Collecting data: tally marks and frequency
Before a school picnic, the teacher asks every child for a favourite drink and makes a mark for each answer. At the end she counts the marks. That is exactly how data is collected and organised.
Data means facts or numbers collected to answer a question. Tally marks are strokes made while counting: four strokes and a fifth stroke across them make one bundle of 5. The frequency of a value is the number of times it occurs. A frequency table lists each value with its frequency.
In the figure: Tea has 2 bundles and 3 strokes (13), Milk has 1 bundle and 3 strokes (8), Juice has 1 bundle (5). Total = 13 + 8 + 5 = 26.
Rule 1: Make and read tally marks — Make one stroke for every item. After four strokes, the fifth crosses the four. To read the tally, count the bundles in fives and add the loose strokes: number = bundles × 5 + loose strokes.
Collecting data: tally marks and frequency
In the app you can move this figure with your finger.
Open in the app- Easy: 3 bundles and 2 strokes: 3 × 5 + 2 = 17.
- Medium: Frequency 23 is drawn as 4 bundles and 3 strokes, since 4 × 5 = 20 and 20 + 3 = 23.
- Exam-type: Tea 13, Milk 8, Juice 5. Total 13 + 8 + 5 = 26. Tea has 13 − 5 = 8 more votes than Juice.
Rule 2: Arrange data and build a frequency table — Write the data in ascending order (smallest to largest). Equal values then sit together, so the smallest, the largest and the frequency of each value are easy to see. The frequencies must add up to the total number of observations.
- Easy: 3, 5, 3, 4, 3 in ascending order is 3, 3, 3, 4, 5. The frequency of 3 is 3.
- Medium: Shoe sizes 5, 6, 5, 7, 6, 5: size 5 occurs 3 times, 6 twice and 7 once. Check: 3 + 2 + 1 = 6 students.
- Exam-type: Red 9, Blue 6, Green 12 gives a total of 27. Green has 12 − 6 = 6 more than Blue.
Count each crossed bundle as 5. The frequencies in a table always add up to the number of children asked.
Counting a crossed bundle as 4 or 6, or counting the loose strokes twice.
Count the bundles by fives, then add the loose strokes. Memory clue: “Four and a Cross make Five”.
Spot the difference
| Term | Meaning | Example |
|---|---|---|
| Data | facts collected | favourite drink of 26 children |
| Tally mark | one stroke, crossed bundle = 5 | 3 bundles and 2 strokes = 17 |
| Frequency | how many times a value occurs | Tea occurs 13 times |
| Frequency table | values with their frequencies | Tea 13, Milk 8, Juice 5 |
Practice questions
- 1How many does one crossed bundle of tally marks stand for?
- 2How many do 2 bundles and 4 strokes show?
- 3Arrange 9, 4, 7, 4 in ascending order.
- 4Tea 13, Milk 8, Juice 5. How many children were asked in all?
- 5What is the frequency of 6 in 6, 2, 6, 5, 6, 2?
- 6In 3, 5, 3, 4, 3, 5 which value occurs most often?
- 7Cricket is shown by 5 bundles and 3 strokes, Hockey by 3 bundles and 1 stroke. How many more children like Cricket?
- 840 children chose among four snacks. Three frequencies are 12, 9 and 11. Find the fourth and draw it as bundles and strokes.
Show answers (try them yourself first)
- 11) 5
- 22) 14 (2 × 5 + 4)
- 33) 4, 4, 7, 9
- 44) 26 (13 + 8 + 5)
- 55) 3
- 66) 3 (it occurs three times)
- 77) 12 (28 − 16)
- 88) 8 = 1 bundle and 3 strokes (40 − 32)
Pictographs and keys
A fruit seller draws one mango on his board for every 10 kg sold. Customers see at a glance which fruit sells the most, without reading any numbers.
A pictograph shows data with pictures or symbols. The key tells how many items one symbol stands for. A half symbol stands for half of the key.
In the figure: Key: 1 circle = 10 students. Tea 2 circles = 20, Milk 3.5 circles = 35, Juice 1.5 circles = 15.
Rule 1: Read a pictograph with its key — First read the key. Then value = number of symbols × key, and symbols = value ÷ key. Add half a key for each half symbol.
Pictographs and keys
In the app you can move this figure with your finger.
Open in the app- Easy: Key 1 symbol = 5. A row with 6 symbols shows 6 × 5 = 30.
