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State Board · Class 6 · Mathematics

Lesson 18: Data Representation

5 concepts · 40 practice questions with answers · Free to read

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Data and frequency tables

From daily life

Your teacher asks the class how many siblings each of you has and writes the answers on the board. Counting by tally strokes is a quick way to turn a messy list into a clear picture.

Simple definition

Data (माहिती / आकडेवारी) is the information collected in a survey or experiment; each value is a score (प्राप्तांक).

Rule 1: Frequency and the frequency table — The frequency (वारंवारता) of a score is how many times it occurs. A frequency table (वारंवारता सारणी) has three columns: score (in ascending order), tally marks (in groups of 5) and frequency. The sum of the frequencies is N, the total number of observations.

Frequency table with tally marks for 30 die throws.Die thrown 30 timesScoreTally marksFrequency162535445565N = 6+5+5+4+5+5 = 30

In the figure : In the figure the frequency table with tally marks.

  • Easy : Easy: A die is thrown 20 times. The faces 1 to 6 appear 3, 4, 3, 2, 5, 3 times. Check: 3 + 4 + 3 + 2 + 5 + 3 = 20.
  • Medium : Medium: Marks of 12 students: 7, 8, 7, 9, 8, 7, 10, 9, 8, 7, 9, 8. Frequencies: 7 → 4, 8 → 4, 9 → 3, 10 → 1; total 4 + 4 + 3 + 1 = 12.
  • Exam-type : Exam: The frequency of score 9 is 3 and N is 12. What fraction of students scored 9? 3 out of 12, which is 1/4.
Remember

Frequencies must add up to N. Tally in groups of five.

Common mistake

Skipping a score or counting it twice. Always check that the total of frequencies is equal to N.

Shortcut

Memory clue: "Four strokes and a slash make five".

Spot the difference

ColumnWhat it shows
ScoreThe value, in ascending order
Tally marksCounting in fives
FrequencyHow many times

Practice questions

Easy
  1. 1What is data?
  2. 2What is a score?
  3. 3What is frequency?
Medium
  1. 4What do the frequencies add up to?
  2. 5How many strokes are drawn in one group of tally marks?
  3. 6Scores 2, 3, 2, 4, 2, 3, 5, 3, 2, 4. Find the frequency of 2.
Challenging
  1. 7In the same data find N and the frequency of 3.
  2. 8Why should scores be written in ascending order in a table?
Show answers (try them yourself first)
  1. 1Information collected in a survey or experiment.
  2. 2Each value in the data.
  3. 3The number of times a score occurs.
  4. 4N, the total number of observations.
  5. 5Five (four lines and a cross line).
  6. 64.
  7. 7N = 10; frequency of 3 = 3.
  8. 8It makes the table neat and easy to read and check.

Pictograph

From daily life

A shopkeeper draws a small mango picture on the board for every 10 kg mangoes sold, so that people see the sales at a glance. That is a pictograph.

Simple definition

A pictograph (चित्रालेख) shows data using pictures or symbols.

Rule 1: Scale and reading a pictograph — A scale (मापन) says what one symbol means, for example 1 symbol = 5 books, so 45 books need 9 symbols. Half or part symbols can be used, but a value like 24 books (4 symbols and a part) is hard to draw exactly.

A pictograph with the scale 1 symbol = 5 books.Books read Scale: 1 symbol = 5 booksKumudini45Aafiya60Sushil?2424 = 4 symbols + a part: hard to draw exactly

In the figure : In the figure the pictograph and its scale.

  • Easy : Easy: Scale 1 symbol = 10 kg. 6 symbols mean 6 × 10 = 60 kg.
  • Medium : Medium: Scale 1 symbol = 5 books. Aafiya read 60 books and Kumudini 45 books. Symbols: 12 and 9. Aafiya read 15 more.
  • Exam-type : Exam: Scale 1 symbol = 20 chairs. 90 chairs need 4 full symbols and a half symbol, since 90 = 4 × 20 + 10.
Remember

Always read the scale first. Number of symbols × value of a symbol = data.

Common mistake

Reading each symbol as 1. Each symbol stands for the scale value.

Shortcut

Memory clue: "Count symbols, then multiply by the scale".

