1. Chapter at a Glance
Every day we meet information in the form of numbers: marks in a test, runs in a match, rainfall in a month, the number of students who like each sport. Such facts are called data. Raw data are often messy, so we collect them carefully, organise them in tables, present them as pictures or graphs, and then interpret them to answer questions and make decisions.
- What data are and why we collect them.
- Organising data with tally marks and frequency tables.
- Pictographs: symbols, keys and half symbols.
- Bar graphs: choosing a scale, drawing and reading them.
- Drawing conclusions and spotting misleading presentations.
2. What is Data?
Data are a collection of facts, such as numbers, words or observations, gathered to give information about something. For example, a teacher asks each student of Class 6 which fruit they like most and writes down the answers: mango, banana, apple, mango, guava, mango, banana, … This list is data.
- Data must be collected for a clear purpose — to answer a question.
- The way of collecting should be fair: ask everyone, record every answer, and do not count anyone twice.
- Each value in the data that we count is called an observation. The number of times a value occurs is its frequency.
3. Organising Data: Tally Marks and Frequency Tables
Raw data are hard to read. To organise them, we use tally marks. We draw one vertical stroke for each observation. The fifth stroke is drawn across the first four, making a bundle of five (𝍸). Bundles make counting quick.
| Favourite fruit | Tally marks | Frequency |
|---|---|---|
| Mango | |||| |||| || | 12 |
| Banana | |||| ||| | 8 |
| Apple | |||| | 5 |
| Guava | |||| | | 6 |
| Orange | ||| | 3 |
| Total | 34 |
(In a notebook, each group of five is drawn as four strokes with a slanting fifth stroke across them.)
- The total of all frequencies equals the total number of observations (34 students here).
- From the table we can answer quickly: the most liked fruit is mango; the least liked is orange; 12 − 5 = 7 more students like mango than apple.
The shoe sizes of 15 students are: 4, 5, 5, 6, 4, 5, 7, 6, 5, 5, 6, 4, 7, 5, 6. Make a frequency table.
Solution: Size 4 → 3; size 5 → 6; size 6 → 4; size 7 → 2. Total = 3 + 6 + 4 + 2 = 15 ✔. The most common size is 5.
4. Pictographs
A pictograph shows data using pictures or symbols. Each symbol stands for a fixed number of items, given by the key (or scale).
- Choose a simple symbol related to the data (a book for books sold, a car for cars, etc.).
- Choose a key that keeps the number of symbols small: for example, 1 symbol = 10 books.
- Half a symbol stands for half the key value (5 books if the key is 10).
| Day | Books sold (📘 = 10 books) | Number |
|---|---|---|
| Monday | 📘📘📘 | 30 |
| Tuesday | 📘📘📘📘 + half | 45 |
| Wednesday | 📘📘 | 20 |
| Thursday | 📘📘📘📘📘📘 | 60 |
| Friday | 📘📘📘📘📘 | 50 |
Pictographs are attractive and easy to understand, but they become hard to draw for large or uneven numbers (like 47). For those we prefer bar graphs.
In a pictograph, 🚗 = 8 cars. How many symbols are needed for 52 cars?
Solution: 52 ÷ 8 = 6 remainder 4. So we need 6 full symbols and one half symbol (half of 8 = 4). That is 6½ symbols.
5. Bar Graphs
A bar graph shows data using bars (rectangles) of equal width with equal gaps between them. The length (height) of each bar shows the value. Bar graphs make comparison very easy.
Parts of a bar graph
- Title — tells what the graph is about.
- Horizontal axis — shows the categories (days, fruits, classes).
- Vertical axis — shows the values, marked with a scale such as 1 unit = 10 students.
- Bars — one for each category, all of the same width.
Steps to draw a bar graph
- Draw two perpendicular axes on graph paper.
- Write the categories along the horizontal axis with equal gaps.
- Choose a suitable scale for the vertical axis based on the largest value (for 60, use 1 unit = 10).
- Draw bars of equal width whose heights match the values.
- Give the graph a title and label both axes.
