Data Handling | ICSE Class 8 Maths Notes

Data Handling covers collecting and organising information, frequency tables, averages, bar graphs, grouped data, pie charts, random experiments, equally likely outcomes, and probability.

What is data, and why do we organise it?

Data is information collected about a situation we want to study. An observation is an individual item in that information. Marks in a Mathematics test, heights of students, and numbers of books read are examples of data.

Data is usually collected in the context of a particular situation. Before collecting it, identify what you want to find out. To study the average height of a class, record the students' heights. Information about another characteristic would answer a different question.

How does organisation help?

Ungrouped data lists individual observations without combining them into groups. A long list can make comparisons difficult. Organising the observations makes repeated values, larger values and smaller values easier to identify, while keeping the meaning of the original information.

A table arranges information in labelled rows and columns. A graph represents information visually. Labels are essential: a number by itself does not tell us whether it counts students, records marks, or measures money.

1. Decide which question the data should answer and what information is relevant to it.
2. Collect or read the observations, keeping their descriptions and measurement units, such as hours or rupees.
3. Organise the observations into a table or suitable groups so that their meaning remains clear.
4. Choose a suitable graph or calculation, then interpret the result in the original situation.

A pictograph uses pictures or symbols to represent quantities. A bar graph uses bars of uniform width, with heights proportional to the quantities represented, meaning that the same height per unit of quantity is used throughout. A scale states the quantity represented by each equal graph interval. Read the labels and scale before comparing information.

Organisation is therefore part of reasoning with data. It connects the collected information to a question and helps us draw meaningful conclusions from it.

How can observations be arranged in a frequency table?

Definition: The frequency of a value is the number of times it occurs. A frequency table lists values or groups alongside their frequencies.

Consider the following monthly watch sales. The observations are the numbers sold in each month. The month labels identify the period to which each number belongs.

What changes when repeated values are collected together?

Instead of asking which month has each sale, we can ask how many months have the same sale. The value 1500 occurs in August, September and December. Its frequency is therefore three months. The other listed sales values each occur once.

Worked example 1. Organise the six monthly sales values 1000, 1500, 1500, 2000, 2500 and 1500 into a frequency table.

Answer: Count each distinct sales value, meaning each different value, and record the number of months in which it occurs.

Monthly sales value	Frequency in months
1000 watches	1
1500 watches	3
2000 watches	1
2500 watches	1
Total	6

The frequencies add to six because there are six monthly observations. They do not add to the number of watches sold: a frequency here counts months, whereas an observation records watches. Keeping these meanings separate prevents a common error.

The two tables answer different questions. The first retains the association between each month and its sales. The second makes repetition easy to see but does not identify the months contributing to each frequency. Choose the arrangement that suits the question.

How do mean, median and mode summarise data?

An average is a representative value used to describe a dataset, meaning a collection of observations. The mean is obtained by adding all the observations and dividing by their number. It represents an equal redistribution of their total.

Result: Mean from the total and the number of observations

In the calculations below, = means equals, + means addition, ÷ and the fraction slash / mean division, and × means multiplication. Parentheses group quantities to be treated together.

Mean = sum of observations ÷ number of observations. Here, the sum is the total obtained by addition. The number of observations counts every entry, including repeated values.

The median is the middle value after arranging the observations in order. With an even number of observations, it is the mean of the two middle values. The mode is the value occurring most frequently.

Worked example 2. Find the mean, median and mode of the watch sales: 1000, 1500, 1500, 2000, 2500 and 1500 watches.

Answer: The total is 1000 + 1500 + 1500 + 2000 + 2500 + 1500 = 10,000 watches. The mean is 10,000 ÷ 6 = 5000/3 watches per month.

In increasing order, the values are 1000, 1500, 1500, 1500, 2000, 2500. The two middle values are both 1500, so the median is (1500 + 1500) ÷ 2 = 1500 watches. The mode is 1500 watches because this value occurs three times.

Why do the three averages differ?

The mean uses every sales value in the total. The median uses the middle positions in the ordered data. The mode identifies the most frequent value. These are different ways of describing a collection, so they need not produce the same answer.

