Login / Register
Follow Us
0
0
Register Free!
Register Free!
Online Mock Tests & Exam Preparation | Target Exams
Select Medium / माध्यम चुनें:
Eng (English) Hindi (हिन्दी)
ICSE • Class 7 • Mathematics • Ch 18
Estimated Time: 45 Mins
Study Progress: In Progress

Data Handling

In ICSE Class 7 Mathematics, "Data Handling" provides an authoritative statistical master study guide analyzing the collection, organization, central tendencies, and graphical representation of discrete data. This comprehensive chapter explores Data Organization (Raw data, arrayed data, frequency, tally marks table, discrete frequency distributions), Measures of Central Tendency (1. Arithmetic Mean: $\bar{x} = \frac{\sum x}{N}$ for ungrouped data and $\bar{x} = \frac{\sum fx}{\sum f}$ for frequency distributions, 2. Median: the middle-most observation of an ordered array; odd $n \to (\frac{n+1}{2})\text{th}$ term; even $n \to \text{mean of } (\frac{n}{2})\text{th}$ and $(\frac{n}{2}+1)\text{th}$ terms, 3. Mode: the observation with the maximum frequency; unimodal vs bimodal distributions), Measure of Dispersion: Range (Difference between the maximum and minimum observations: $\text{Range} = X_{\max} - X_{\min}$), Graphical Representation: Bar Graphs and Double Bar Graphs (Choosing a uniform scale, drawing uniform width bars with equal spacing, interpreting comparative double bar graphs), and Probability Concepts (Experiments, outcomes, equally likely events, probability $P(E) = \frac{\text{Favourable Outcomes}}{\text{Total Outcomes}}$) aligned with the 2026–27 CISCE curriculum.

Take Chapter CBT Test View All Chapters
Overview Core Concepts Formulas & Identities Summary Self Check Take MCQ Test

How Did a London Physician Use a Simple Tally Chart of Deaths on a Street Map to Halt a Deadly Cholera Epidemic in 1854?

In the scorching summer of 1854, a horrifying cholera outbreak erupted in the crowded Soho neighborhood of London, killing hundreds of residents within days. Doctors claimed cholera was spread by "miasma"—poisonous bad air rising from rotting sewage. But a solitary, skeptical physician named Dr. John Snow suspected otherwise. He took a map of the neighborhood and began recording Data: he marked every single cholera death with a black bar directly on the street grid! When Dr. Snow examined the resulting frequency distribution, an unmistakable statistical pattern leaped out: almost every death was tightly clustered around a single public water pump on Broad Street! Dr. Snow walked to the pump, removed its brass handle, and the epidemic stopped dead in its tracks! This was the birth of Modern Data Analytics! Why is the Mean (average) easily distorted by a single billionaire in a room of workers? Why is the Median a more trustworthy measure of wealth? How do Double Bar Graphs compare multi-year trends? Let's master data handling.

Why This Chapter Matters

Data handling is the foundation of data science, artificial intelligence algorithms, clinical medicine, sports analytics (batting averages, strike rates), and economic census planning. Mastering frequency tables, mean, median, mode, and bar graphs is an essential skill for ICSE examinations and navigating an information-driven world.

Before You Begin (Prerequisites)

  • Basic arithmetic: Addition, multiplication, and division.
  • Converting fractions and decimals.
  • Graph plotting on grid paper.

What You Will Learn (Core Objectives)

  • Organize raw numerical data into structured frequency distribution tables using tally marks.
  • Calculate the Range of a given dataset ($X_{\max} - X_{\min}$).
  • Calculate the Arithmetic Mean for ungrouped and grouped frequency distributions.
  • Find the Median for both odd and even numbers of observations after arraying data.
  • Identify the Mode of a dataset from frequency distributions.
  • Construct and interpret single and double Bar Graphs with appropriate scaling.
  • Calculate basic empirical probabilities: $P(E) = \frac{n(E)}{n(S)}$.

