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I. Chapter Summary:
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.