Here, the mean describes sales per month if the total is shared equally across the six months. The mode answers which monthly sales value occurred most often. The median identifies the centre of the ordered observations.

The frequency table also supports the mean calculation: 1000 × 1 + 1500 × 3 + 2000 × 1 + 2500 × 1 = 10,000. Divide by the total frequency, six, rather than by the number of different values.

How do we construct and interpret a bar graph?

A bar graph compares quantities using bars of equal width. The bar heights are proportional to their values: with the same scale, a larger value has a taller bar. Bars representing separate categories have equal gaps between them.

The axes are the reference lines used to organise the graph. For a vertical bar graph, the horizontal axis names the categories, and the vertical axis shows the quantities. A scale tells us the quantity represented by each equal interval on an axis.

How should the watch sales be displayed?

1. Write a title indicating that the graph shows the number of watches sold each month.
2. Label the horizontal axis with July, August, September, October, November and December.
3. Label the vertical axis with the number of watches sold, using a consistent scale.
4. Draw equal-width bars with heights representing 1000, 1500, 1500, 2000, 2500 and 1500 respectively.

Worked example 3. Watch sales from July to December are respectively 1000, 1500, 1500, 2000, 2500 and 1500. Identify the greatest sale and compare it with the smallest.

Answer: November has the greatest sale, 2500 watches, and July has the smallest, 1000 watches. Using − to mean subtraction, their difference is 2500 − 1000 = 1500 watches. August, September and December have equal sales of 1500 watches.

Read labels and values together. Equal bar heights indicate equal quantities, but they do not make the category names interchangeable. A graph of monthly sales must keep each month paired with its correct value.

A double bar graph shows two sets of data together for comparison. Each category has two related bars, and a key identifies the sets. Compare corresponding bars using the same vertical scale before deciding which quantity is greater.

How is grouped data represented and interpreted?

Grouped data combines observations into groups. When observations are numerous, grouping can make the distribution, meaning how observations are spread among values or groups, easier to read. A class interval is a range used as one such group.

The lower limit is an interval's lower endpoint; its upper limit is its upper endpoint. For adjoining intervals, state which interval receives a boundary value. Under the convention of including the lower limit and excluding the upper limit, the interval 75 to 100 includes 75 but excludes 100. The next interval receives 100, so it is counted once.

How are individual observations grouped?

1. Choose intervals that together cover the observations and state the boundary convention.
2. Read each observation and identify the interval to which it belongs.
3. Count each observation once in its interval, recording the resulting frequency beside that interval.
4. Check that the total frequency equals the original number of observations before representing the groups by bars.

What does a grouped income table show?

In the table below, ₹ denotes Indian rupees. Each row groups workers according to their daily income. The frequency is the number of workers in that income interval, rather than the amount of money they earn.

Worked example 4. Using the income table, identify the group with the greatest frequency and the group with the smallest frequency.

Answer: The ₹225 to ₹250 group has the greatest frequency, 140 workers. The ₹150 to ₹175 group has the smallest frequency, 30 workers. These answers compare the numbers of workers, not the numerical sizes of the incomes.

To represent these groups by bars, place the income intervals in order along the horizontal axis and the numbers of workers on the vertical axis. Each bar height must match its group's frequency. Label both axes so the two different quantities remain clear.

A grouped table does not display each worker's exact income. It tells us the interval containing the observation. Do not read an interval limit as the income of every person in that group.

What does a pie chart tell us about a whole?

A pie chart, also called a circle graph, represents a whole and its parts. A radius is a line segment from the centre to the circle. The plural of radius is radii. An arc is a portion of the circle's boundary.

The circle is divided into sectors, regions bounded by two radii and the arc between them. The central angle of a sector is the angle between its radii at the centre. Sector sizes are proportional to the quantities represented: equal shares have equal central angles.

Result: A sector's share determines its central angle

Central angle = (part ÷ whole) × 360°. A full turn at the centre measures 360°. The symbol ° means degrees, the angle unit used here. The part and whole must describe quantities measured in the same units.

