Data Handling and Presentation | CBSE Class 6 Maths Notes

This note covers collecting and organising data, tally marks, frequencies, ordered lists, pictographs, bar graphs, choosing scales, calculating totals, comparing quantities, and presenting information clearly through graphs and infographics.

What is data, and how do we collect it?

Definition: Data is a collection of facts, numbers, measurements, observations or descriptions that gives information about things.

A list of classmates’ favourite colours is data. So is a collection of their weights. Data need not consist entirely of numbers: the names of colours or games also communicate information. What we collect depends on the question we want to answer.

How can a question guide collection?

Navya and Naresh want to identify their classmates’ most popular game. A personal preference cannot settle this question. They ask each student for a favourite game and record the responses. This gives them evidence covering the class rather than just their own choices.

Collecting data means gathering the relevant information. Finding the most popular television show among classmates requires their responses. Finding how much water is wasted in a locality requires relevant observations or measurements. These questions call for information about the particular group or place being studied.

After collection, a long list may still be difficult to interpret. Organising data means arranging the information so that it is easier to understand. We can group identical responses and count how many times each occurs. A category is one group, such as hockey or cricket.

Worked example 1. The game responses give hockey 8, cricket 6, kabaddi 6, Satoliya (Pittu) 5, football 4 and badminton 2. Which game is most popular?

Answer: Hockey is most popular because its count, 8, is greater than every other game’s count. Cricket and kabaddi have equal counts of 6. The conclusion follows from comparing all the categories, rather than choosing a familiar game.

Interpretation means making sense of the information. Here, counting the responses turns an unorganised list into a clear comparison. The question, the information collected and the conclusion should remain connected: this result describes the preferences in this class.

How do tally marks and frequencies organise responses?

A tally mark is a stroke used to record one occurrence. The symbol | represents one tally. Make four separate strokes, then draw the fifth stroke across them. This creates a group of five, making a longer count easier to read.

Frequency is the number of times a value or response occurs. A frequency table places each category beside its count. In a sweet-preference table, the category is the sweet and the frequency is the number of students choosing it.

How does the sweets table work?

Shri Nilesh asks students to choose among jalebi, gulab jamun, gujiya, barfi and rasgulla. He records their choices using tallies. The completed frequencies are shown below. Each row connects one sweet with the number of students who prefer it.

Worked example 2. Gujiya has two complete tally groups and three additional strokes. How many students chose it?

Answer: Each complete group represents five students. Add five, five and three to obtain 13 students. The tally groups help count the responses without treating the crossing stroke as an extra mark beyond the fifth.

The table helps purchase the required numbers of sweets. However, it records counts rather than individual names, so it cannot identify which sweet a particular student selected. Grouping students according to their choices is one way to help distribute the sweets correctly.

How can tallies be recorded systematically?

1. Write the categories that the investigation concerns.
2. Record each response with a tally beside the correct category.
3. Complete each group of five before starting another group.
4. Count the groups and remaining strokes to obtain each frequency.

Tabular form means an arrangement in rows and columns. It keeps the categories and their frequencies together, helping us compare preferences and use the results for a decision.

How does arranging data in order help?

Ascending order means arranging numbers from smallest to largest. When repeated values sit together, their frequencies are easier to count. The smallest value appears at the beginning and the largest at the end, so questions about the extremes become easier to answer.

Sushri Sandhya records her students’ shoe sizes and arranges them in ascending order. The sizes in the list are 3, 4, 5, 6 and 7. Repeated sizes are retained because each occurrence represents a student.

What does the organised shoe-size data show?

The same information can be summarised in a frequency table. This table is obtained by counting the occurrences in the ordered list. The shoe size is the value being observed; the number of students tells us how frequently that value occurs.

Worked example 3. Using these frequencies, find the smallest and largest shoe sizes, the number wearing size 5, and the number wearing sizes larger than 4.

