count bundles of 5 instead of single marks.
How do you draw a pictograph for class 6?
Choose a symbol to represent a set number of items -- for example, one book picture = 5 books. Then for each category, draw the right number of symbols. If cricket got 20 votes and one symbol = 5 votes, you draw 4 symbols for cricket. Always write the key so readers know what one symbol means.
How do you draw a bar graph step by step?
First draw two lines at a right angle -- one going across (horizontal) and one going up (vertical). Label the horizontal line with the categories (like sports names) and the vertical line with numbers. For each category, draw a rectangle (bar) as tall as its count. All bars should be the same width and have equal gaps between them.
What is the difference between a pictograph and a bar graph?
A pictograph uses small pictures or symbols to show data -- it looks fun and is easy to read but can be hard to draw for large numbers. A bar graph uses solid rectangles whose height shows the value -- it is easier to draw and more accurate, especially when numbers are big or do not divide neatly by the symbol value.