A percentage expresses a share out of one hundred; its symbol is %. Thus 50% means 50 out of 100. To use a percentage in the angle calculation, divide it by 100 before multiplying by 360°.

Worked example 5. In a school, 50% of students prefer chocolate ice-cream, 25% prefer vanilla, and 25% prefer other flavours. Find the angles for a pie chart.

Answer: Chocolate occupies 50/100 = 1/2 of the circle, giving 180°. Vanilla occupies 25/100 = 1/4, giving 90°. Other flavours also occupy 1/4, giving 90°. The angles total 180° + 90° + 90° = 360°.

The chart shows that chocolate accounts for half the whole group. It does not give the actual number of students unless the group's total is supplied. A share and a count answer different questions.

How do we calculate and draw a complete pie chart?

Begin with the total of the quantities to be represented. Convert each quantity into a fraction of that total, then calculate its central angle. Keep each category attached to its own quantity, fraction and angle throughout the calculation.

How are a baker's sales converted into angles?

Worked example 6. A baker's sales are ₹320 for ordinary bread, ₹120 for biscuits, ₹160 for cakes and pastries, ₹80 for fruit bread, and ₹40 for other items. Prepare the angles for a pie chart.

Answer: Total sales are ₹720. Divide each item's sales by 720, then multiply by 360° to obtain its central angle.

Item	Sales in rupees	Fraction of total	Central angle
Ordinary bread	320	4/9	160°
Biscuits	120	1/6	60°
Cakes and pastries	160	2/9	80°
Fruit bread	80	1/9	40°
Others	40	1/18	20°

The central angles add to 160° + 60° + 80° + 40° + 20° = 360°, accounting for the complete circle.

1. Draw a circle with a convenient radius and mark its centre.
2. Draw a starting radius from the centre to the circle.
3. Use a protractor, an instrument for measuring angles, to mark the first central angle and draw the next radius.
4. Measure each remaining angle from the preceding radius, then label the sectors with their categories and values.

Five sectors meet at the centre. Ordinary bread is labelled 160°, biscuits 60°, cakes and pastries 80°, fruit bread 40°, and others 20°. Ordinary bread occupies the largest sector.

See Fig. 4.5 in your NCERT textbook

The radius chosen for the drawing changes the size of the circle, but the calculated central angles remain the same. A correct chart preserves the shares. Compare the labels with the calculation table and check that the final sector completes the circle.

How can we find amounts from a pie chart?

Reading a pie chart can involve more than identifying its largest sector. If the total amount is known, multiply that total by the required fraction. If one sector's amount and share are known, use them to work out another sector's amount.

What can we infer from the family budget?

The circle labels food 25%, others 20%, savings 15%, education for children 15%, clothes 10%, house rent 10%, and transport 5%. Food is the largest sector; savings and education have equal shares.

See Fig. 4.4 in your NCERT textbook

The whole budget includes both expenditure and savings. Savings must therefore remain part of the whole when interpreting this chart. Comparing expenditure categories alone does not change the percentages printed against the sectors.

Worked example 7. A family's savings are 15% of its monthly budget and amount to ₹3000. Clothes account for 10% of the same budget. Find the expenditure on clothes.

Answer: Since 15% corresponds to ₹3000, the amount corresponding to 10% is ₹3000 × 10/15 = ₹2000. The comparison uses two percentages of the same whole.

Do not calculate 10% of ₹3000: ₹3000 is the amount saved, rather than the whole budget. The required clothes expenditure is 10/15 of the savings amount because their shares of the whole are 10% and 15% respectively.

Equal percentages of the same whole represent equal amounts. Thus education and savings represent equal amounts in this chart. The conclusion depends on their common whole; identifying that whole is the first step in any calculation from a pie chart.

What are random experiments, outcomes and events?

A random experiment is an experiment whose outcome cannot be predicted exactly in advance. An outcome is a possible result of the experiment. Tossing a coin gives head or tail, but we cannot choose which result appears on a particular toss.

An event consists of one or more outcomes. Getting a head is an event in a coin toss. Getting an even number is an event when a die is thrown. An even number is an integer divisible by two.

