Chapter 13 of 14

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Odisha (BSE)Class 10 Mathematics★ Unit VII — Statistics and Probability: 11 marks, shared with Probability · CBSE Class 10 Maths (Chapter 13)

Statistics

Statistics — mean, median and mode of grouped data by the direct, assumed-mean and step-deviation methods, the empirical relationship between them, and cumulative frequency. CBSE Class 10 Maths Chapter 13.

⏱ 22 min readMedium difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Find the mean of grouped data by the direct, assumed-mean, and step-deviation methods, and choose the quickest one for the numbers given
2Build a cumulative frequency column and use it to locate the median class
3Apply the median formula, including the 'below/above' data format that needs converting to class intervals first
4Apply the mode formula, and identify the modal class correctly
5Use the empirical relationship to check mean, median and mode against each other

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Why this chapter matters

Raw data is useless until it is summarised — a mean, a median, a mode. This chapter extends those three ideas from single numbers you can list to grouped data with hundreds of entries, which is how real surveys, census figures and lab results actually arrive.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Surface Areas and Volumes

What CBSE examines here (2026-27). Mean, median and mode of grouped data — including all three methods for the mean (direct, assumed mean, step-deviation) and the cumulative-frequency idea the median formula depends on. Both of these survived rationalisation intact; do not trust a source that tells you otherwise. The chapter runs to three exercises — 13.1, 13.2 and 13.3. What did change: there is no longer a dedicated exercise on constructing an ogive by hand. The concept is still named in the book's own closing note, so it is kept here for context, clearly marked as not its own tested skill.

"Without data, you are just another person with an opinion." — W. Edwards Deming

1. About the Chapter

Statistics = collecting, organising, analysing, interpreting data.

Class 10 focuses on grouped data — large datasets organised into class intervals.

Three Measures of Central Tendency

Mean (average)
Median (middle value)
Mode (most frequent value)

Why Important

Election polls and analyses
Economic data (inflation, GDP)
Medical research
Sports statistics
Climate data
Business intelligence

2. Grouped Data — Recap

Class Intervals

Data is organised into ranges (classes):

0-10
10-20
20-30
etc.

Frequency

Number of data items in each class.

Frequency Distribution Table

Class	Frequency (f)
0-10	5
10-20	12
20-30	8
30-40	7

Important Terms

Class mark = (Lower limit + Upper limit) / 2
Class size = Upper limit − Lower limit (constant)
Cumulative frequency = running total of frequencies

3. Mean of Grouped Data

Direct Method

Mean = Σ(fᵢxᵢ) / Σfᵢ

where:

fᵢ = frequency of i-th class
xᵢ = class mark of i-th class

Example

Class	f	x (mark)	fx
0-10	5	5	25
10-20	12	15	180
20-30	8	25	200
30-40	7	35	245
Total	32		650

Mean = 650 / 32 = 20.3125

Assumed Mean Method (Easier Calculation)

For large numbers: Mean = a + (Σfᵢdᵢ / Σfᵢ)

where:

a = assumed mean
dᵢ = xᵢ − a

This simplifies arithmetic.

Step-Deviation Method

Mean = a + (Σfᵢuᵢ / Σfᵢ) × h

where:

a = assumed mean
uᵢ = (xᵢ − a) / h
h = class size

Useful for clean numbers.

4. Median of Grouped Data

Definition

Median = middle value (when data is arranged in order).

For grouped data, the median lies within a specific class — the MEDIAN CLASS.

Finding Median Class

Calculate cumulative frequency (CF)
Find class where CF ≥ N/2 first (where N = total frequency)
That's the median class.

Formula

Median = l + ((N/2 − cf) / f) × h

where:

l = lower limit of median class
N = total frequency
cf = cumulative frequency of class BEFORE median class
f = frequency of median class
h = class size

Example

Class	f	CF
0-10	5	5
10-20	12	17
20-30	8	25
30-40	7	32

N = 32, N/2 = 16. First CF ≥ 16 is 17 (in class 10-20).

l = 10, cf = 5, f = 12, h = 10
Median = 10 + ((16 − 5) / 12) × 10 = 10 + 110/12 = 10 + 9.17 = 19.17

5. Mode of Grouped Data

Definition

Mode = most frequent value.

For grouped data, the mode is in the MODAL CLASS — class with highest frequency.

