Data Handling | ICSE Class 6 Maths Notes

This note covers collecting and organising data, tally marks, frequency tables, interpreting and drawing pictographs, choosing scales, constructing and reading bar graphs, and finding the mean and median of small collections of observations.

What is data, and why do we collect it?

Data is a collection of facts, numbers, measurements or descriptions that gives information about something. An observation is one recorded item in that collection. Favourite colours and measured weights are both examples of data, although they give different kinds of information.

Data handling involves collecting information, organising it and presenting it so that we can interpret it. To interpret data means to explain what the recorded information shows. A useful starting point is a question that the information can help us answer.

How can data help us examine a hypothesis?

A hypothesis is an idea that we examine using evidence. Navya thinks cricket is the most popular game in her class. Naresh is unsure. Their own preferences cannot settle the question, so they ask the students in their class about their favourite games.

They first obtain a list of responses. A category is a group into which responses are placed, such as hockey or cricket. Counting the responses in each category makes the most popular game easier to identify than repeatedly reading the original list.

1. State the question: which game is most popular among the classmates?
2. Ask the students which game they prefer and record their responses.
3. Organise the responses by game and count how many belong to each category.
4. Compare these counts and use them to answer the original question.

In Navya and Naresh’s collected data, hockey has the highest count, with 8 responses. The evidence therefore identifies hockey as the most popular game in that class. The answer concerns the classmates whose preferences were collected.

Note: A guess made before collection and a conclusion drawn after counting are different stages. Keep the question in view while deciding what information to collect.

Collecting information about favourite games helps answer a question about preferences. Recording shoe sizes helps answer a question about sizes. The information collected must match the question being investigated.

How do tally marks and frequency tables organise data?

The frequency of a response is the number of times it occurs. A frequency table lists categories or values alongside their counts. This arrangement makes it easier to compare responses without searching through the whole original list each time.

Tally marks are strokes used to keep count. The symbol | represents one recorded response. Draw four separate strokes, then draw the fifth stroke across those four. This crossed group represents five responses and can be counted as one group of five.

How are groups of tally marks counted?

Count the complete groups of five, then add any separate strokes. Shri Nilesh records the sweet preferences of his students in this way. The table below gives the completed frequencies. A preference means the choice a student makes from the available sweets.

Worked example 1. The gujiya row contains two complete groups of five tally marks and three separate strokes. Find its frequency.

Answer: The two groups count as 10 responses. Adding the remaining 3 gives 13 students. In calculations, + means addition and = means “is equal to”: 5 + 5 + 3 = 13.

The table shows that gujiya was selected by the greatest number of students and barfi by the smallest number. It also helps Shri Nilesh purchase the correct numbers of each sweet.

What information does the table leave out?

A table of counts does not identify which particular student chose each sweet. To distribute the sweets to the correct students, their individual choices must also be available. A summary can be useful without containing every detail of the original responses.

When checking a tally table, compare each written frequency with its tally row. A crossed group counts as five, not four, because the crossing stroke records the fifth response.

How do we read a pictograph and its key?

A pictograph displays data using pictures or symbols. A key, also called a scale in a pictograph, states how many objects or people each complete picture represents. The picture count and the actual frequency need not be the same.

Read the category labels and the key before counting pictures. If one picture stands for several children, each complete picture contributes that many children to the category. A half-picture contributes half the number represented by a complete picture.

How does a key change the meaning of a picture?

Nand Kishor collects responses about how often children sleep for at least 9 hours at night. “At least 9 hours” means 9 hours or more. In his pictograph, one complete triangular picture represents 10 children.

Worked example 2. One complete picture represents 10 children. The “Sometimes” row contains two complete pictures and a half-picture. How many children does it represent?

Answer: The symbol × means multiplication. The two complete pictures represent 2 × 10 = 20 children. The half-picture represents 5 children. Together they represent 20 + 5 = 25 children.

The “Always” response means the children always sleep at least 9 hours at night. The “Never” response means they never sleep at least 9 hours, so they always sleep less than 9 hours. Keep these response labels distinct when describing the data.

Interpretation includes explaining both the number and its meaning. Saying “25 children” gives a count; saying that 25 children sometimes sleep at least 9 hours at night connects the count to the question asked.

A pictograph can reveal the largest and smallest categories quickly. For exact answers, use its key and account for every complete or partial picture.

How do we construct a pictograph with a suitable scale?

Drawing one picture for every person can take considerable time and space when frequencies are large. A suitable scale lets one picture represent several people while keeping the data clear. State the chosen key beside the pictograph so that its pictures can be interpreted.