The number of students in Classes 6 to 10 is 45, 60, 55, 70 and 50. What scale would you choose for a bar graph drawn on a page that has room for about 8 units?
Solution: The largest value is 70. If 1 unit = 10 students, the tallest bar is 7 units — it fits in 8 units. So choose 1 unit = 10 students. Then 45 is 4.5 units, 55 is 5.5 units, and so on.
6. Interpreting Data and Drawing Conclusions
After presenting data, we read it to answer questions:
- Which category has the highest or lowest value?
- What is the difference between two categories?
- What is the total? What fraction of the total is one category?
- Is there a trend — rising, falling or steady?
From the fruit bar graph, what fraction of the students like mango?
Solution: 12 out of 34 students like mango. Fraction = 12/34 = 6/17, a little more than one-third.
Misleading presentations
Graphs can be made to look unfair. Watch for:
- A vertical axis that does not start at 0, making small differences look huge.
- Bars of different widths, which trick the eye.
- Pictures where the symbol is enlarged in both length and width, so a value twice as big looks four times as big.
- Missing scales or labels.
Before trusting a graph, read its title, axis labels and scale. Make sure bars have equal width and the scale starts at zero.
7. Master Formula Table
| Idea | Rule | Example |
|---|---|---|
| Frequency | Number of times a value occurs | Mango liked by 12 |
| Total observations | Sum of all frequencies | 12 + 8 + 5 + 6 + 3 = 34 |
| Tally bundle | 4 strokes + 1 across = 5 | |||| |||| || = 12 |
| Pictograph value | symbols × key | 4½ × 10 = 45 |
| Symbols needed | value ÷ key | 52 ÷ 8 = 6½ |
| Bar height | value ÷ scale | 45 ÷ 10 = 4.5 units |
| Difference | larger − smaller | 12 − 5 = 7 |
| Fraction of total | part ÷ total | 12/34 |
8. Master Summary Table
| Tool | Best for | Limitation |
|---|---|---|
| Raw list | Recording answers | Hard to read |
| Tally and frequency table | Counting and organising | Not visual |
| Pictograph | Small, round numbers; attractive | Awkward for uneven values |
| Bar graph | Comparing categories accurately | Needs a sensible scale |
Chart: books sold in a week
9. Mind Map
- Clear question
- Fair method
- Observations
- Tally marks
- Frequency table
- Totals
- Pictograph and key
- Bar graph and scale
- Titles and labels
- Highest, lowest
- Differences, fractions
- Misleading graphs
10. Tips and Tricks
- Cross out each observation in the raw list as you tally it — no item is missed or counted twice.
- Always check that the frequencies add up to the total number of observations.
- Choose a scale so that the tallest bar fits comfortably on the page and values fall on easy marks.
- In a pictograph, choose a key that divides most values exactly to avoid many partial symbols.
- Drawing bars of different widths or with unequal gaps.
- Forgetting the key in a pictograph, so values cannot be read.
- Reading the bar height without using the scale (3 units with scale 10 means 30, not 3).
- Drawing the fifth tally stroke as a separate vertical line.
11. Important One-liners
- Data are facts collected to answer a question.
- Frequency is how many times a value occurs.
- Tally marks are grouped in fives with the fifth mark across the four.
- The sum of frequencies equals the total number of observations.
- A pictograph uses symbols; its key tells the value of one symbol.
- Half a symbol stands for half the key value.
- Bars in a bar graph have equal widths and equal gaps.
- The scale tells how many items one unit of bar height represents.
- Every graph needs a title and labelled axes.
- A vertical axis that does not start at zero can mislead.
- Bar graphs are best for comparing categories.
12. Exam-pattern Questions with Detailed Answers
How many observations are shown by the tally |||| |||| |||| |||?
- 15
- 17
- 18
- 19
Show answer
Three bundles of 5 = 15, plus 3 single marks = 18.
Answer: (C) 18
In a pictograph, ⚽ = 4 footballs. How many footballs do 7 symbols show?
Show answer
Value = number of symbols × key = 7 × 4 = 28.