Why must we list the possible outcomes?

A die is the numbered cube used here, with one face for each number 1, 2, 3, 4, 5 and 6. A throw has six possible outcomes. The event of getting an even number consists of the outcomes 2, 4 and 6.

Equally likely outcomes each have the same chance of occurring. In the coin and die calculations here, we use a fair coin and a fair die: fairness means that the coin's two outcomes, or the die's six outcomes, are equally likely.

Note: Two possible results do not automatically have equal chances. A train can be on time or late, but those possibilities need not be equally likely. Identify equal likelihood before using an outcome-counting calculation.

Keep the experiment and event separate. The experiment tells us what action is performed; its possible outcomes list what can happen. The event selects the result or collection of results whose chance we want to measure.

This distinction matters for a die. “Getting an even number” is one event, but it contains three outcomes. Counting the event's name once would miss the different die faces that make the event occur.

How do we calculate probabilities using equally likely outcomes?

Probability is a number expressing how likely an event is to occur. Its values run from 0 to 1, including the endpoints. Larger probabilities indicate greater likelihood. The calculation below applies when the individual outcomes are equally likely.

Result: Probability from equally likely outcomes

Probability of an event = number of outcomes making the event ÷ total number of equally likely outcomes. The outcomes making the event are also called favourable outcomes. “Favourable” means they satisfy the event, rather than that they are personally desirable.

Worked example 8. A fair die has faces numbered 1, 2, 3, 4, 5 and 6. Find the probabilities of getting 2, getting an even number, getting 7, and getting a number from 1 through 6.

Answer: Getting 2 has probability 1/6. Getting an even number has probability 3/6 = 1/2 because 2, 4 and 6 qualify. Getting 7 has probability 0/6 = 0. Getting a number from 1 through 6 has probability 6/6 = 1.

An impossible event cannot occur in the stated experiment, so its probability is 0. A certain event must occur, so its probability is 1. These descriptions depend on the experiment and its possible outcomes.

Worked example 9. A bag contains 4 red balls and 2 yellow balls, identical except for colour. One ball is drawn without looking. Compare the probabilities of drawing red and drawing yellow.

Answer: There are 6 equally likely individual balls. Red corresponds to 4 outcomes, so its probability is 4/6 = 2/3. Yellow corresponds to 2 outcomes, so its probability is 2/6 = 1/3. Red is more likely than yellow.

The two colours are not equally likely even though each ball is equally likely to be selected. Count the individual balls first, then collect the outcomes belonging to the required colour. Different numbers of balls produce different colour probabilities.

Worked example 10. Ten separate slips carry the numbers 1 to 10, one number per slip. They are mixed well, and one is drawn without looking. Find the probability of a number less than 6.

Answer: The favourable numbers are 1, 2, 3, 4 and 5. There are 5 favourable outcomes among 10 equally likely slips, so the probability is 5/10 = 1/2.

What can repeated experiments tell us about chance?

Repeating an experiment allows us to collect observations about its outcomes. A trial is one performance of the experiment, such as one toss of a coin. Recording the results helps us compare how often different outcomes occur.

How should the results of coin tossing be read?

The table gives recorded numbers of heads and tails for different numbers of tosses. Read each row with its own number of tosses, rather than adding the rows together as if they described one stated experiment.

In the row for 100 tosses, 48 heads and 52 tails account for all the results. Equal likelihood does not require an exactly equal count in this recorded set. The probability of a head on a fair toss remains 1/2.

A probability near 0 indicates an unlikely event. A probability around 1/2 indicates an event that is neither unlikely nor likely. A probability near 1 indicates a likely event. Nearness to an endpoint is different from being exactly at it.

Repeated observations can help form a hypothesis, a proposed expectation to examine against evidence, about future chances. However, a random outcome cannot be predicted exactly in advance. A run of previous results does not make a particular next coin result certain.

When a die is thrown a large number of times, the counts of its six outcomes become almost equal to each other. Keep the qualification: almost equal does not mean exactly equal. Recording, organising and comparing such results connects data handling with the study of chance.

