This chapter introduces the statistical tools used to analyze and interpret grouped data. Students learn to compute mean, median, and mode for grouped frequency distributions using different methods. It also emphasizes graphical representation and the interpretation of real-life data. These concepts help students develop logical reasoning, critical thinking, and data analysis skills.
II. Key Concepts Covered:
Concept
Description
Grouped Data
Data presented in intervals (e.g., 0–10, 10–20, etc.)
Mean (Average)
Represents the central tendency of data; calculated using: – Direct method – Assumed Mean method – Step Deviation method
Median
The middle value of the data when arranged; found using the formula with cumulative frequency
Mode
The value that appears most frequently in a dataset; formula used for grouped data
Cumulative Frequency Table
Table showing the sum of frequencies up to a certain class
Class Mark
The average of the lower and upper boundaries of a class: (lower limit + upper limit) / 2
III. Important Questions:
(A) Multiple Choice Questions (1 Mark):
In a frequency distribution, the mode is the value that: a) Occurs least often b) Occurs most often ✅ c) Is in the middle d) Is the average
If the mean of the data is 25 and the total number of observations is 10, the sum of all observations is: a) 250 ✅ b) 35 c) 2.5 d) 25
The median class is identified using: a) Modal formula b) Mean formula c) Cumulative frequency ✅ d) Class mark
If the mode is 36, mean is 30, what is the value of the median (using empirical formula)? a) 32 b) 33 c) 34 ✅ d) 31
(B) Short Answer Questions (2/3 Marks):
Find the mean of the following data using direct method:
Class
0–10
10–20
20–30
30–40
Freq.
5
8
15
12
Calculate the mode for the following data:
Class
5–15
15–25
25–35
35–45
45–55
Freq.
3
8
15
9
5
A cumulative frequency table is given. Identify the median class and calculate the median.
Find the mean of a frequency distribution using the assumed mean method. (PYQ 2020)
(C) Long Answer Questions (5 Marks):
The following table shows the marks obtained by 50 students. Calculate the mean, median, and mode. (Include a sample table with frequencies) (PYQ 2022)
From the following data, find the missing frequency if mean = 50. (PYQ 2021)
Class
0–20
20–40
40–60
60–80
80–100
Freq.
5
8
?
6
2
Calculate the mean of the following frequency distribution using the step deviation method. (PYQ 2019)
The marks of 100 students are grouped in a frequency distribution. Find the median and interpret the result in context.
(D) HOTS (Higher Order Thinking Skills):
A student incorrectly calculates the mean by using class limits instead of class marks. How will this affect the result? Discuss the impact and correction.
A grouped data shows a bimodal distribution. How would you identify it, and what practical steps would you take while interpreting the data?
IV. Key Formulas/Concepts:
Concept
Formula
Class Mark (xᵢ)
$frac{text{Lower Limit} + text{Upper Limit}}{2}$
Mean (Direct Method)
$bar{x} = frac{sum f_i x_i}{sum f_i}$
Mean (Assumed Mean Method)
$bar{x} = A + frac{sum f_i u}{sum f_i} times h, quad text{where } u = frac{x_i – A}{h}$
Median
$text{Median} = l + left( frac{frac{n}{2} – F}{f} right) times h$
Where:
l = lower boundary of median class
n = total frequency
F = cumulative frequency before median class
f = frequency of median class
h = class width | | Mode | $text{Mode} = l + left( frac{f_1 – f_0}{2f_1 – f_0 – f_2} right) times h$ Where:
$l = $ lower boundary of modal class
$f_1 = $ frequency of modal class
$f_0 = $ frequency before modal class
$f_2 = $ frequency after modal class
$h = $ class width | | Empirical Relation $text{Mode} = 3 times text{Median} – 2 times text{Mean}$
V. Deleted Portions (CBSE 2025–2026):
No portions have been deleted from this chapter as per the rationalized NCERT textbooks.
VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–26):
Unit/Chapter
Estimated Marks
Type of Questions Typically Asked
Statistics
6–8 Marks
1 Long Answer + 1 Short or HOTS Question
VII. Previous Year Questions (PYQs):
Year
Marks
Question
2019
5M
Mean using step deviation method
2020
3M
Assumed mean-based question
2021
5M
Missing frequency based on given mean
2022
5M
Full calculation: mean, median, mode
VIII. Real-World Application Examples to Connect with Topics:
Scenario
Statistical Concept
School result analysis
Mean/Median of student marks
Business sales data
Mode to find most sold product
Public health surveys
Median income or age group analysis
Sports analytics
Mean and mode of runs scored or wickets taken
IX. Student Tips & Strategies for Success (Class-Specific):
Time Management:
Allocate 20–30 minutes twice a week to solve complete tables.
Keep a dedicated notebook for formulas and solved examples.
Exam Preparation:
Memorize all formulas and know when to apply which method.
Use proper layout: write frequency table, method, and formula clearly.
Practice at least 10 problems from each method.
Stress Management:
Don’t panic if numbers look big — use approximation to verify.
Practice simplifying calculations with step deviation and assumed mean.
X. Career Guidance & Exploration (Class-Specific):
For Classes 9–10:
Stream
Career Paths
Relevant Exams
Science
Data Scientist, Statistician, Biostatistician
NTSE, Olympiads
Commerce
Actuary, Market Analyst, Financial Planner
NSE Exams, CUET
Arts
Sociologist, Political Analyst, Researcher
Humanities Aptitude Tests
Tip: Learn to interpret graphs and data tables — these are essential in NEET, JEE (Data Interpretation), CUET, and general aptitude sections.
XI. Important Notes:
Always check latest updates on https://cbseacademic.nic.in and https://ncert.nic.in
Use graphs and tables for visual understanding.
Mastering mean, median, and mode helps in competitive exams and real-world data handling.
Revise all formulas regularly and maintain a neat formula chart.
