📊 CBSE · Class 10 · Mathematics · Chapter 14

Chapter 14
Statistics

Complete chapter resources for CBSE Class 10 Maths Statistics — topic breakdown, key formulas for mean, median, mode and ogive, sample questions, previous year board questions, and instant AI question paper generation.

3Topics
6–8Board marks
8Sample questions
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Key Formulas — Chapter 14
  • Mean (Direct): x̄ = Σfᵢxᵢ / Σfᵢ
  • Mean (Step Dev.): x̄ = a + (Σfᵢuᵢ / Σfᵢ) × h
  • Median: M = l + ((n/2 − cf) / f) × h
  • Mode: Mo = l + ((f₁ − f₀) / (2f₁ − f₀ − f₂)) × h
  • Empirical relation: Mode = 3 Median − 2 Mean
Chapter Overview

What this chapter covers

Chapter 14 of NCERT Class 10 Mathematics extends the study of statistics from ungrouped to grouped frequency distributions. Students learn to calculate the three measures of central tendency — mean, median, and mode — for data presented in class intervals, using specialised formulas suited to each measure. The chapter establishes that these three averages may differ for the same data set, and each captures a different aspect of the distribution.

The mean can be found by three equivalent methods: the Direct Method (Σfᵢxᵢ / Σfᵢ), the Assumed Mean Method (which reduces arithmetic by choosing a central value a), and the Step Deviation Method (which further simplifies by dividing deviations by class width h). The median is located using the cumulative frequency table and the formula M = l + ((n/2 − cf) / f) × h, identifying the median class where the running total first reaches n/2. The modal class is the class with the highest frequency, and the mode formula uses the frequencies of the class before and after it.

The chapter also introduces the cumulative frequency curve (ogive) — a graphical tool where less-than and more-than ogives are plotted on the same axes. The x-coordinate of their intersection gives the median directly from the graph. Board exams regularly test all three averages from a single frequency table, the empirical relationship Mode = 3 Median − 2 Mean, and the construction and reading of ogives.

Topics

What's inside Chapter 14

As per NCERT Class 10 Mathematics (CBSE syllabus)

Topic 1
Mean of Grouped Data
Three methods: Direct Method (Σfᵢxᵢ / Σfᵢ), Assumed Mean Method using deviations dᵢ = xᵢ − a, and Step Deviation Method using uᵢ = dᵢ / h. All three give identical results; the step-deviation method is preferred when class widths are equal and values are large.
Topic 2
Median and Mode of Grouped Data
Median uses the cumulative frequency table to locate the median class, then applies M = l + ((n/2 − cf) / f) × h. Mode identifies the modal class (highest frequency) and applies Mo = l + ((f₁ − f₀) / (2f₁ − f₀ − f₂)) × h. The empirical relation Mode = 3 Median − 2 Mean links all three.
Topic 3
Cumulative Frequency Curve (Ogive)
Less-than ogive plots cumulative frequency against upper class boundaries; more-than ogive plots against lower boundaries. Plotting both on the same graph gives the median at their intersection. Ogives are also used to read off percentiles and quartiles.
Connections

How this chapter fits in

Useful for setting question difficulty and cross-chapter papers.

Builds on
Class 9 · Statistics
Ungrouped data — mean, median, mode; bar graphs, histograms, frequency polygons
Ch 13 · Surface Areas & Volumes
Real-world data contexts — measurement data presented in frequency tables
Chapter 14 Statistics
Leads to
Ch 15 · Probability
Experimental probability uses frequency distributions; relative frequency ties back to statistics
Class 11 · Statistics
Measures of dispersion — variance, standard deviation, frequency distribution analysis
Board Blueprint

Marks & question-type breakdown

Typical pattern based on CBSE Class 10 board papers from the last five years.

Question type Marks Typical count What's usually tested
MCQ / Objective 1 1–2 Empirical relation, modal class identification, or median class selection
Very Short Answer 2 1 Find missing frequency given mean/median, or apply empirical relation
Short Answer 3 1 Calculate mean using step deviation method from a frequency table
Long Answer / Graph 4–5 1 Find median or mode from grouped data, or draw and interpret an ogive
Total (approximate) 6–8 4–5 Weightage varies across paper sets and years
Sample Questions

8 sample questions — generated by MarksZen AI

Aligned to CBSE Class 10 Maths Chapter 14. Covers all question types across Easy, Medium, and Hard difficulty.