Spot the difference

ScaleSymbolsValue
1 symbol = 5945
1 symbol = 10660
1 symbol = 204 and a half90

Practice questions

Easy
  1. 1What is a pictograph?
  2. 2What does a scale tell?
  3. 3Scale 1 symbol = 5. How many symbols for 35?
Medium
  1. 4Scale 1 symbol = 10. How many are shown by 8 symbols?
  2. 5Scale 1 symbol = 20. Show 70.
  3. 6Scale 1 symbol = 5. Ravi has 7 symbols and Meena has 4. Find the difference.
Challenging
  1. 7Scale 1 symbol = 10 kg. Sales are 40, 70, 30 kg. Find the total symbols and the total kg.
  2. 8Why is a pictograph difficult for a value like 24 with 1 symbol = 5?
Show answers (try them yourself first)
  1. 1Data shown by pictures or symbols.
  2. 2The value of one symbol.
  3. 37.
  4. 480.
  5. 53 full symbols and a half symbol.
  6. 63 × 5 = 15.
  7. 74 + 7 + 3 = 14 symbols; 140 kg.
  8. 824 is not a multiple of 5, so a part symbol has to be drawn.

Drawing a bar graph

From daily life

A graph paper looks like a net of tiny squares. You can place a column on it exactly the way a builder places bricks, row by row, to the right height.

Simple definition

A bar graph (स्तंभालेख) shows data with columns of equal width whose heights show the values.

Rule 1: Axes and scale — On graph paper the thick lines are 1 cm apart and the faint lines are 1 mm apart. The horizontal base line is the X-axis (X-अक्ष) and the perpendicular line on the left is the Y-axis (Y-अक्ष).

A bar graph with X-axis, Y-axis, scale and equal-width columns.XY051015202530354040Farmer25Merchant35Service15Daily wageScale: Y-axis 1 cm = 5 persons

In the figure : In the figure axes, scale and columns of a bar graph.

  • Easy : Easy: Scale 1 cm = 5 persons. 40 persons need 40 ÷ 5 = 8 cm.
  • Medium : Medium: Scale 1 cm = 10 books. A column of 6.5 cm means 6.5 × 10 = 65 books.
  • Exam-type : Exam: Values 15, 30, 25, 40 with scale 1 cm = 5: heights 3, 6, 5 and 8 cm.

Rule 2: Steps to draw a bar graph — Draw both axes. Choose a scale (for values in multiples of 5 take 1 cm = 5). Write the scale. Draw columns of equal width at equal distances, with heights by the scale.

  • Easy : Easy: For marks 10, 20, 15, take 1 cm = 5 marks: columns of 2 cm, 4 cm and 3 cm.
  • Medium : Medium: For values up to 80, take 1 cm = 10 so that the tallest column is 8 cm and fits on the page.
  • Exam-type : Exam: Why should all columns have equal width? So that only the height is used to compare.
Live figure

Drawing a bar graph

In the app you can move this figure with your finger.

Open in the app
Remember

Draw axes, choose scale, write scale, equal columns, height = value ÷ scale.

Common mistake

Choosing a scale so big that the graph does not fit on the page. Pick a smart scale first.

Shortcut

Memory clue: "X names, Y numbers".

Spot the difference

AxisShows
X-axis (horizontal)Items
Y-axis (vertical)Quantity by scale

Practice questions

Easy
  1. 1What is a bar graph?
  2. 2Which axis is horizontal?
  3. 3What is the distance between thick lines on graph paper?
Medium
  1. 4Scale 1 cm = 10. Find the height for 70.
  2. 5Scale 1 cm = 5. Find the height for 35.
  3. 6Scale 1 cm = 20. A column is 4.5 cm. What is the value?
Challenging
  1. 7Which scale suits the values 12, 24, 36, 48 – 1 cm = 12 or 1 cm = 1?
  2. 8Why is a scale written on the graph?
Show answers (try them yourself first)
  1. 1A graph with columns whose heights show values.
  2. 2X-axis.
  3. 31 cm.
  4. 47 cm.
  5. 57 cm.
  6. 690.
  7. 71 cm = 12, as the columns are 1, 2, 3, 4 cm.
  8. 8So that a reader can find the true value from the height.

Reading a bar graph

From daily life

Looking at four columns of fruit sales, you can tell at once which fruit sold most and which least, without reading a single number.

Simple definition

A bar graph makes comparison easy: the tallest column is the largest value.

Rule 1: Reading a bar graph — Always read the axes and the scale first: the X-axis names the items and the Y-axis shows the quantity. Fruit sold: Mango 40, Banana 55, Orange 30, Apple 45 kg. Tallest column: Banana; shortest: Orange.

Reading a bar graph: tallest column is Banana, shortest is Orange.01530456040Mango55Banana30Orange45AppleFruit sold (kg)Banana − Orange = 55 − 30 = 25 kg

In the figure : In the figure reading a bar graph.