- Medium: Key 1 symbol = 10. A row with 2 full and 1 half symbol shows 20 + 5 = 25.
- Exam-type: Key 1 symbol = 20 cars. To show 70 cars we need 70 ÷ 20 = 3.5 symbols.
Rule 2: Choose a good key — Pick a key that divides all the values nicely, so you need only whole and half symbols. A bigger key saves space when the numbers are big. Every symbol must be the same size.
- Easy: Data 20, 40, 60: the key 1 symbol = 20 gives 1, 2 and 3 symbols.
- Medium: 250 students with 1 symbol = 1 would need 250 symbols; with 1 symbol = 10 only 25.
- Exam-type: With 1 symbol = 5, the number 17 would need 3.4 symbols, so it cannot be shown exactly. Only multiples of 2.5 can: round 17 to 15 or 20.
Always write the key. A symbol is just a picture; its meaning is the key.
Counting every symbol as 1, or drawing bigger symbols for bigger values.
Value = symbols × key. Symbols = value ÷ key. Memory clue: “Key First, then Count”.
Spot the difference
| Key | Symbols | Value |
|---|---|---|
| 1 = 5 | 4 symbols | 20 |
| 1 = 10 | 2 full + 1 half | 25 |
| 1 = 20 | 3.5 symbols | 70 |
| 1 = 50 | 6 symbols | 300 |
Practice questions
- 1Key: 1 symbol = 10. A row has 4 symbols. What value is shown?
- 2Key: 1 symbol = 5. How many does a half symbol show?
- 3Key: 1 symbol = 5. How many symbols show 30?
- 4Key: 1 symbol = 10. A row has 3 full symbols and 1 half symbol. What value is shown?
- 5Key: 1 symbol = 4 books. Monday has 3 symbols and Tuesday 5. How many books on the two days?
- 6Key: 1 symbol = 20. How many symbols show 90?
- 7Key: 1 symbol = 50 mangoes. A farmer sold 3.5 symbols to A, 2 to B and 4.5 to C. How many mangoes in all?
- 8Key: 1 symbol = 10 plants. Rose has 4 symbols and Marigold 5.5. How many more Marigold plants are there?
Show answers (try them yourself first)
- 11) 40 (4 × 10)
- 22) 2.5
- 33) 6 (30 ÷ 5)
- 44) 35 (3 × 10 + 5)
- 55) 32 ((3 + 5) × 4)
- 66) 4.5 (90 ÷ 20)
- 77) 500 (10 symbols × 50)
- 88) 15 (55 − 40)
Bar graphs: reading and comparing
A teacher draws a bar for each class's attendance on the board. The tallest bar tells her at once which class is the most regular.
A bar graph uses bars of the same width with equal gaps. The height (or length) of a bar shows the value, read with the scale on the axis. Vertical bars make a column graph; bars lying sideways make a horizontal bar graph.
In the figure: Scale: 1 unit = 10 students. Kabaddi (50) is the tallest bar, Hockey (20) the shortest, and Football is 45.
Rule 1: Read the value of a bar — Look at the scale first. Then value = height in units × scale. If the top of a bar lies between two lines, take the middle value.
Bar graphs: reading and comparing
In the app you can move this figure with your finger.
Open in the app- Easy: The tallest bar is Kabaddi, with value 50.
- Medium: Total = 35 + 50 + 20 + 45 = 150 students.
- Exam-type: A bar ends halfway between the lines for 40 and 50. Its value is (40 + 50) ÷ 2 = 45.
Rule 2: Compare the bars — Tallest bar = greatest value, shortest bar = least value. Difference = subtract the values. A bar of height 0 means the frequency is zero.
- Easy: If Class V has no bar (height 0), no student of Class V was absent.
- Medium: Bars of 6 and 2 units: 6 ÷ 2 = 3, so the first is 3 times the second.
- Exam-type: Tallest 50 and shortest 20: the difference is 50 − 20 = 30 students.
Bars have the same width and gaps; only the height changes. Always check the scale before reading.
Reading the value without the scale, or assuming each unit is 1 when it is 10.
Value = height × scale. Tallest bar = most. Memory clue: “Scale first, then the Top of the bar”.