Glossary

• Data — Information collected about a situation so that it can be organised, represented and interpreted.
• Observation — An individual recorded item or value within a collection of data.
• Frequency — The number of times a value occurs, or the number of observations in a group.
• Class interval — A range of values used to collect observations into a group.
• Mean — The sum of all observations divided by the total number of observations.
• Median — The middle value of ordered data, or the mean of its two middle values.
• Mode — The value that occurs most frequently in a collection of observations.
• Bar graph — A representation using bars of uniform width whose heights are proportional to the quantities represented.
• Pie chart — A circle divided into sectors showing how parts relate to a whole.
• Central angle — The angle formed at a circle's centre by the two radii bounding a sector.
• Random experiment — An experiment whose particular outcome cannot be predicted exactly in advance.
• Outcome — A possible result of a random experiment, such as head in a coin toss.
• Event — One outcome or a collection of outcomes of a random experiment.
• Equally likely outcomes — Outcomes that each have the same chance of occurring in an experiment.
• Probability — A number from zero to one expressing the likelihood of an event occurring.

Common errors and misconceptions

• Misconception: The frequency is the value of an observation. Correct: Frequency counts occurrences. A monthly sale of 1500 watches has frequency three when three months have that sale.
• Misconception: The median can be read from an unordered list. Correct: Arrange the observations first, then find the middle value or the mean of the two middle values.
• Misconception: The mean, median and mode must agree. Correct: They describe data in different ways. In the watch-sales example, the mean differs from the median and mode.
• Misconception: A sector representing 25% must have angle 25°. Correct: Its angle is 25/100 of 360°, which equals 90°.
• Misconception: Clothes costing 10% of the budget must cost 10% of the savings. Correct: Both percentages refer to the whole budget, so compare their shares before calculating amounts.
• Misconception: Two possible results must each have probability 1/2. Correct: Equal likelihood must be established. Red and yellow balls have different probabilities when their numbers differ.
• Misconception: Getting an even number on a die counts as one favourable outcome. Correct: It is one event containing three outcomes: 2, 4 and 6.
• Misconception: A fair coin must produce exactly equal head and tail counts in every experiment. Correct: Equal chances do not guarantee identical recorded frequencies in a particular set of tosses.

Exam-style questions with model answers

Q1. The monthly watch-sales values are 1000, 1500, 1500, 2000, 2500 and 1500. What is the frequency of 1500, and what does that frequency count? [2 marks]

1. The frequency of the sales value 1500 is three, because it occurs three times in the list.
2. This frequency counts months with that sales value, rather than the total number of watches sold.

Q2. Find the mean, median and mode of the six monthly sales values 1000, 1500, 1500, 2000, 2500 and 1500 watches. Show your method. [3 marks]

1. The total is 10,000 watches. Dividing by the six observations gives a mean of 10,000 ÷ 6 = 5000/3 watches per month.
2. In increasing order the values are 1000, 1500, 1500, 1500, 2000, 2500. The two middle values are 1500 and 1500, so the median is 1500 watches.
3. The value 1500 occurs three times, more frequently than any other value. Therefore the mode is 1500 watches.

Q3. A school's ice-cream preferences are chocolate 50%, vanilla 25% and other flavours 25%. Calculate all three central angles for a pie chart and check their total. [4 marks]

1. Chocolate represents half of the students, so its central angle is 50/100 × 360° = 180°.
2. Vanilla represents a quarter of the students, so its central angle is 25/100 × 360° = 90°.
3. Other flavours have the same share as vanilla. Their central angle is also 25/100 × 360° = 90°.
4. The angles total 180° + 90° + 90° = 360°, correctly accounting for the complete circle.