Answer: The smallest size is 3 and the largest is 7. Size 5 occurs 10 times. Sizes larger than 4 are 5, 6 and 7, with counts of ten, four and one. Adding those counts gives 15 students.

Notice the difference between a value and its frequency. Shoe size 7 is the largest size, but it occurs just once. Shoe size 5 occurs most frequently. The largest value and the greatest frequency therefore answer different questions.

Ordering is also useful when comparing counts of letters in a news item. Count each selected letter first, then arrange the letters by frequency. A letter’s position in the alphabet is different from its position in an order based on how often it appears.

How do we read a pictograph correctly?

Definition: A pictograph presents data through pictures of objects or parts of pictures. A scale or key explains the quantity represented by each complete picture.

Property: Each pictograph symbol has a value set by the key

A symbol is the picture used to stand for a quantity. The same symbol may represent one person in one pictograph and several people in another. Read the key before counting. Counting pictures alone does not necessarily give the number of people or objects.

Nand Kishor collects responses about how often children sleep at least 9 hours at night. At least 9 hours includes 9 hours and longer periods. The categories are Always, Sometimes and Never, and each complete triangle represents 10 children.

The rows are labelled Always, Sometimes and Never. Always has five complete triangles, Sometimes has two complete triangles and a half triangle, and Never has four complete triangles. The key assigns 10 children to a complete triangle.

Reference: NCERT Class 6, page 80

How do complete and half symbols combine?

Worked example 4. In this sleep pictograph, determine the numbers of children in the three response categories.

Answer: Always has five groups of ten, giving 50 children. Sometimes has two groups of ten and half of ten, giving 25 children. Never has four groups of ten, giving 40 children. Here, a half triangle represents five children.

Never must be interpreted with the full question. The 40 children in that row never sleep at least 9 hours, so they always sleep less than 9 hours. It does not mean that they never sleep.

A pictograph helps compare frequencies at a glance. The largest group of symbols indicates the greatest frequency when all symbols follow the same key. Both the row label and the key are necessary for a complete statement about what the picture means.

How do we choose a key and draw a pictograph?

A useful key makes the pictograph manageable while representing the data accurately. One picture per student works for small counts, but larger counts require many pictures. Letting a picture represent several students can save time and space.

Jarina and Sangita use the same class-attendance data with different keys. Jarina uses one face for 5 students; Sangita uses one face for 10 students. With Sangita’s key, a half face represents 5 students, making counts such as 25 and 35 possible to show.

Both drawings list Classes I to VIII vertically, meaning Classes 1 to 8. The upper drawing uses complete faces worth 5 students each. The lower uses faces worth 10 students each and includes half faces.

Reference: NCERT Class 6, page 82

A multiple of a number is obtained by multiplying it by a whole number. Frequencies that are exact multiples of the key can be represented by complete pictures. Other frequencies may require parts of pictures, which can be harder to draw clearly.

How can the kite data be represented?

Worked example 5. Use one kite picture for 100 kites. How should Rani’s and Chaman’s purchases be shown?

Answer: Rani’s 300 kites need three complete pictures. Chaman’s 250 kites need two complete pictures and a half picture, since the half represents 50 kites. Poonam Ben’s 700 kites would need seven complete pictures.

1. List the categories in rows or columns.
2. Choose a simple picture and state its value in the key.
3. Work out how many complete pictures and parts each frequency requires.
4. Draw the pictures beside their categories, using the same key throughout.

Large frequencies and frequencies that are not exact multiples of the chosen key can make pictographs more challenging. Choosing an attractive symbol is useful, but the symbol must still communicate the quantity clearly.

What does a bar graph show?

Definition: A bar graph represents quantities using equally spaced rectangular bars of uniform width. Each bar’s height or length represents its category’s frequency or quantity.

Property: Bar graphs have equal bar widths and equal spacing

Uniform width means that the bars are equally wide. The spaces between them are also equal. The quantity is communicated by height for upright bars or by length for sideways bars. Categories remain separate, with a gap between neighbouring bars.