Spot the difference
| Question | How to find | Example | Answer |
|---|---|---|---|
| Highest | tallest bar | Kabaddi 50 | Kabaddi |
| Total | add all values | 35 + 50 + 20 + 45 | 150 |
| Difference | subtract | 50 − 20 | 30 |
| Value from height | height × scale | 4 units, scale 10 | 40 |
Practice questions
- 1In a bar graph, what does the tallest bar show?
- 2Scale: 1 unit = 5 students. A bar is 4 units tall. How many students?
- 3Bars show 12, 8, 12 and 5. Which two bars are equal in height?
- 4Books borrowed: Mon 4, Tue 7, Wed 5, Thu 7, Fri 2. How many in all?
- 5Scale: 1 unit = 20 trees. The Neem bar is 7 units tall. How many Neem trees?
- 6Rice 40, Wheat 25, Maize 15 kg. How many more kg of rice than maize?
- 7Bars A, B, C and D are in the ratio 1 : 2 : 3 : 4 and bar A shows 15. What is the total of the four bars?
- 8Runs in 5 overs: 6, 4, 9, 7, 12. In which over were the runs more than double the previous over?
Show answers (try them yourself first)
- 11) the greatest value
- 22) 20 (4 × 5)
- 33) the first and the third
- 44) 25 (4 + 7 + 5 + 7 + 2)
- 55) 140 (7 × 20)
- 66) 25 (40 − 15)
- 77) 150 (15 + 30 + 45 + 60)
- 88) Over 3 (9 is more than 2 × 4 = 8)
Drawing a bar graph and choosing a scale
A notice board is only 8 cm tall, but the biggest value to show is 400. We cannot draw 400 cm! So we agree that 1 cm will stand for 50. That agreement is a scale.
A scale tells how much one unit of length stands for. To draw a bar graph: give a title, draw the axes, mark the scale from 0 in equal steps, and draw bars of equal width and equal gaps. Bar height = value ÷ scale.
In the figure: Scale 1 unit = 10 students, equal widths and equal gaps. Cricket: 35 ÷ 10 = 3.5 units tall.
Rule 1: Choose a scale that fits — The largest bar must fit on the paper, and every height should be easy to mark (whole or half units). Divide the largest value by the paper height to see which scale works; a larger scale gives shorter bars.
Drawing a bar graph and choosing a scale
In the app you can move this figure with your finger.
Open in the app- Easy: Scale 1 unit = 5. For 30 students the bar is 30 ÷ 5 = 6 units tall.
- Medium: Largest value 240 with 1 cm = 20 gives 240 ÷ 20 = 12 cm.
- Exam-type: Data up to 400 on paper 8 cm tall: with 1 cm = 50 the largest bar is 400 ÷ 50 = 8 cm. Scales 10, 20 and 25 give 40, 20 and 16 cm, which are too tall.
Rule 2: Draw the bars step by step — 1) Write a title. 2) Draw two axes: categories on one, scale on the other. 3) Mark the scale from 0 in equal steps. 4) For each category draw a bar of height value ÷ scale. 5) Write the values on the bars.
- Easy: Scale 1 cm = 2 books: 8, 12 and 6 books make bars of 4, 6 and 3 cm. The total height is 13 cm.
- Medium: Scale 1 unit = 10: the bar for 35 is 3.5 units, ending halfway between the lines for 3 and 4.
- Exam-type: Scale 1 cm = 50 students: bars of 4.2 cm and 6.5 cm show 210 and 325 students, a difference of 115.
Start the scale at 0 and keep the steps equal. Use the same scale for all the bars.
Using different steps on the same axis, or leaving out the title and the labels.
Height = value ÷ scale. Value = height × scale. Memory clue: “Title, Axes, Scale, Bars”.
Spot the difference
| Value | Scale | Bar height |
|---|---|---|
| 30 | 1 unit = 5 | 6 units |
| 90 | 1 cm = 20 | 4.5 cm |
| 250 | 1 unit = 100 | 2.5 units |
| 240 | 1 cm = 20 | 12 cm |
Practice questions
- 1Scale: 1 unit = 5 students. How many units tall is the bar for 30 students?
- 2Scale: 1 cm = 10 books. How tall is the bar for 70 books?
- 3What should every graph have so that readers know what it shows?
- 4Scale: 1 cm = 20 kg. How tall is the bar for 90 kg?
- 5Scale: 1 unit = 4 students. A bar is 9 units tall. How many students?
- 6Data 12, 18 and 30. Which scale, 1 unit = 4, 5, 6 or 7, gives whole units for all three?