Chapter Roadmap & Progression

1 1. Data Organization & The Range
2 2. Measures of Central Tendency: Me...
3 3. Graphical Representation: Bar Gr...
4 4. Elementary Probability

Complete Concept Guide (100% Curriculum Coverage)

1. Data Organization & The Range

Understand
A. Statistical Terminology:
  • Raw Data: The initial numerical observations collected in their original unorganized form.
  • Array / Ordered Data: Raw data arranged in ascending or descending numerical order.
  • Frequency ($f$): The number of times a specific observation occurs in a dataset.
  • Tally Marks: Vertical tally lines counted in groups of five (four vertical lines crossed by a fifth diagonal stroke: $\cancel{||||}$).
B. Range of Data (Measure of Dispersion):

The difference between the highest and lowest values in a dataset:

$$\mathbf{\text{Range} = X_{\max} - X_{\min}}$$

Example: For data $12, 45, 8, 24, 67, 31$, $\text{Range} = 67 - 8 = \mathbf{59}$.

2. Measures of Central Tendency: Mean, Median & Mode

Central Tendencies
1. Arithmetic Mean (Average, $\bar{x}$):
$$\bar{x} = \frac{\text{Sum of all observations}}{\text{Total number of observations}} = \frac{\sum x}{N}$$

For frequency distributions: $\bar{x} = \frac{\sum (f \cdot x)}{\sum f}$.

2. Median ($M$): The Middle Value:

The value of the middle observation when data is arranged in ascending order:

  • If $n$ is ODD: $$\text{Median} = \text{Value of } \left( \frac{n + 1}{2} \right)^{\text{th}} \text{ observation}$$
  • If $n$ is EVEN: $$\text{Median} = \frac{\text{Value of } \left( \frac{n}{2} \right)^{\text{th}} + \text{Value of } \left( \frac{n}{2} + 1 \right)^{\text{th}}}{2}$$
3. Mode: The Most Frequent Value:

The observation that occurs with the highest frequency in the dataset. A dataset with two modes is called *bimodal*.

3. Graphical Representation: Bar Graphs & Double Bar Graphs

Graphs
A. Construction of Bar Graphs:
  • Consists of rectangular bars of uniform width drawn with equal spacing between them on horizontal or vertical axes.
  • The height of each bar is directly proportional to the frequency / value it represents.
  • Always choose an unambiguous scale on the numerical axis (e.g., $1\text{ cm} = 10\text{ students}$).
B. Double Bar Graphs:

Used to display two sets of data simultaneously on the same graph for immediate side-by-side visual comparison (e.g., comparing marks of boys vs girls, or sales in 2024 vs 2025 across different branches).

4. Elementary Probability

Probability

The mathematical measure of the likelihood of an event occurring:

$$\mathbf{P(E) = \frac{\text{Number of Favourable Outcomes } n(E)}{\text{Total Number of Possible Outcomes } n(S)}}$$
  • $0 \le P(E) \le 1$.
  • $P(\text{Impossible Event}) = 0$; $P(\text{Certain / Sure Event}) = 1$.
  • Example: Rolling an even number on a standard die: Favourable $= \{2, 4, 6\} \implies P(\text{even}) = \frac{3}{6} = \frac{1}{2}$.

Key Formulas, Identities & Theorems

Arithmetic Mean Formulation
$$\bar{x} = \frac{\sum x_i}{N} \quad \text{or} \quad \bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$
Primary measure of central location.
Median Formulation for Even Observations
$$M = \frac{1}{2} \left[ X_{n/2} + X_{(n/2)+1} \right]$$
Average of the two central terms.

Central Tendencies & Comparative Double Bar Graph

Data Handling: Central Tendencies & Double Bar Graphs MEASURES OF CENTRAL TENDENCY • 1. Mean (Average, x̄): x̄ = ∑x / N or ∑fx / ∑f • 2. Median (M - Middle Value): Arrange in Ascending Order first! Odd n: Term ((n+1)/2) Even n: Average of (n/2) and (n/2 + 1) • 3. Mode (Most Frequent): Observation with HIGHEST frequency • Range = Maximum - Minimum DOUBLE BAR GRAPH (COMPARISON) Math Sci Eng Term 1 Term 2 • Uniform width & spacing • Direct side-by-side comparison MEDIAN MANDATES ASCENDING ORDER FIRST • PROBABILITY: P(E) = FAVORABLE / TOTAL