Q4. A baker sells ordinary bread for ₹320, biscuits for ₹120, cakes and pastries for ₹160, fruit bread for ₹80, and other items for ₹40. Calculate total sales and the five central angles needed for a pie chart. [6 marks]

1. Total sales are ₹320 + ₹120 + ₹160 + ₹80 + ₹40 = ₹720. This is the whole represented by the circle.
2. Ordinary bread represents 320/720 = 4/9 of total sales. Its central angle is 4/9 × 360° = 160°.
3. Biscuits represent 120/720 = 1/6 of total sales. Their central angle is 1/6 × 360° = 60°.
4. Cakes and pastries represent 160/720 = 2/9 of total sales. Their central angle is 2/9 × 360° = 80°.
5. Fruit bread represents 80/720 = 1/9 of total sales. Its central angle is 1/9 × 360° = 40°.
6. Other items represent 40/720 = 1/18 of total sales. Their central angle is 1/18 × 360° = 20°.

Q5. A family's monthly savings of ₹3000 represent 15% of its budget. Clothes represent 10% of the same budget. Find the expenditure on clothes and explain your calculation. [2 marks]

1. The clothes amount is 10/15 of the savings amount because both percentages refer to the same whole budget.
2. Clothes therefore cost ₹3000 × 10/15 = ₹2000, rather than 10% of the amount saved.

Q6. A fair die has six faces numbered 1, 2, 3, 4, 5 and 6. Find the probabilities of getting 2, an even number, 7, and any number from 1 through 6. Explain each count. [4 marks]

1. Exactly one of the six equally likely faces shows 2. Therefore the probability of getting 2 is 1/6.
2. The even outcomes are 2, 4 and 6. Three of the six faces qualify, giving probability 3/6 = 1/2.
3. No face shows 7. There are zero favourable outcomes, giving probability 0/6 = 0, so this event is impossible.
4. All six faces show numbers from 1 through 6. The probability is 6/6 = 1, so this event is certain.

Q7. A bag contains 4 red and 2 yellow balls, identical except for colour. One ball is drawn without looking, with each ball equally likely to be selected. Find both colour probabilities and state which is greater. [3 marks]

1. There are 4 + 2 = 6 equally likely individual balls. Four are red, so the probability of drawing a red ball is 4/6 = 2/3.
2. Two of the six balls are yellow, so the probability of drawing a yellow ball is 2/6 = 1/3.
3. Since 2/3 is greater than 1/3, red is more likely. The two colour names do not represent equally likely events.

Q8. Ten separate slips are numbered 1 to 10, one number per slip. They are mixed well and one is drawn without looking. Find the probability of drawing a number less than 6, listing the favourable outcomes. [2 marks]

1. The favourable outcomes are 1, 2, 3, 4 and 5, giving five qualifying slips.
2. There are ten equally likely slips altogether, so the required probability is 5/10 = 1/2.

Key takeaways

• Organise data around a clear question, retaining labels and units so that observations keep their meaning.
• A frequency counts observations or occurrences; it is different from the numerical value being observed.
• Mean, median and mode describe data differently, so choose and interpret the representative value carefully.
• Bar heights represent quantities on a consistent scale; grouped bars compare the frequencies of labelled groups.
• A pie chart represents one whole, with sector angles proportional to the shares of its parts.
• Calculate each central angle by multiplying the part's fraction of the whole by 360°.
• Probability by counting requires equally likely outcomes, with favourable outcomes counted within the complete outcome list.
• Repeated experiments produce data about chance, but equal likelihood does not require exactly equal recorded frequencies.

Test yourself

What does the frequency of a value tell you?

It tells you how many times that value occurs among the observations.

What must you do before finding a median?

Arrange the observations in order before identifying the middle value or the two middle values.

What is the difference between a bar graph and a double bar graph?

A bar graph displays quantities using bars; a double bar graph displays two sets together for comparison.

What central angle represents 25% of a whole?

The angle is 25/100 × 360° = 90°, representing a quarter of the circle.

Why is a random experiment called random?

Its particular outcome cannot be predicted exactly before the experiment is performed.

For a die numbered 1 to 6, which outcomes make the event “an even number”?

The outcomes 2, 4 and 6 together make this event.

When can probability be found by dividing favourable outcomes by total outcomes?

This counting method applies when the individual outcomes are equally likely to occur.

Must 100 fair coin tosses produce exactly 50 heads?

No. Equal likelihood does not guarantee exactly equal counts in a particular set of tosses.