Chapter 12 of 12

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Rajasthan (RBSE)Class 9 Mathematics★ 6 marks · CBSE Class 9 board

Statistics

Collecting and presenting data — frequency tables, bar graphs, histograms and frequency polygons. Plus mean, median and mode for ungrouped data.

⏱ 11 min readEasy difficulty✓ Free · NCERT-aligned

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

1Distinguish primary and secondary data; raw and processed data
2Build a frequency distribution table in exclusive and inclusive forms
3Draw bar graphs, histograms and frequency polygons accurately
4Compute mean, median and mode for ungrouped data
5Pick the appropriate measure of central tendency for a given dataset

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Why this chapter matters

Statistics is the easiest 'high marks' chapter in Class 9 — most questions are direct computation with no tricks. The vocabulary (frequency, class, mean, median, mode) and the histogram/frequency-polygon construction recur in Class 10 with one extra formula layer. Master the basics here and Class 10 Statistics becomes revision, not learning.

Before you start — revise these

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

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Data Handling (Class 8)

Basic bar graphs and pie charts — direct prerequisite

Statistics

After eleven chapters of pure mathematics, the last chapter of Class 9 introduces data — how to collect it, organise it into tables and graphs, and summarise it with single numbers like the average.

1. Types of data

Term	Meaning
Primary data	Collected first-hand by the investigator (e.g. you survey your class)
Secondary data	Collected by someone else and reused (e.g. census tables)
Raw data	Untouched, unsorted observations
Range	Maximum value − minimum value

2. Presenting data

Frequency distribution table. Lists each value (or class interval) and how often it occurs.

Marks (class)	Number of students (frequency)
0 – 10	4
10 – 20	6
20 – 30	12
30 – 40	8

Two conventions for class boundaries:

Exclusive form (e.g. 0–10, 10–20): a value of 10 falls in 10–20, not 0–10.
Inclusive form (e.g. 0–9, 10–19): boundaries don't overlap. Convert to exclusive by adding/subtracting 0.5 from limits before drawing histograms.

3. Graphical representations

Bar graph: bars with gaps; for discrete or categorical data.
Histogram: bars without gaps; for continuous (grouped) data. Bar height = frequency density (frequency / class width) — but when all classes are equal width, height = frequency directly.
Frequency polygon: join the midpoints of the tops of histogram bars with straight lines. Useful for comparing two distributions.

4. Measures of central tendency (for raw / ungrouped data)

For $n$ observations $x_{1}, x_{2}, \dots, x_{n}$ :

Mean

$\overset{x}{ˉ} = \frac{x _{1} + x _{2} + \dots + x _{n}}{n} = \frac{\sum x _{i}}{n}$

Median

Arrange the data in ascending order. Then:

If $n$ is odd: median = middle value $= (\frac{n + 1}{2})$ th observation.
If $n$ is even: median = average of the two middle values $= \frac{1}{2} [(\frac{n}{2}) th + (\frac{n}{2} + 1) th]$ .

Mode

The value(s) that occur most often. A dataset can be:

Unimodal (one mode), bimodal (two modes), multimodal, or have no mode (all values different).

5. Worked example

Find the mean, median and mode of: 4, 8, 7, 4, 6, 5, 4, 9, 7, 6.

Sorted: 4, 4, 4, 5, 6, 6, 7, 7, 8, 9 ( $n = 10$ ).

Mean = (4+4+4+5+6+6+7+7+8+9)/10 = 60/10 = 6.

Median ( $n = 10$ , even): average of 5th and 6th values = (6 + 6)/2 = 6.

Mode = value occurring most often = 4 (occurs 3 times).

6. When does each measure win?

Mean is sensitive to extreme values (a single outlier moves it a lot).
Median is robust against outliers — preferred for income, house prices.
Mode is the only measure that works for categorical data (e.g. most common favourite colour).

What's next

In Class 10 you'll meet the mean of grouped data, median formula ( $l + \frac{n /2 - c f}{f} \times h$ ) and mode formula for class intervals. The intuition you build here makes those formulas instantly clickable.

Key formulas & results

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

Mean (ungrouped)

x̄ = (Σ xᵢ) / n

Sum of all observations ÷ count. Sensitive to outliers.

Median (odd n)

Median = ((n+1)/2)-th observation (data sorted)

ALWAYS sort first.

Median (even n)

Median = average of (n/2)-th and (n/2 + 1)-th obs

Data must be sorted.

Mode

Value with the highest frequency

Can be undefined if all values appear equally often.

Range

Range = Maximum value − Minimum value

Simplest measure of spread.

Frequency density (histogram)

Frequency density = frequency / class width

Use when class widths are unequal.

Class midpoint (frequency polygon)

Midpoint = (lower limit + upper limit) / 2

Plot frequency vs midpoint to draw frequency polygon.

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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

✗ Computing the median without sorting the data first

✓ ALWAYS sort the observations in ascending order before finding the median position. Median of an unsorted list is meaningless — you will get a wrong answer every time.

