CBSE Class 8 Maths Chapter 5: Data Handling Notes

Visual Representation & Grouping of Data

In our daily lives, we encounter a huge amount of unorganized information called raw data. To draw meaningful conclusions, we must categorize and display it visually. The three basic pictorial forms are Pictographs (using pictures), Bar Graphs (using separate rectangular bars), and Double Bar Graphs (used for comparing two sets of data side-by-side).

When dealing with very large datasets, individual data point listings become highly confusing. To simplify this, we organize data using grouped frequency distributions. By partitioning the data into uniform classes called class intervals (such as 0–10, 10–20, etc.), we can log counts efficiently with tally marks. The number of times a data entry falls in an interval is called its frequency. A grouped frequency distribution is plotted visually as a histogram—a special type of bar graph with no gaps between consecutive bars, indicating continuous numerical data. If the intervals do not start at zero, we draw a broken line (jagged line/kink) along the horizontal axis.

Important Terminology Glossary

Raw Data

The original, unorganized collection of facts or numbers gathered during an observation.

Frequency

The number of times a particular observation or data entry occurs in a dataset.

Class Interval

A sub-group or range into which a large set of raw data is grouped (e.g., 20–30).

Class Limits

The boundary values of a class interval. For 10–20, 10 is the lower class limit and 20 is the upper class limit.

Range

The difference between the highest value and the lowest value in a given dataset.

Pie Chart

A circular graph divided into sectors, where the angle of each sector is proportional to the size of the category it represents.

Random Experiment

An experiment or trial where the exact outcome cannot be predicted beforehand, though all possible outcomes are known.

Probability

A numerical measure ranging from 0 to 1 that quantifies the likelihood of a specific event occurring.

Step-by-Step: How to Construct a Pie Chart

1. Sum the Total Data Values —
2. Calculate Central Angles —
3. Draw the Outline Circle —
4. Measure and Plot Sectors —
5. Label the Diagram —

Comparison: Grouped vs. Ungrouped Data

Key Points & Must-Remember Rules

• The sum of all central angles in any Pie Chart is always exactly $360^\circ$.
• In a grouped interval (like 20–30), the upper limit (20) of the lower interval is excluded, while it is included as the lower limit of the next interval (20–30). Thus, the number '20' belongs in 20–30.
• Class Mark is the mid-point of a class interval: $\text{Class Mark} = \frac{\text{Lower Class Limit} + \text{Upper Class Limit}}{2}$.
• The width of all class intervals in a histogram must be uniform (equal width).
• A broken line (or kink/wiggly line) on the horizontal axis of a histogram indicates that the data values do not start immediately at zero.
• The probability of any sure event is always 1, and the probability of an impossible event is 0.
• The sum of probabilities of all elementary outcomes of an experiment is equal to 1.
• Probability Formula: $P(E) = \frac{\text{Number of outcomes favorable to event } E}{\text{Total number of possible outcomes}}$.

Solved Mini-Examples for Exam Practice

• Example 1: Finding a Pie Chart Central Angle Question: Out of 36 students, 9 prefer playing Cricket. Calculate the central angle representing Cricket in a pie chart. Solution: 1. Value of component (Cricket) = 9 2. Total value (Total Students) = 36 3. Central Angle formula: $\text{Angle} = \left( \frac{\text{Component Value}}{\text{Total Value}} \right) \times 360^\circ$ 4. Calculation: $\text{Angle} = \left( \frac{9}{36} \right) \times 360^\circ = \frac{1}{4} \times 360^\circ = 90^\circ$ Answer: The central angle for Cricket is $90^\circ$.
• Example 2: Calculating Event Probability Question: A normal six-sided die is thrown once. What is the probability of getting a number greater than 4? Solution: 1. Possible outcomes when a die is thrown: $\{1, 2, 3, 4, 5, 6\}$ (Total possible outcomes = 6) 2. Favorable outcomes (numbers greater than 4): $\{5, 6\}$ (Number of favorable outcomes = 2) 3. Probability $P(\text{Number } > 4) = \frac{\text{Favorable outcomes}}{\text{Total possible outcomes}} = \frac{2}{6} = \frac{1}{3}$ Answer: The probability is $\frac{1}{3}$.

Common Board Exam Traps & Tips

1. The Boundary Overlap Trap: If your data contains the value 30, and your classes are 20-30 and 30-40, make sure you place the value 30 in the 30-40 class interval. Counting it in both or the wrong group is a common mistake.

2. Missing Broken Line (Kink): If your class intervals start from a non-zero value, like 150-160, 160-170, do not forget to draw a kink (broken line) on the horizontal axis of your histogram. You will lose 1 mark for not including this.

3. Probability Range Check: Probability can never be a negative number or a number greater than 1 (or greater than 100%). If your answer is outside $[0, 1]$, re-check your fractional division!

Practice Questions with Solutions

• What is the class mark of the class interval 45–55? The Class Mark is calculated as: $\frac{\text{Lower limit} + \text{Upper limit}}{2} = \frac{45 + 55}{2} = \frac{100}{2} = 50$.
• If a coin is tossed twice, write down all the possible outcomes. The possible outcomes when two coins are tossed are: Head-Head (HH), Head-Tail (HT), Tail-Head (TH), and Tail-Tail (TT). Total possible outcomes = 4.
• A bag has 3 red balls and 5 black balls. If a ball is drawn at random, what is the probability of getting a red ball? Total outcomes = 3 (red) + 5 (black) = 8. Favorable outcomes (red balls) = 3. Therefore, $P(\text{Red Ball}) = \frac{3}{8}$.
• What is the sum of the central angles of all sectors in a circle graph? The sum of the central angles of all sectors in a circle graph (or pie chart) is always $360^\circ$.

Frequently Asked Questions