A vertical bar rises upwards; a horizontal bar extends sideways. Both arrangements can present data. The graph’s scale explains how much each unit of length represents. A unit length is one equal step in the graph’s measurement scale.

How does the absence graph work?

Lakhanpal’s numbers of absent students in Classes 1 to 8 are, in order, 3, 5, 4, 2, 0, 1, 5 and 7. With one unit length representing one student, the heights follow these counts directly.

Classes 1 to 8 are labelled along the horizontal direction. The vertical scale runs from 0 to 8 students in equal steps. Class 8 has the tallest bar, while Class 5 has no raised bar.

Reference: NCERT Class 6, page 86

Worked example 6. Read the absence counts above. How many students were absent in Class 2, which class had the most absentees, and which had full attendance?

Answer: Class 2 had 5 absentees. Class 8 had the most, with 7. Class 5 had full attendance because its absence count was 0. A zero-height bar represents zero in the measured quantity.

Maximum means the greatest value and minimum the smallest. Comparisons use the bar heights or lengths, but the answer should name the category and state what is counted. A bar for zero absentees means full attendance, rather than an empty class.

How does the scale change the size of a graph?

A scale connects a drawn length with a numerical value. It must be stated so the reader can translate the bars into quantities. Choose it using the data, including the minimum and maximum, and the space available for the graph.

Smriti’s scores in eight matches are 80, 50, 10, 100, 90, 0, 90 and 50 runs, respectively. The minimum score is 0 and the maximum is 100. Marking every run separately would require many steps, so one unit length can represent 10 runs.

How are bar heights calculated?

Worked example 7. At a scale of one unit length for 10 runs, find the heights for Smriti’s scores of 80, 100 and 0 runs.

Answer: Divide each score by ten. The heights are eight units for 80 runs, ten units for 100 runs and zero units for 0 runs. The scale changes the drawn lengths while preserving the given scores.

Property: Equal scale steps represent equal increases in quantity

The scale markings start from zero. Equal steps along the scale must represent equal increases in the measured quantity. For Smriti’s graph, consecutive labelled steps increase by ten runs. The two scores of 90 therefore have equal bar heights.

A graph of traffic at a Delhi road crossing uses horizontal bars, with one unit length representing 100 vehicles. Its longest bar is for 7 to 8 a.m., when 1200 vehicles passed. The next longest is for 8 to 9 a.m., showing 1000 vehicles.

The shortest traffic bar, for 6 to 7 a.m., represents about 150 vehicles. The 11 a.m. to noon bar represents about 600 vehicles. Keeping “about” matters because it preserves the precision of the readings being reported.

For the two-hour interval from 8 to 10 a.m., the graph gives about 1000 plus 800, or about 1800 vehicles. Read the relevant bars first, then combine their quantities. The time interval determines which categories belong in the total.

How do we construct a clear bar graph?

Start from an organised table, then decide how its categories and quantities will appear. A graph title states what the graph shows. Labels identify the categories and the quantity being measured. The scale tells the reader how to interpret lengths.

What steps turn a table into bars?

1. Draw a horizontal and a vertical line perpendicular to each other, meaning that they meet at a right angle.
2. Place the category names at equal spacing along the horizontal line.
3. Choose and state a suitable scale on the vertical line, beginning at zero.
4. Calculate each bar’s height from its quantity and the scale.
5. Draw bars of equal width with equal gaps, and label the graph clearly.

For the sweet-preference data, one unit length represents one student. Jalebi therefore needs a height of six units, gulab jamun nine, gujiya thirteen, barfi three and rasgulla seven. The bar heights reproduce the frequencies from the table.

How can larger quantities fit on the page?

Expenditure means money spent. Imran’s family’s monthly expenditure is listed below. The symbol ₹ means Indian rupees. The category “Miscellaneous” groups other expenses. One unit length represents ₹200, allowing the amounts to be shown with manageable bar heights.