  • Easy : Easy: Banana − Orange = 55 − 30 = 25 kg.
  • Medium : Medium: Total = 40 + 55 + 30 + 45 = 170 kg.
  • Exam-type : Exam: How many kg more are Banana and Apple together than Mango and Orange? (55 + 45) − (40 + 30) = 100 − 70 = 30 kg.

Rule 2: Differences and averages — Use subtraction to find the difference between two columns and addition for the total. The total divided by the number of columns gives the average height.

  • Easy : Easy: Total 170 kg, 4 fruits: average = 170 ÷ 4 = 42.5 kg.
  • Medium : Medium: Which fruits sold more than the average 42.5? Banana (55) and Apple (45).
  • Exam-type : Exam: If Mango sales rise by 10 kg, the new total is 170 + 10 = 180 kg.
Remember

Read axes and scale first. Tallest = most. Difference = subtract. Total = add.

Common mistake

Reading the number from the wrong axis. The quantity is on the Y-axis.

Shortcut

Memory clue: "Look — Read — Compare".

Spot the difference

FruitSold (kg)
Mango40
Banana55
Orange30
Apple45

Practice questions

Easy
  1. 1Which fruit sold the most?
  2. 2Which fruit sold the least?
  3. 3Find Apple − Mango.
Medium
  1. 4Find the total sale of all fruits.
  2. 5What does the Y-axis show in this graph?
  3. 6By how many kg is Banana more than the average?
Challenging
  1. 7Which two fruits together sold 85 kg?
  2. 8Why do we read the scale first?
Show answers (try them yourself first)
  1. 1Banana.
  2. 2Orange.
  3. 345 − 40 = 5 kg.
  4. 4170 kg.
  5. 5Quantity in kg.
  6. 655 − 42.5 = 12.5 kg.
  7. 7Mango and Apple (40 + 45).
  8. 8The scale tells what each unit of height means.

Infographics and data problems

From daily life

A poster of the highest mountains with pictures, colours and numbers tells the story better than a plain table. This kind of picture-rich display is called an infographic.

Simple definition

An infographic (माहितीचित्र) presents information in an artistic, attractive way with text, pictures, symbols and colours.

Rule 1: What an infographic does — An infographic is easy to understand and easy to remember. A column graph only shows heights; an infographic of mountain peaks can show their shape and height together.

An infographic of peaks in Maharashtra.Peaks in Maharashtra (m)1646Kalsubai1567Salher1438Mahabales.1424Harishcha.1416Saptashru.1404TornaShape + height shown together

In the figure : In the figure an infographic of mountain peaks.

  • Easy : Easy: A weather poster with a sun picture and temperature numbers is an infographic.
  • Medium : Medium: A bus timetable with route colours and icons is easier to read than plain text.
  • Exam-type : Exam: Name any two things an infographic uses: pictures and colours.

Rule 2: Data problems — Practice: the frequency of Sports among 25 hobbies is 11. The primes from 1 to 50 in groups of ten are 4, 4, 2, 2, 3 (N = 15).

  • Easy : Easy: Primes in 1–10 are 2, 3, 5, 7, which is 4 primes.
  • Medium : Medium: Add 4 + 4 + 2 + 2 + 3 = 15 primes up to 50.
  • Exam-type : Exam: In 25 hobbies, 11 chose Sports. The others are 25 − 11 = 14.
Remember

Infographic = data + pictures + colours. Make the data clear and attractive.

Common mistake

Filling an infographic with so many pictures that the numbers cannot be read.

Shortcut

Memory clue: "Info + Graphic = Infographic".

Spot the difference

ToolBest for
Frequency tableCounting data
PictographQuick look with symbols
Bar graphComparing values
InfographicAttractive story

Practice questions

Easy
  1. 1What is an infographic?
  2. 2Why is an infographic easy to remember?
  3. 3How many primes are there between 1 and 10?
Medium
  1. 4How many primes between 1 and 50 as per the data?
  2. 5What does a column graph not show about mountains?
  3. 6Among 40 students, 18 choose cricket. How many do not?
Challenging
  1. 7Choose the best tool: count how many times each score occurs.
  2. 8Make a short plan for an infographic about your class's favourite fruits.
Show answers (try them yourself first)
  1. 1Information presented with text, pictures, symbols and colours.
  2. 2Pictures and colours attract attention and stay in mind.
  3. 34 (2, 3, 5, 7).
  4. 415.
  5. 5Their shape.
  6. 640 − 18 = 22.
  7. 7Frequency table.
  8. 8Collect data, make a tally table, choose pictures and colours for each fruit and show the counts.