Chapter Summary & 10 Key Takeaways

Takeaway 1
Raw data is unorganized data; an array is data sorted in ascending or descending order.
Takeaway 2
The Range is the difference between the maximum and minimum observations: $X_{\max} - X_{\min}$.
Takeaway 3
Arithmetic Mean is the sum of observations divided by total count: $\bar{x} = \frac{\sum x}{N}$.
Takeaway 4
To find the Median, ALWAYS arrange data in ascending numerical order first!
Takeaway 5
If $n$ is odd, Median is the $(\frac{n+1}{2})$th term; if $n$ is even, Median is the average of $(\frac{n}{2})$th and $(\frac{n}{2}+1)$th terms.
Takeaway 6
Mode is the observation occurring with the highest frequency.
Takeaway 7
Bar graphs feature uniform width bars with equal spaces; height is proportional to frequency.
Takeaway 8
Double bar graphs allow immediate visual comparison of two related datasets side by side.
Takeaway 9
Probability measures the chance of an event occurring: $P(E) = \frac{n(E)}{n(S)}$.
Takeaway 10
The probability of a sure event is 1; the probability of an impossible event is 0.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
The marks obtained by 10 students in a mathematics test out of 50 are: $34, 45, 28, 45, 39, 45, 23, 17, 48, 36$. Find: (a) The Range, (b) The Mean, (c) The Median, (d) The Mode.
Reveal Answer & Explanation
Answer:

Step 1: Arrange data in ascending numerical order:

$$17, 23, 28, 34, 36, 39, 45, 45, 45, 48$$


• (a) Range: Maximum value $-$ Minimum value $= 48 - 17 = \mathbf{31}$.
• (b) Mean ($\bar{x}$): Sum of all observations divided by 10:

$$\text{Sum} = 17 + 23 + 28 + 34 + 36 + 39 + 45 + 45 + 45 + 48 = 360$$


$$\bar{x} = \frac{360}{10} = \mathbf{36}$$

.
• (c) Median: Here $n = 10$ (even). The two middle observations are the $5^{\text{th}}$ ($36$) and $6^{\text{th}}$ ($39$) terms:

$$\text{Median} = \frac{36 + 39}{2} = \frac{75}{2} = \mathbf{37.5}$$

.
• (d) Mode: The mark $45$ occurs most frequently (3 times):

$$\text{Mode} = \mathbf{45}$$

.


Sort data first: Range $= 48 - 17 = 31$; Mean $= 360/10 = 36$; Median $= (36+39)/2 = 37.5$; Mode $= 45$.
2
The mean of five numbers is $27$. If one number is excluded, their mean becomes $25$. Find the excluded number.
Reveal Answer & Explanation
Answer: Step 1: Calculate the total sum of the original five numbers:
$$\text{Total Sum} = \text{Mean} \times N = 27 \times 5 = 135$$
Step 2: When one number is excluded, $N = 4$ and new mean $= 25$:
$$\text{New Sum of 4 numbers} = 25 \times 4 = 100$$
Step 3: Find the excluded number:
$$\text{Excluded Number} = 135 - 100 = \mathbf{35}$$.
Original sum $= 27 \times 5 = 135$. New sum $= 25 \times 4 = 100$. Excluded number is $135 - 100 = 35$.
3
Find the mean of the following discrete frequency distribution:
Variate ($x$): $10, 15, 20, 25, 30$
Frequency ($f$): $4, 6, 8, 7, 5$.
Reveal Answer & Explanation
Answer: Compute $\sum f$ and $\sum (f \cdot x)$:
• $10 \times 4 = 40$
• $15 \times 6 = 90$
• $20 \times 8 = 160$
• $25 \times 7 = 175$
• $30 \times 5 = 150$
$$\sum f = 4 + 6 + 8 + 7 + 5 = 30$$
$$\sum (f \cdot x) = 40 + 90 + 160 + 175 + 150 = 615$$
$$\text{Mean } \bar{x} = \frac{\sum fx}{\sum f} = \frac{615}{30} = \mathbf{20.5}$$.
Calculate $f \times x$ for each row, sum them to get 615, and divide by total frequency $\sum f = 30$.
4
A standard fair six-sided die is thrown once. Find the probability of getting: (a) A prime number, (b) A number greater than 4, (c) A number divisible by 3.
Reveal Answer & Explanation
Answer:

Sample space $S = \{1, 2, 3, 4, 5, 6\} \implies n(S) = 6$.
• (a) Prime Number: Prime outcomes are $\{2, 3, 5\} \implies n(E) = 3$:

$$P(\text{prime}) = \frac{3}{6} = \mathbf{\frac{1}{2}}$$

.
• (b) Number Greater Than 4: Outcomes are $\{5, 6\} \implies n(E) = 2$:

$$P(> 4) = \frac{2}{6} = \mathbf{\frac{1}{3}}$$

.
• (c) Number Divisible by 3: Outcomes are $\{3, 6\} \implies n(E) = 2$:

$$P(\text{div by 3}) = \frac{2}{6} = \mathbf{\frac{1}{3}}$$

.