Jarina and Sangita use the following data for the number of students present in different classes. Roman class labels I, II, III, IV, V, VI, VII and VIII mean Classes 1, 2, 3, 4, 5, 6, 7 and 8 respectively.

How can the same data use different keys?

Jarina chooses one picture for 5 students. Sangita chooses one picture for 10 students. Both keys can represent the same class counts. Sangita needs half-pictures to show 25 and 35, because a half-picture in her key represents 5 students.

Worked example 3. Show the 35 students present in Class II using one complete picture for 10 students.

Answer: Three complete pictures represent 30 students. A half-picture represents the remaining 5 students. Draw three complete pictures and a half-picture, giving 30 + 5 = 35 students.

1. Write a title explaining what the pictograph represents.
2. List the classes and choose a key that suits their frequencies.
3. Draw the required number of complete or partial pictures beside each class.
4. Write the key and check that each row represents its original count.

Draw rows labelled I to VIII. With one face picture representing 10 students, use respectively 3, 3 and a half, 2, 2 and a half, 3, 2 and a half, 3, and 2 pictures.

Pictographs can become more challenging when frequencies are not exact multiples of the key. Numbers such as 33 or 27 are less convenient to represent with a 10-student picture than numbers requiring complete pictures or halves.

What does a bar graph show, and how do we read it?

A bar graph represents values using rectangular bars of uniform width. Their heights or lengths show the corresponding frequencies. Bars can be vertical, extending upwards, or horizontal, extending sideways. Keep equal spacing between consecutive bars.

The labelled reference lines of a graph are its axes; one such line is an axis. In a vertical bar graph, categories can be written along the horizontal axis and counts along the vertical axis. The scale explains the quantity represented by each unit length.

How is a bar converted into a count?

Read the graph’s title, labels and scale. Then follow the top of a vertical bar across to the count axis. Equal steps on the count axis must represent equal numerical increases. A bar’s width does not give its frequency.

Lakhanpal records students absent from Classes 1 to 8. The counts are 3, 5, 4, 2, 0, 1, 5 and 7 respectively. His bar graph uses one unit length for one student, so each bar height directly matches its class count.

Worked example 4. Classes 1 to 8 have respectively 3, 5, 4, 2, 0, 1, 5 and 7 students absent. Identify the greatest absence count and the class with full attendance.

Answer: Class 8 has the greatest absence count, 7 students. Class 5 has 0 students absent and therefore full attendance. The Class 5 category remains labelled even though it has no positive-height bar.

Label the horizontal axis “Class” and the vertical axis “Number of students”. Use one unit length for one student. For Classes 1 to 8, draw equal-width bars with heights 3, 5, 4, 2, 0, 1, 5 and 7 units.

A maximum is the greatest value and a minimum is the smallest value in a collection. In this absence data, the maximum is 7 and the minimum is 0. Classes 2 and 7 have equal counts, represented by equal bar heights.

How do we draw a bar graph when the values are larger?

A scale of one unit length for one count is not convenient for every data set. Larger values may need a scale in which each unit length represents several counts. A unit length is the fixed interval used to measure bar heights or lengths.

How does the scale determine bar height?

Smriti’s runs in eight cricket matches are shown below. Runs are the points scored in cricket. The scores extend from 0 to 100. A scale of one unit length for 10 runs gives a more compact graph than one unit length for one run.

Worked example 5. Draw the bar for Match 1, when Smriti scored 80 runs, using one unit length for 10 runs.

Answer: The symbol ÷ means division. Divide the score by the runs represented by each unit length: 80 ÷ 10 = 8. The bar must be 8 unit lengths high, reaching the value 80 on the runs axis.

1. Draw a horizontal axis for the matches and a vertical axis for runs.
2. Mark equally spaced values from 0 to 100 in steps of 10 on the vertical axis.
3. Draw equal-width bars for the matches, leaving equal gaps between them.
4. Use the title “Runs scored by Smriti” and state the scale clearly.

Match 6 remains on the graph even though its score is zero. Matches 5 and 7 both have scores of 90, so their bars reach the same level. Match 4 reaches 100 and is the tallest bar.

Check the labels after drawing. With this scale, a height of eight units means 80 runs, not 8 runs. Changing the scale changes the required bar height on paper; it does not change the score being represented.

How do we compare values shown in a graph?

Graph questions can ask for a category, a frequency, a comparison or a total. Decide which is required before calculating. The highest bar identifies the greatest value, while addition combines values. A comparison should name the quantities being compared.