- 7Sales are 1200, 800, 2000 and 1600 and the scale is 1 cm = 400. What is the total height of the four bars?
- 8The paper allows a bar up to 10 cm. The largest value is 750. Which of 1 cm = 50, 75, 100, 150 is the smallest scale that fits?
Show answers (try them yourself first)
- 11) 6 (30 ÷ 5)
- 22) 7 cm (70 ÷ 10)
- 33) a title
- 44) 4.5 cm (90 ÷ 20)
- 55) 36 (9 × 4)
- 66) 1 unit = 6 (the units are 2, 3 and 5)
- 77) 14 cm (3 + 2 + 5 + 4)
- 88) 1 cm = 75 (750 ÷ 75 = 10)
Choosing a display and reading graphs wisely
A shopkeeper wants to show that sales rose a little. If his graph starts the axis at a high number, even a tiny rise looks huge. Reading graphs carefully protects us from such tricks.
The same data can be shown as a table, a pictograph, a bar graph or an infographic (pictures, words and numbers together). We choose the display that suits our purpose. A graph is misleading if its axis does not start at 0, if bars have different widths, or if symbols have different sizes.
In the figure: Same data: 42 and 46. With the axis from 0 the bars look nearly equal. With the axis from 40 the second bar looks 6 ÷ 2 = 3 times as tall.
Rule 1: Pick the right display — Ask what you want to show. Exact numbers: table. A few small counts: pictograph. Comparing many categories: bar graph. Long category names: horizontal bars. A story with pictures: infographic.
- Easy: Number of students in each of 6 sports: a bar graph compares them at a glance.
- Medium: The category "Visit to the science museum" is long. Horizontal bars give the name room on the left.
- Exam-type: A pictograph with 1 symbol = 20 shows 5 symbols = 100. A bar graph with 1 cm = 10 draws it 100 ÷ 10 = 10 cm tall.
Rule 2: Spot a misleading graph — Check three things: does the axis start at 0? Are all bars equally wide? Are all symbols the same size? If not, the picture exaggerates. Heights above a cut-off axis do not show the real ratio.
Choosing a display and reading graphs wisely
In the app you can move this figure with your finger.
Open in the app- Easy: Bars 52 and 54 with the axis from 50 are drawn 2 and 4 tall. The second looks 2 times as tall, but 54 ÷ 52 is only about 1.04.
- Medium: Jan 40 and Feb 45 with the axis from 30 give heights 10 and 15. Feb looks 1.5 times Jan, but the real ratio is 45 ÷ 40 = 1.125.
- Exam-type: Using 1 cm = 5 instead of 1 cm = 10 from 0 doubles both bars (3 cm and 6 cm become 6 cm and 12 cm), yet the ratio is still 2.
A fair graph starts at 0, has bars of equal width and symbols of equal size, and has a title and a scale.
Believing that a much taller picture always means a much bigger value, without checking where the axis starts.
Check: axis at 0? widths equal? symbols equal? Memory clue: “Zero, Width, Size”.
Spot the difference
| Need | Best display | Why |
|---|---|---|
| Exact numbers | table | every value is written |
| Few small counts | pictograph | quick to read with a key |
| Many categories | bar graph | bars are easy to compare |
| Long names | horizontal bars | names fit on the left |
Practice questions
- 1What is a bar graph with sideways bars called?
- 2Which display needs a key like 1 symbol = 5?
- 3What helps readers know what a graph is about?
- 4Bars 52 and 54 are drawn with the axis starting at 50. How many times as tall does the second bar look?
- 5Why is it misleading to draw some bars wider than others?
- 6A pictograph with 1 symbol = 20 shows 5 symbols. A bar graph uses 1 cm = 10. How tall is the bar?
- 7Sales are 40 in January and 45 in February, but the axis starts at 30. How many times as tall does the February bar look?
- 8A horizontal bar graph has 1 cm = 15 students and bars 4, 6, 2.5 and 3.5 cm long. How many students in all?
Show answers (try them yourself first)
- 11) a horizontal bar graph
- 22) a pictograph
- 33) the title
- 44) 2 times (4 ÷ 2)
- 55) wider bars look bigger even if the value is not
- 66) 10 cm (100 ÷ 10)
- 77) 1.5 times (15 ÷ 10); the real ratio is 1.125
- 88) 240 (16 cm × 15)