Total outcomes $n(S) = 6$. Primes are $\{2,3,5\}$ ($3/6 = 1/2$). Numbers $>4$ are $\{5,6\}$ ($2/6 = 1/3$).
5
A bag contains 5 red, 8 white, and 7 green balls. A ball is drawn at random. Find the probability that the ball drawn is: (a) White, (b) Not red.
Reveal Answer & Explanation
Answer:

Total balls $n(S) = 5 + 8 + 7 = 20$.
• (a) White Ball: Number of white balls $n(W) = 8$:

$$P(W) = \frac{8}{20} = \mathbf{\frac{2}{5} = 0.4}$$

.
• (b) Not Red Ball: Balls that are not red (white or green) $= 8 + 7 = 15$ (or $20 - 5 = 15$):

$$P(\text{not red}) = \frac{15}{20} = \mathbf{\frac{3}{4} = 0.75}$$

.


Total balls $= 20$. White is $8/20 = 2/5$. Not red is $(8+7)/20 = 15/20 = 3/4$.
6
Why is the "Median" often considered a better measure of central tendency than the "Mean" when analyzing household income?
Reveal Answer & Explanation
Answer:

• The Mean is severely distorted by extreme values (outliers). If 9 workers earn $\text{Rs. } 20,000$ and one CEO earns $\text{Rs. } 50,00,000$, the arithmetic mean is over $\text{Rs. } 5,00,000$, giving a misleading representation of typical income.
• The Median represents the actual middle observation. It is completely unaffected by extreme wealth outliers, accurately reflecting what the typical citizen in the middle of society earns.


The mean is distorted by extreme high outliers (billionaires); the median accurately reflects the middle citizen.
7
Find the median of the following set of observations: $15, 6, 11, 23, 8, 19, 14, 21, 9$.
Reveal Answer & Explanation
Answer:

Step 1: Arrange in ascending order:

$$6, 8, 9, 11, 14, 15, 19, 21, 23$$


Step 2: Count number of observations: $n = 9$ (odd).
Step 3: Apply odd median formula:

$$\text{Median} = \left( \frac{n + 1}{2} \right)^{\text{th}} \text{ observation} = \left( \frac{9 + 1}{2} \right)^{\text{th}} = 5^{\text{th}} \text{ observation}$$


The $5^{\text{th}}$ term in the array is $14$.

$$\text{Median} = \mathbf{14}$$

.


Sort data: $6, 8, 9, 11, 14, 15, 19, 21, 23$. The 5th term is 14.
8
What is the purpose of a "Double Bar Graph"? Give a real-world example of where it is used.
Reveal Answer & Explanation
Answer:

• Purpose: A Double Bar Graph displays two related sets of data simultaneously using paired side-by-side vertical or horizontal bars with a common scale and distinct colors/hatchings.
• Real-World Utility: It enables immediate visual comparison between two categories across time or groups.
Example: Comparing the academic test scores of students in Term 1 vs Term 2 across subjects, or comparing annual rainfall in two different cities across 12 months.


Displays two related datasets side by side on a single graph for immediate comparative visual analysis.
Finished Studying This Chapter?
READY TO PRACTICE?

Timed CBT Practice Tests (Exam Simulator)

Put your concepts to the test with official curriculum-aligned Foundation and Advanced practice tests. Get instant accuracy scores, time metrics, and step-by-step verified explanations.

Foundation Test
15 MCQs • Core Concepts
Advanced Test
15 MCQs • HOTS & Traps

All Class 7 Mathematics Chapters

AI Study Friend

Instant Concept Tutor

Have a doubt in Data Handling? Ask our AI study tutor for rapid explanations or customized quizzes.