Imran’s family records its monthly expenditure, meaning money spent, on different items. The symbol ₹ denotes Indian rupees. “Miscellaneous” groups other expenses together. This table supplies the numbers for a bar graph of the family’s spending.

How can we check a graph against its table?

Worked example 6. Use one unit length for ₹200. Find the bar heights for house rent, food, education, electricity, transport and miscellaneous spending.

Answer: Divide each expenditure by 200. The heights are 3000 ÷ 200 = 15, 3400 ÷ 200 = 17, 800 ÷ 200 = 4, 400 ÷ 200 = 2, 600 ÷ 200 = 3 and 1200 ÷ 200 = 6 unit lengths respectively.

The food bar is the highest and the house-rent bar is the second highest. Electricity spending is about one-half of education spending. Education spending is less than one-fourth of food spending. Read each comparison alongside the amounts and their category names.

A graph and its table should communicate the same values. If a bar height seems surprising, read its scale before comparing it with the printed number. One unit represents ₹200 here, whereas one unit represents 10 runs in Smriti’s graph.

What should remain clear in the presentation?

Equal bar widths keep attention on the heights or lengths being compared. Clear labels identify each category. A title identifies the whole collection. These features make a graph easier to read and help avoid interpreting a decorative feature as numerical information.

A pictograph and a bar graph can both display frequencies. Pictographs use repeated pictures, while bar graphs use lengths or heights measured against a scale. In either case, the key or scale is essential to an exact reading.

What is the arithmetic mean, and how do we calculate it?

A representative value summarises a collection using one value. The arithmetic mean, often called the mean or average, is found by adding all the numerical observations and dividing by the number of observations. Here, “sum” means the total obtained by addition.

Definition: Mean = sum of all observations ÷ number of observations. Count every recorded observation when finding the number by which to divide.

The mean can be understood as an equal share: the amount each observation would have if the total were redistributed equally. It is also a balance point for the data. The examples here use collections of no more than ten observations.

Result: How is the arithmetic mean calculated?

For numerical observations, $\text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}}$. This gives the equal share when the total is divided equally among all members of the group.

Worked example 7. Shreyas’s group collects 3, 8, 10, 5 and 4 guavas. Parag’s group collects 5, 4, 6, 3, 4 and 8 guavas. Find each group’s equal share.

Answer: Shreyas’s group has 3 + 8 + 10 + 5 + 4 = 30 guavas shared among 5 people, giving 30 ÷ 5 = 6 each. Parag’s group has 5 + 4 + 6 + 3 + 4 + 8 = 30 guavas shared among 6 people, giving 30 ÷ 6 = 5 each.

The groups have the same total but different numbers of people. Shreyas’s group gives each member one more guava. Comparing totals alone would miss the difference between their equal shares.

How can an average describe daily counts?

Worked example 8. Vaishnavi records 2, 7, 9, 4 and 3 hibiscus flowers blooming on five days. Find the mean number blooming per day.

Answer: Add the counts: 2 + 7 + 9 + 4 + 3 = 25 flowers. Divide by the five days: 25 ÷ 5 = 5 flowers per day.

The mean of 5 flowers does not say that exactly 5 flowers bloomed on every day. It describes the daily number that would result if the total were spread equally across the five days. Keep the average separate from the individual recorded counts.

For a mean calculation, write the full sum before dividing. Then state what the answer represents, such as guavas per person or flowers per day, so the result retains its meaning.

How do we find the median of an odd number of observations?

The median describes the middle of ordered data. Begin by sorting the values in ascending order, which means from smallest to largest. With an odd number of observations, there is one middle position. An odd number is not divisible by two without a remainder.

The median measures centre in the sense that it is roughly the middle value. A data distribution, meaning the way values are arranged or spread, may not have a definite centre. Different ways of measuring centre can give different values.

Result: What is the median of an odd number of observations?

After arranging an odd number of observations in ascending order, the median is the single middle value. There are equally many observations before and after its position in the ordered list.

The middle of the original writing order need not be the middle of the sizes of the observations. Sorting places smaller values before larger values. It lets us locate the central observation by its position in the ordered collection.

Worked example 9. Poovizhi’s family members have heights of 170, 173, 165, 118 and 175 centimetres. The abbreviation cm means centimetres, a unit of length. Find the median height.

Answer: Put the heights in ascending order: 118, 165, 170, 173, 175 cm. There are five observations. The third observation has two observations on either side, so the median height is 170 cm.

1. Copy every observation from the question, including any repeated values.
2. Arrange the values from smallest to largest.
3. Locate the single middle position when the number of observations is odd.
4. Read the value in that position and attach the appropriate unit.