Worked example 8. Find the bar heights for Imran’s family’s expenditure when each unit length represents ₹200.

Answer: Divide each amount by 200. House rent needs 15 units, food 17, education 4, electricity 2, transport 3 and miscellaneous expenses 6. Food has the tallest bar, followed by house rent, matching the order of the two largest amounts.

Checking the completed graph involves comparing its bars with the original table. Check that each category is represented, the heights match the scale, and all bars have the same width. This connects the finished picture back to the data used to construct it.

How do frequencies help us calculate totals correctly?

A frequency table may list values together with the number of times each value occurs. Adding the values alone can give the wrong total because it ignores repetition. First decide whether the question asks for the number of observations or the total quantity recorded.

Faiz’s table records wickets taken by Jaspreet Bumrah in 30 matches. A wicket here means a batter’s dismissal counted in cricket. The first column gives wickets in a match; the second gives how many matches had that wicket count.

Which calculation answers which question?

To count matches, add the frequencies in the second column. To count wickets, multiply each wicket value by its frequency before adding. A product is the result of multiplication. Each product gives the contribution from one row of the table.

Worked example 9. Use Faiz’s table to calculate the total number of wickets in the 30 matches.

Answer: The row contributions are zero, four, twelve, twenty-four, twelve, twenty-five, six and seven wickets, respectively. Adding these gives 90 wickets. In particular, three wickets in each of eight matches contribute 24 wickets, rather than just three.

The frequency of four wickets is three matches. This differs from the number of wickets contributed by that row, which is twelve. State the unit of your answer, such as matches or wickets, to make the meaning clear.

Comparisons also require the appropriate calculation. Poonam Ben buys 700 kites and Rani buys 300. Double Rani’s purchase is 600 kites, so Poonam Ben buys more than double that amount. “More than double” compares a quantity with twice another, rather than simply checking which is larger.

These methods use the same organised data for different questions. A table can show the most frequent category, the number of observations and the total quantity, but each result requires reading the headings and selecting the relevant counts.

How can attractive graphs avoid misleading the reader?

Good presentation makes data easier to understand and fits it into the intended space. Scale, orientation, colour and accompanying pictures all affect the result. The aim is to communicate the quantities clearly while making their relationships easy to see.

A column graph is a bar graph with vertical bars. In general, heights measured upwards from the ground are more intuitively shown by columns. Lengths parallel to the ground are usually best shown by horizontal bars. These are presentation choices, not a ban on the other orientation.

What can mountain graphs teach us?

The mountain-height data give Everest as 8848 metres and Elbrus as 5642 metres. A metre is the unit of length used for these heights. Keeping bars equally wide makes clear that the comparison concerns height.

The upper graph uses horizontal rectangular bars. The lower graph uses vertical columns for the same mountains. Each graph keeps its bars equally wide, with lengths or heights representing the mountain heights.

Reference: NCERT Class 6, page 102

An infographic, short for information graphic, combines a data visualisation with more extensive artistic imagery. It aims to make information engaging and easy to understand. However, making a presentation more appealing can sometimes introduce misleading suggestions.

In the mountain infographic using triangles, the taller triangles are also wider. That adds a suggestion about mountain width even though the intended comparison concerns height. The implied extra information may be misleading and may or may not be correct.

A mountain illustration places seven labelled peaks together. Everest appears to be twice as tall as Elbrus in the picture. Their printed numerical heights provide a way to check that visual impression.

Reference: NCERT Class 6, page 105

Worked example 10. Everest is listed at 8848 metres and Elbrus at 5642 metres. Does Everest have twice Elbrus’s listed height?

Answer: Twice 5642 is 11284 metres, which is greater than 8848 metres. Everest therefore does not have twice Elbrus’s listed height. The numerical comparison checks the impression created by the illustration.

Accurate presentation requires care both when making graphics and when reading them. Compare what the picture suggests with what the data actually support. Visual appeal is useful when it helps the information remain clear and correctly understood.