Formula

Mode = l + ((f₁ − f₀) / (2f₁ − f₀ − f₂)) × h

where:

l = lower limit of modal class
f₁ = frequency of modal class
f₀ = frequency of class BEFORE modal class
f₂ = frequency of class AFTER modal class
h = class size

Example

From the same table:

Modal class: 10-20 (highest f = 12)
l = 10, f₁ = 12, f₀ = 5, f₂ = 8, h = 10
Mode = 10 + ((12 − 5) / (24 − 5 − 8)) × 10
= 10 + (7 / 11) × 10
= 10 + 6.36 = 16.36

6. Empirical Formula

For most distributions: Mode = 3 × Median − 2 × Mean

Or: Mean − Mode = 3(Mean − Median)

These are approximate but useful relationships.

Verification (from above example)

Mean = 20.31, Median = 19.17, Mode = 16.36
3 × Median − 2 × Mean = 3(19.17) − 2(20.31) = 57.51 − 40.62 = 16.89 ≈ Mode ✓

7. Ogives (Cumulative Frequency Curves)

Not a separate exercise in the current book. Ogives are described here for context — the chapter's closing note still mentions them, and cumulative frequency itself is central to the median formula above. But there is no exercise asking you to plot one by hand; nothing in Exercises 13.1-13.3 requires drawing a curve.

What is an Ogive?

A graph of CUMULATIVE FREQUENCY against class boundaries.

Two Types

Less Than Ogive:

Plot (upper limit of each class, less than CF)
Curve rises gradually
Used when 'how many less than this value'

More Than Ogive:

Plot (lower limit, more than CF)
Curve falls gradually
Used when 'how many more than this value'

Finding Median from Ogive

Both ogives intersect at the median (graphical method).

8. Worked Examples

Example 1: Mean

Heights of 20 students (cm):

Class	f	x	fx
130-140	4	135	540
140-150	8	145	1160
150-160	6	155	930
160-170	2	165	330
Total	20		2960

Mean = 2960 / 20 = 148 cm

Example 2: Median

From Example 1 data:

CF: 4, 12, 18, 20
N/2 = 10. First CF ≥ 10 is 12 (in class 140-150).
Median class: 140-150
l = 140, cf = 4, f = 8, h = 10
Median = 140 + ((10 − 4) / 8) × 10 = 140 + 7.5 = 147.5 cm

Example 3: Mode

Modal class: 140-150 (f = 8 highest)
l = 140, f₁ = 8, f₀ = 4, f₂ = 6, h = 10
Mode = 140 + ((8 − 4) / (16 − 4 − 6)) × 10
= 140 + (4/6) × 10 = 140 + 6.67 = 146.67 cm

9. Common Mistakes

Class mark wrong
- Class mark = (lower + upper) / 2. For 10-20, it's 15, not 20.
Cumulative frequency confusion
- Just keep adding frequencies.
Median class wrong
- Find FIRST CF ≥ N/2. Don't pick last.
Empirical formula misuse
- 3 × Median − 2 × Mean = Mode (when needed).
Class size assumption
- Class size is the WIDTH (upper − lower), constant across classes.

10. Real-World Applications

Census Data

Indian census every 10 years uses statistics.

Economic Indicators

Inflation rate, GDP, unemployment — all statistics
Indian Statistical Service handles national data

Health

Average heights, weights of children
Disease prevalence rates
COVID-19 statistics

Education

Class average marks
School performance comparisons
Indian SSA, NCERT use statistical tools

Sports

Cricket batting averages
Football scores

Indian Context

Indian Statistical Institute (founded 1931) — world-leading
PM Modi government uses statistics for policy

11. Indian Heritage

Statistical Methods in India

Aryabhata (5th c.) used statistical methods in astronomy
C.R. Rao (1920-2023) — world-renowned Indian statistician
P.C. Mahalanobis (1893-1972) — founder of ISI, Indian planning statistician

12. Conclusion

Statistics is the language of MODERN ANALYSIS:

Politics (polling)
Economics (data)
Science (research)
Daily life (sports, weather)

Master:

Three measures: mean, median, mode — and all three methods for the mean
Cumulative frequency, since the median formula is built on it
Empirical formula: Mode = 3 × Median − 2 × Mean

In Class 11-12, you'll learn variance, standard deviation, regression.

Statistics: the science of making sense of data.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Mean — direct method

Σfᵢxᵢ / Σfᵢ

xᵢ is the class mark (midpoint) of each class

Mean — assumed mean method

a + Σfᵢdᵢ/Σfᵢ, where dᵢ = xᵢ − a

Pick a near the middle of the data to keep dᵢ small

Mean — step-deviation method

a + h × Σfᵢuᵢ/Σfᵢ, where uᵢ = (xᵢ − a)/h

Fastest when every dᵢ shares a common factor h — still in the syllabus

Mode

l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h

f₁ = modal class frequency; f₀, f₂ = frequencies just before/after it

Median

l + [(n/2 − cf)/f] × h

cf = cumulative frequency before the median class; f = median class's own frequency

Empirical relationship

Mode = 3 × Median − 2 × Mean

An approximate check, not an identity — use it to sanity-test your three answers against each other

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Using the class's raw values instead of the class mark

✓ Every formula in this chapter works on the class mark xᵢ = (lower + upper)/2, treating the whole class as if it were concentrated at its midpoint — never the lower or upper limit alone.