Position and value are different. In the family example, the middle position is third, but the median value is 170 cm. The answer must state the height, rather than merely saying “the third observation”.

Sorting is a preparation step, not a change to the measurements. Keep every observation in the ordered list. After finding the median, look back at the list to check that the same number of positions lies on each side of the selected middle position.

How do we find an even-data median and distinguish it from the mean?

An even number of observations can be divided into two equal groups without a remainder. Such an ordered list has two central positions. Its median is the arithmetic mean of the values in those two positions.

Result: What is the median of an even number of observations?

After arranging an even number of observations in ascending order, take the mean of the two middle values: $\text{Median} = \frac{\text{First middle value} + \text{Second middle value}}{2}$.

Worked example 10. Yaangba’s family heights are 169, 173, 155, 165, 160 and 164 cm. Find the median height.

Answer: Sort the heights: 155, 160, 164, 165, 169, 173 cm. The two middle values are 164 and 165 cm. Their sum is 329 cm. The median is 329 ÷ 2 = 164.5 cm.

There is no need to choose one of the two central observations and discard the other. Both contribute to the median. In this example, the median lies between the two middle heights and is not one of the recorded family heights.

Must the two representative values agree?

Poovizhi’s family heights are 170, 173, 165, 118 and 175 cm. Their mean is 160.2 cm, while their median is 170 cm. The mean is below the heights of four of the five family members because one child is much shorter than the others.

Here, the average does not seem to represent the data very well. The mean uses the total of all the heights, while the median is found from their ordered positions. Their different methods can therefore provide different views of the same collection.

Note: Do not assume that a representative value is the value of every observation. Read it alongside the original data and the question it is intended to answer.

For small data sets, showing the ordered list and the mean calculation makes the distinction clear. It also helps check whether an incorrect result came from missing an observation, dividing by the wrong count or selecting the wrong middle position.

Glossary

• Data — A collection of facts, measurements or descriptions that communicates information about something.
• Observation — One recorded item or value belonging to a collection of data.
• Hypothesis — An idea examined using collected evidence to see what the data shows.
• Category — A group used to organise responses that share the same description.
• Frequency — The number of times a value or response occurs in collected data.
• Tally marks — Counting strokes arranged in groups of five, with the fifth crossing four earlier strokes.
• Frequency table — A table showing categories or values together with their corresponding counts.
• Pictograph — A representation of data using pictures whose numerical meaning is given by a key.
• Scale — The quantity represented by each picture or each unit length on a graph.
• Bar graph — A display using equal-width rectangular bars whose lengths or heights represent values.
• Ascending order — An arrangement of numerical observations from the smallest value to the largest value.
• Arithmetic mean — The total of all numerical observations divided by the number of observations.
• Median — The middle value of ordered data, or the average of its two middle values.
• Representative value — A single value used to summarise a collection of numerical observations.

Common errors and misconceptions

• Misconception: A crossed group of tally marks represents four. Correct: The crossing stroke records the fifth response, so the group represents five.
• Misconception: Each pictograph picture represents one object. Correct: Read the key; a picture can represent one object or several objects.
• Misconception: A half-picture has no meaning. Correct: It represents half the quantity assigned to a complete picture by the key.
• Misconception: A wider bar should represent a larger count. Correct: Use equal-width bars and show counts through their heights or lengths.
• Misconception: Finding the mean requires dividing by the largest value. Correct: Divide the sum by the number of recorded observations.
• Misconception: The median is the middle entry as originally written. Correct: First sort the data; use its middle position or two middle positions.
• Misconception: Mean and median must agree. Correct: They use different methods and can give different representative values for the same observations.

Exam-style questions with model answers

Q1. A pictograph uses one complete picture for 10 children. Its “Sometimes” row has two complete pictures and a half-picture. Calculate the number of children represented, showing the two contributions. [2 marks]

1. The two complete pictures represent 2 × 10 = 20 children because each complete picture stands for ten children.
2. The half-picture represents 5 children. The complete row therefore represents 20 + 5 = 25 children.

Q2. The frequencies of sweet preferences are jalebi 6, gulab jamun 9, gujiya 13, barfi 3 and rasgulla 7. Explain how to tally gujiya, identify the greatest and smallest preferences, and state whether the table identifies individual students’ choices. [3 marks]

1. For gujiya, draw two crossed groups of five tally marks and three separate marks. Together they record 5 + 5 + 3 = 13 students.
2. Gujiya has the greatest frequency, 13, while barfi has the smallest frequency, 3. These counts identify the most and least selected sweets.
3. The table does not identify the choice of each individual student. It records category totals, so individual choices must also be available for correct distribution.