Glossary

• Data — A collection of facts, numbers, measurements, observations or descriptions that communicates information about things.
• Category — A group in which similar responses or observations are placed for comparison.
• Tally mark — A counting stroke recording one occurrence, with every fifth stroke crossing the previous four.
• Frequency — The number of times a particular value, response or observation occurs in collected data.
• Frequency table — An arrangement showing categories or values alongside the number of times each occurs.
• Ascending order — An arrangement of numerical values from the smallest to the largest, retaining repeated values.
• Pictograph — A representation of data using pictures or parts of pictures with a stated key.
• Key — The explanation of how many people or objects each complete picture represents.
• Bar graph — A representation using equally wide, equally spaced bars whose lengths or heights show quantities.
• Scale — The relationship between a drawn unit length and the quantity it represents.
• Column graph — A bar graph with upright bars, whose heights represent the quantities being compared.
• Infographic — A data visualisation enriched with artistic imagery to communicate information attractively and clearly.

Common errors and misconceptions

• Misconception: Data must be numbers. Correct: Descriptions such as favourite colours and games are also data because they convey information.
• Misconception: The fifth tally starts a new group. Correct: The fifth stroke crosses the previous four, completing a group of five.
• Misconception: A sweet-preference frequency table identifies every student’s choice. Correct: It provides counts; distributing sweets to particular students requires connecting students with their choices.
• Misconception: Every picture represents one object. Correct: Its value comes from the key. A half picture represents half the quantity assigned to a complete picture.
• Misconception: A larger quantity needs a wider bar. Correct: Bars have uniform width. Their heights or lengths represent the quantities on the chosen scale.
• Misconception: Adding wicket values without their frequencies gives total wickets. Correct: Multiply each wicket value by its number of matches, then add those products.
• Misconception: An attractive infographic must show accurate comparisons. Correct: Artistic shapes can sometimes mislead. Check the numerical data before accepting an impression of height or width.

Exam-style questions with model answers

Q1. What is data? Give one example involving classmates’ preferences. [2 marks]

1. Data is a collection of facts, numbers, measurements, observations or descriptions that conveys information.
2. A list of classmates’ favourite games is data, even though the responses are game names rather than numbers.

Q2. Gujiya has two groups of five tally marks and three additional marks. Find its frequency and explain how the fifth tally is drawn. [2 marks]

1. The frequency is 13 students: two groups of five and three more make thirteen.
2. The fifth tally is drawn across the preceding four strokes, creating one complete group of five.

Q3. A pictograph records how often children sleep at least 9 hours. One complete triangle represents 10 children. Always has five triangles; Sometimes has two and a half; Never has four. Find each category’s count and explain Never. [3 marks]

1. Always represents 50 children because five complete triangles each stand for ten children. These children always sleep at least nine hours.
2. Sometimes represents 25 children. Two complete triangles give twenty children, and the half triangle adds five children.
3. Never represents 40 children, from four groups of ten. They never sleep at least nine hours, meaning that they always sleep less than nine hours.

Q4. Shoe sizes 3, 4, 5, 6 and 7 have frequencies 3, 9, 10, 4 and 1, respectively. State the smallest size, largest size, number wearing size 5 and number wearing sizes larger than 4. [4 marks]

1. The smallest shoe size is 3. It is the lowest size listed, and its frequency is three students.
2. The largest shoe size is 7. This describes the size, even though just one student wears it.
3. Ten students wear shoe size 5, as shown by the frequency paired with that size.
4. Fifteen students wear sizes larger than 4. Add the frequencies for sizes 5, 6 and 7: ten, four and one.