Answer: 28 footballs
In a bar graph with scale 1 unit = 5 students, a bar is 6 units tall. How many students does it show?
Show answer
Students = 6 × 5 = 30.
Answer: 30
The marks (out of 10) of 20 students are: 7, 8, 6, 9, 7, 10, 5, 8, 7, 6, 9, 8, 7, 7, 6, 8, 9, 7, 5, 10. Make a frequency table and find the mark scored by the most students.
Show answer
5 → 2; 6 → 3; 7 → 6; 8 → 4; 9 → 3; 10 → 2. Total = 2 + 3 + 6 + 4 + 3 + 2 = 20 ✔.
The highest frequency is 6, for the mark 7.
Answer: 7 marks (scored by 6 students)
A pictograph uses 🌳 = 20 trees. A village planted 130 trees. How should this be shown?
Show answer
130 ÷ 20 = 6 remainder 10. 10 is half of 20.
So we draw 6 full symbols and 1 half symbol.
Answer: 6½ symbols
Rainfall (in mm) in four months: 120, 240, 300, 90. Which scale is most suitable for a bar graph if the tallest bar should be about 6 units?
- 1 unit = 10 mm
- 1 unit = 25 mm
- 1 unit = 50 mm
- 1 unit = 100 mm
Show answer
With 1 unit = 50 mm, the tallest bar (300 mm) is 6 units. 10 mm would need 30 units (too tall); 100 mm gives only 3 units and 90 mm becomes 0.9 units, hard to draw.
Answer: (C) 1 unit = 50 mm
From the fruit data (Mango 12, Banana 8, Apple 5, Guava 6, Orange 3), how many students did not choose mango or banana?
Show answer
Total = 34. Mango + banana = 20. Others = 34 − 20 = 14 (check: 5 + 6 + 3 = 14).
Answer: 14 students
A shop's bar graph shows sales starting the vertical axis at 90 instead of 0. Sales were 95 and 100. Why could this be misleading?
Show answer
Above 90 the bars are 5 and 10 units, so the second bar looks twice as tall, suggesting double the sales. In reality 100 is only about 5% more than 95.
Answer: Cutting the axis exaggerates a small difference
In a pictograph with key 📦 = 12 parcels, Monday shows 3½ symbols and Tuesday shows 5 symbols. How many more parcels were delivered on Tuesday, and what is the total?
Show answer
Monday = 3.5 × 12 = 42; Tuesday = 5 × 12 = 60.
Difference = 60 − 42 = 18; total = 42 + 60 = 102.
Answer: 18 more; total 102 parcels
A class of 40 students voted for one of four games. Cricket got 15 votes, football got 10, and kabaddi got twice as many votes as kho-kho. Find the votes for kabaddi and kho-kho.
Show answer
Votes left for kabaddi and kho-kho = 40 − 15 − 10 = 15.
If kho-kho = k, kabaddi = 2k, so 3k = 15 and k = 5.
Answer: Kabaddi 10, kho-kho 5
A bar graph uses 1 unit = 25 students. Bars for Schools A, B and C are 4.4, 3.2 and 5 units. Find the total number of students and the fraction in School C.
Show answer
A = 4.4 × 25 = 110; B = 3.2 × 25 = 80; C = 5 × 25 = 125. Total = 315.
Fraction in C = 125/315 = 25/63.
Answer: 315 students; 25/63 in School C
Arun says, "In our survey, 8 students like chess and 4 like carrom, so chess is twice as popular in the whole school." Is his conclusion reliable? Explain.
Show answer
Not necessarily. The survey covered only 12 students; the result may change for a larger, more varied group. A fair conclusion needs a bigger sample chosen from all classes.
Answer: No — the sample is too small to represent the whole school
13. Quick Revision Summary
- Collect data for a clear question; count every observation once.
- Organise with tally marks (bundles of 5) and a frequency table; frequencies add to the total.
- Pictograph: symbol × key = value; half symbol = half the key.
- Bar graph: equal widths and gaps, sensible scale, title and labels.
- Interpret: highest/lowest, differences, fractions, trends; watch for misleading scales.