WATCH OUT

✗ Picking the modal class as 'whichever class looks central'

✓ The modal class is simply the one with the highest frequency — nothing about its position in the table matters.

WATCH OUT

✗ Finding the median class by picking the wrong cumulative frequency

✓ Find the *first* class whose cumulative frequency is ≥ n/2, scanning from the top. Picking a later class, or the one right at n/2 exactly without checking it's the first, gives the wrong class.

WATCH OUT

✗ Forgetting to convert 'less than' data into class intervals before using the median formula

✓ The formula needs frequencies per class, not running totals. Subtract consecutive cumulative frequencies to recover each class's own frequency first.

WATCH OUT

✗ Applying the median/mode formulas to non-continuous classes

✓ Classes like 118-126, 127-135 have gaps between them. Adjust by subtracting 0.5 from every lower limit and adding 0.5 to every upper limit before applying any formula — the textbook's own hint for exactly this case.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Ex 13.1

Exercise 13.1

Mean of grouped data — direct, assumed-mean and step-deviation methods

Questions

Ex 13.2

Exercise 13.2

Mode of grouped data, with mean for comparison

Questions

Ex 13.3

Exercise 13.3

Median of grouped data, including cumulative-frequency ('below') data and non-continuous classes

Questions

Practice problems

Work through this chapter's problems as a readiness check — reveal each solution, mark yourself honestly, and get your gap report at the end.

Readiness check

Are you exam-ready for Statistics?

3 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

3 questions~2 min worth ~10 marks in Odisha (BSE) exams

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Class mark = (lower limit + upper limit) / 2 — every formula in this chapter is built on it
•Three methods for the mean: direct, assumed mean, step-deviation — step-deviation is fastest when the dᵢ share a common factor
•Cumulative frequency: running total of frequencies, either 'less than' or 'more than' type
•Median class: the first class whose cumulative frequency reaches n/2
•Modal class: simply the class with the highest frequency
•Empirical relationship: Mode = 3 × Median − 2 × Mean — approximate, useful as a check
•Non-continuous classes must be converted to continuous ones (±0.5 adjustment) before the median or mode formula applies
•This chapter has three exercises: 13.1 (mean, 9 Q), 13.2 (mode, 6 Q), 13.3 (median, 7 Q)

Odisha (BSE) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 8-10 marks

Question type	Marks each	Typical count	What it tests
MCQ	1	2	Definitions
Short	2-3	1-2	Single measure
Long	5	1	Find all three measures

Prep strategy

Memorise all 3 measures formulas
Practice grouped data problems
Master ogive plotting

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Indian Census

Mean, median, mode used in census data analysis (every 10 years).

Cricket statistics

Batting averages, run rates, strike rates all use Class 10 statistics.

Inflation calculation

CPI, WPI inflation indices use grouped data statistics.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Build the class mark and cumulative frequency columns first, before touching any formula

State which method you're using for the mean and why — markers give credit for the choice, not just the arithmetic

Double-check the median class by confirming it is the *first* one whose cumulative frequency reaches n/2

Use the empirical relationship at the end as a plausibility check on all three answers together

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

STRETCH

Variance, standard deviation (Class 11)

STRETCH

Probability distributions

STRETCH

Regression analysis

STRETCH

Indian statistician C.R. Rao's work

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 10 BoardVery High

Maths OlympiadMedium

JEE FoundationHigh

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

No — and this is worth being precise about, because some study guides say otherwise. The NCERT chapter (Reprint 2026-27) still teaches all three methods for the mean, including step-deviation, and the chapter's own summary states the step-deviation formula explicitly. If a source tells you step-deviation was dropped, it is wrong for this chapter.

The concept of cumulative frequency is central and fully examinable — the median formula depends on it. But there is no longer an exercise asking you to actually plot a 'less than' or 'more than' ogive by hand and read off the median graphically. The idea is mentioned in the chapter's closing note, but it is not one of the three exercises' own questions.

Direct method when the numbers are small and the arithmetic is manageable as-is. Assumed mean when the xᵢ values are large, to keep the numbers you're multiplying smaller. Step-deviation when, on top of that, every deviation shares a common factor h — it's the same answer either way, just less arithmetic.

Practice more, ask doubts, find a tutor

Odisha (BSE) Class 10 mock tests

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