Q3. Smriti scores 80, 50, 10, 100, 90, 0, 90 and 50 runs in Matches 1 to 8 respectively. Describe the axes and scale for a vertical bar graph using one unit length for 10 runs, give all bar heights, and identify the maximum. [4 marks]

1. Label the horizontal axis “Matches” and the vertical axis “Runs”. Place Matches 1 to 8 at equally spaced positions.
2. State the scale as one unit length for 10 runs. Mark the vertical axis from 0 to 100 in equal steps of 10.
3. Draw equal-width bars of heights 8, 5, 1, 10, 9, 0, 9 and 5 units respectively, leaving equal gaps between them.
4. Match 4 has the maximum score, 100 runs, and the tallest bar. Keep the Match 6 label even though its bar has zero height.

Q4. Shreyas’s group collects 3, 8, 10, 5 and 4 guavas. Parag’s group collects 5, 4, 6, 3, 4 and 8 guavas. Each number records one person’s collection. Calculate the mean for each group, compare their equal shares and explain why comparing totals alone is insufficient. [5 marks]

1. Shreyas’s group contains five observations, representing five people. Its total collection is 3 + 8 + 10 + 5 + 4 = 30 guavas.
2. Divide this total by five people. The mean is 30 ÷ 5 = 6 guavas per person, which is each person’s equal share.
3. Parag’s group contains six observations, representing six people. Its total collection is 5 + 4 + 6 + 3 + 4 + 8 = 30 guavas.
4. Divide Parag’s total by six people. The mean is 30 ÷ 6 = 5 guavas per person, giving each member a smaller equal share.
5. Shreyas’s group gives each person one more guava. Although both totals are 30, the different group sizes produce different shares, so totals alone do not settle the comparison.

Q5. Poovizhi’s family heights are 170, 173, 165, 118 and 175 cm, where cm means centimetres. Find the median height by showing the ordered list and identifying the middle position. [3 marks]

1. Arrange all five heights in ascending order: 118, 165, 170, 173, 175 cm. This places the measurements from the smallest to the largest.
2. There are five observations, so the ordered list has one middle position. The third entry has two observations before it and two after it.
3. The third entry is 170 cm, so the median height is 170 cm. State the height itself, rather than giving only its position.

Q6. Yaangba’s family heights are 169, 173, 155, 165, 160 and 164 cm, where cm means centimetres. Find the median and explain how the even number of observations affects the method. [4 marks]

1. First arrange every height in ascending order: 155, 160, 164, 165, 169, 173 cm. There are six observations in the ordered list.
2. Six is even, so there is no single middle entry. The two middle entries are the third and fourth values, 164 and 165 cm.
3. Take the arithmetic mean of these two values: their sum is 164 + 165 = 329 cm, and 329 ÷ 2 = 164.5 cm.
4. The median height is therefore 164.5 cm. Averaging both middle values gives the median between them; choosing only one would omit the other central value.

Key takeaways

• Collect information that answers the question being investigated, then organise the responses so that their patterns become easier to see.
• Frequency means the count of a response; tally marks organise these counts into convenient groups of five.
• Read a pictograph’s key before counting, because one complete picture can represent several people or objects.
• Bar graphs use equal-width bars and equal gaps, with heights or lengths determined by the stated scale.
• Calculate the mean by dividing the sum of all observations by the number of observations in the collection.
• Sort data before finding its median; use the middle value or the mean of the two middle values.
• Mean and median can differ because one uses the total while the other uses positions in ordered data.

Test yourself

What does the frequency of a sweet preference tell us?

It tells us how many students chose that particular sweet in the collected responses.

Why is the fifth tally mark drawn across the preceding four?

It completes a group of five, making the recorded responses easier to count in groups.

If one pictograph picture represents 10 students, what does a half-picture represent?

A half-picture represents 5 students, which is half the value of a complete picture.

A bar graph uses one unit length for 10 runs. What height represents 80 runs?

The height is 8 unit lengths, because dividing 80 runs by 10 runs per unit gives 8.

Find the mean of the daily flower counts 2, 7, 9, 4 and 3.

The total is 25 flowers across five days, so the mean is 25 ÷ 5 = 5 flowers per day.

What is the median of the ordered heights 118, 165, 170, 173 and 175 cm?

The median is 170 cm, the third value, with two observations on either side.

The two middle heights in an ordered list of six observations are 164 and 165 cm. What is the median?

The median is their mean: add 164 and 165, then divide 329 by 2 to obtain 164.5 cm.