Q5. Imran’s family spends ₹3000 on house rent, ₹3400 on food, ₹800 on education, ₹400 on electricity, ₹600 on transport and ₹1200 on miscellaneous items. Here ₹ means rupees. Using one unit length for ₹200, give six labelled bar heights. [6 marks]

1. House rent needs a bar of 15 units. Dividing the expenditure of 3000 rupees by 200 gives this height.
2. Food needs a bar of 17 units. Dividing 3400 rupees by 200 gives the tallest of these six bars.
3. Education needs a bar of 4 units. Its expenditure of 800 rupees contains four groups of 200 rupees.
4. Electricity needs a bar of 2 units. Its expenditure of 400 rupees contains two groups of 200 rupees.
5. Transport needs a bar of 3 units. Dividing its expenditure of 600 rupees by the same scale value gives three.
6. Miscellaneous items need a bar of 6 units. Dividing 1200 rupees by 200 gives six; use the same bar width and spacing throughout.

Q6. Wicket counts 0, 1, 2, 3, 4, 5, 6 and 7 have match frequencies 2, 4, 6, 8, 3, 5, 1 and 1. Explain the columns, find total matches, explain why adding wicket values alone is wrong, show row contributions and find total wickets. [5 marks]

1. The wicket counts give the number of wickets taken in a match. The frequencies give how many matches had each corresponding wicket count.
2. The total number of matches is 30, obtained by adding two, four, six, eight, three, five, one and one.
3. Adding the wicket values alone ignores repeated performances. For example, three wickets occurred in eight matches, contributing twenty-four wickets rather than three.
4. Multiply each wicket count by its frequency. The row contributions are zero, four, twelve, twenty-four, twelve, twenty-five, six and seven wickets.
5. Adding these row contributions gives 90 wickets. This total measures wickets, whereas the sum of the frequencies measures matches.

Q7. An illustration makes Everest appear twice as tall as Elbrus. Their listed heights are 8848 metres and 5642 metres, respectively. Check the impression and explain the lesson for reading infographics. [3 marks]

1. Double Elbrus’s height is 11284 metres, obtained by doubling the given value of 5642 metres.
2. Everest’s listed height of 8848 metres is less than 11284 metres. Therefore, Everest is not twice as tall as Elbrus using these values.
3. The illustration creates a misleading height comparison. Check the numerical values as well as the appearance when reading an infographic, because visually appealing pictures can sometimes suggest unsupported information.

Key takeaways

• Data includes numerical measurements and descriptive responses; collect information that directly addresses the question being investigated.
• Tally marks organise counting into groups of five, while frequencies state how often each response or value occurs.
• Ordering numerical data helps identify extremes and count repeated values without confusing a value with its frequency.
• A pictograph’s key determines every complete symbol’s value; parts of symbols represent corresponding parts of that value.
• Bar graphs use uniform bar widths and equal spacing, with quantity represented by height or length.
• Choose a suitable scale, state it clearly, and begin its markings at zero to communicate the data accurately.
• When values repeat, calculate their contributions by multiplying each value by its frequency before adding the products.
• Artistic presentation can improve communication, but check that shapes and visual impressions do not suggest misleading comparisons.

Test yourself

Why does a list of favourite colours count as data?

It contains descriptions that convey information about the preferences of the people asked.

What does a frequency of 9 for gulab jamun mean?

It means that nine students selected gulab jamun as their preferred sweet.

One picture represents 10 students. What does half a picture represent?

Half a picture represents five students, which is half the value of a complete picture.

Rani buys 300 kites. One picture stands for 100 kites. How many pictures are needed?

Three complete pictures are needed because each represents one hundred of the three hundred kites.

Class 5 has zero absentees. What does this tell us?

Class 5 had full attendance that day because none of its students was absent.

A score is 90 runs and one unit length represents 10 runs. What is the bar height?

The height is nine unit lengths, found by dividing ninety runs by ten runs per unit length.

Three wickets are taken in each of eight matches. How many wickets do these matches contribute?

They contribute twenty-four wickets in total, because the three-wicket performance occurs eight times.

Why might mountain-shaped triangles suggest more than a height comparison?

If taller triangles are also wider, they suggest information about width that the height data do not establish.