Q1 Easy 1 mark MCQ
The empirical relationship between mean (M̄), median (Md), and mode (Mo) for a moderately skewed distribution is: (a) Mo = 3M̄ − 2Md (b) Mo = 3Md − 2M̄ (c) M̄ = 3Md − 2Mo (d) Md = 3Mo − 2M̄
Q2 Easy 2 marks Short Answer
The mean and median of a frequency distribution are 26.8 and 27.2 respectively. Find the mode using the empirical relationship.
Q3 Medium 2 marks Short Answer
Find the missing frequency f in the following distribution if the mean is 18: Class: 0–6 6–12 12–18 18–24 24–30 Frequency: 4 5 f 4 2
Q4 Medium 3 marks Short Answer
The following table shows the ages of 100 patients admitted to a hospital. Calculate the mean age using the step deviation method: Age (years): 10–20 20–30 30–40 40–50 50–60 60–70 No. of patients: 5 25 35 20 10 5
Q5 Medium 3 marks Short Answer
Find the mode of the following frequency distribution: Class: 10–20 20–30 30–40 40–50 50–60 Frequency: 5 12 20 11 6
Q6 Hard 4 marks Long Answer
The distribution below gives the weight of 30 students of a class. Find the median weight: Weight (kg): 40–45 45–50 50–55 55–60 60–65 65–70 No. of students: 2 3 8 6 6 5
Q7 Hard 4 marks Long Answer
The following distribution gives the daily income of 50 workers of a factory: Daily income (₹): 100–120 120–140 140–160 160–180 180–200 Number of workers: 12 14 8 6 10 Convert the distribution to a less-than type cumulative frequency distribution and draw its ogive. From the ogive, find the median.
Q8 Hard 5 marks Case-Based
A school surveyed 80 students about the number of hours they study per day: Hours per day: 0–2 2–4 4–6 6–8 8–10 No. of students: 5 12 23 25 15 (i) Find the mean number of hours studied per day using the assumed mean method (take a = 5). (ii) Find the median number of hours. (iii) Identify the modal class and find the mode.
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Previous Year Questions

From CBSE board examinations

Actual questions from past Class 10 Maths board papers — Statistics chapter.

Board 20223 marks
The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median of the data. Monthly consumption (units): 65–85 85–105 105–125 125–145 145–165 165–185 185–205 Number of consumers: 4 5 13 20 14 8 4 (CBSE All India 2022)
Board 20234 marks
The following table gives the distribution of the life-time (in hours) of 400 neon lamps. Draw a less-than ogive for the given data and use it to find the median. Lifetime (hours): 1500–2000 2000–2500 2500–3000 3000–3500 3500–4000 4000–4500 4500–5000 Number of lamps: 14 56 60 86 74 62 48 (CBSE 2023)
Board 20203 marks
If the mean of the following frequency distribution is 54, find the value of p: Class: 0–20 20–40 40–60 60–80 80–100 Frequency: 7 p 10 9 13 (CBSE Delhi 2020)
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FAQ

Questions teachers ask

How many marks does Statistics carry in the CBSE Class 10 board exam? +
Statistics typically carries 6–8 marks in the CBSE Class 10 Maths board paper, usually appearing as one 1-mark MCQ, one 2-mark short answer, and one 4-mark long answer involving mean, median, or mode from a grouped frequency table. The chapter has appeared consistently in CBSE board papers for over a decade.
Which method for finding the mean is most important for CBSE board exams? +
All three methods — Direct Method, Assumed Mean Method, and Step Deviation Method — are in the CBSE syllabus, but the Step Deviation Method is the most commonly tested in board long-answer questions because it simplifies large values and demonstrates deeper understanding. Students should be comfortable with all three and choose based on the data given.
What is the empirical relationship between mean, median, and mode? +
The empirical relationship is: Mode = 3 Median − 2 Mean. This formula is directly stated in NCERT and is tested in 1-mark and 2-mark board questions where two measures are given and the third must be calculated. Students must memorise this formula as it is a guaranteed high-frequency board item.
What is a cumulative frequency curve (ogive) and how is it tested in boards? +
An ogive is a smooth curve drawn by plotting cumulative frequencies against upper (less-than ogive) or lower (more-than ogive) class boundaries. Board questions ask students to draw both types on the same graph and locate the median at the intersection of the two curves, or read off a value at n/2 on the y-axis. This is a standard 4–5 mark question.
How do I generate a custom question paper for Statistics using MarksZen? +
Sign up for a free MarksZen account, choose CBSE Class 10 Mathematics, select Chapter 14 (Statistics), set your preferred question-type mix and total marks — the AI generates a complete board-aligned paper with answer key in under 2 minutes, ready for PDF export.