WATCH OUT

✗ Mixing inclusive and exclusive class limits in one frequency table

✓ Pick one convention and stick with it. Convert inclusive to exclusive (true class limits) BEFORE drawing a histogram — subtract 0.5 from lower limits and add 0.5 to upper limits.

WATCH OUT

✗ Drawing a histogram with gaps between bars

✓ Histograms have NO gaps — the data is continuous. Gaps belong on a BAR GRAPH (discrete or categorical data). This is a 1-mark trap in every exam.

WATCH OUT

✗ Claiming 'no mode' when there is a single most-frequent value

✓ 'No mode' only applies when every value appears exactly the same number of times. If 4 appears 3 times and everything else appears twice, mode = 4.

NCERT exercises (with solutions)

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

Ex 12.1

Exercise 12.1

Frequency tables, bar graphs, histograms and frequency polygons — graphical representation

Questions

Ex 12.2

Exercise 12.2

Mean, median, mode for ungrouped data — direct computation

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Mean

Find the mean of 25, 30, 35, 40, 45.

▶ Show solution

Step 1 — Sum all observations.
  25 + 30 + 35 + 40 + 45 = 175.

Step 2 — Divide by n = 5.
  Mean = 175 / 5 = 35.

✦ Answer: Mean = 35.

Q2EASY· Median

Find the median of 11, 7, 14, 8, 21, 5, 17.

▶ Show solution

Step 1 — Sort in ascending order.
  5, 7, 8, 11, 14, 17, 21.

Step 2 — n = 7 (odd). Middle position = (7+1)/2 = 4th observation.

Step 3 — 4th observation = 11.

✦ Answer: Median = 11.

Q3EASY· Mode

Find the mode of 2, 3, 3, 4, 5, 3, 6, 7.

▶ Show solution

Step 1 — Count frequencies: 2→1, 3→3, 4→1, 5→1, 6→1, 7→1.

Step 2 — 3 occurs 3 times; all others occur once.

✦ Answer: Mode = 3.

Q4MEDIUM· Mean — reverse problem

The mean of 5 numbers is 18. If one number is added and the new mean becomes 20, find the added number.

▶ Show solution

Step 1 — Find original sum.
  Original sum = 5 × 18 = 90.

Step 2 — Find new sum.
  New sum = 6 × 20 = 120.

Step 3 — Added number = 120 − 90 = 30.

✦ Answer: The added number is 30.

Q5MEDIUM· Frequency table + mode

20 students scored: 7, 8, 9, 7, 6, 8, 9, 5, 7, 8, 9, 7, 6, 5, 8, 9, 7, 7, 6, 8. Build a frequency table and find the mode.

▶ Show solution

Step 1 — Build the frequency table.
  Score 5: appears 2 times
  Score 6: appears 3 times
  Score 7: appears 6 times
  Score 8: appears 5 times
  Score 9: appears 4 times
  Total: 20 ✓

Step 2 — Identify highest frequency: 6 (for score = 7).

✦ Answer: Mode = 7.

Q6MEDIUM· Histogram concept

Explain the difference between a bar graph and a histogram. When would you use each?

▶ Show solution

Step 1 — Bar graph: bars WITH GAPS; used for discrete or CATEGORICAL data (e.g., favourite colours of students). The gap indicates that values in between are not possible.

Step 2 — Histogram: bars WITHOUT GAPS; used for CONTINUOUS (grouped/interval) data (e.g., heights in cm class-wise). The no-gap shows continuity of the data.

✦ Answer: Use bar graph for discrete/categorical data. Use histogram for continuous grouped data.

Q7HARD· Combined mean

The mean of 30 observations is 20 and the mean of 20 other observations is 25. Find the combined mean of all 50 observations.

▶ Show solution

Step 1 — Find the sum of the first group.
  Sum₁ = 30 × 20 = 600.

Step 2 — Find the sum of the second group.
  Sum₂ = 20 × 25 = 500.

Step 3 — Combined sum = 600 + 500 = 1100. Combined n = 30 + 20 = 50.

Step 4 — Combined mean = 1100 / 50 = 22.

✦ Answer: Combined mean = 22.

5-minute revision

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

•Primary data: collected by you. Secondary data: collected by someone else.
•Range = max − min.
•Frequency table: exclusive form (10–20, 20–30) or inclusive form (10–19, 20–29).
•Histogram = continuous data, bars touching. Bar graph = discrete data, gaps allowed.
•Frequency polygon = join midpoints of histogram-bar tops with straight lines.
•Mean = Σx / n. Sensitive to outliers.
•Median: SORT first, then pick middle (odd n) or average of two middles (even n).
•Mode = most-frequent value. Can be multiple (bimodal) or none (all equal frequency).
•Median is robust to outliers; mean is not. Mode works for categorical data.

CBSE marks blueprint

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

Typical chapter weightage: 6 marks

Question type	Marks each	Typical count	What it tests
MCQ	1	1	Definition or formula identification
Short answer (2-mark)	2	1–2	Direct formula application
Long answer (3-mark)	3	1–2	Multi-step problem solving
Long answer (4-mark)	4	1	Proof or complex application

Prep strategy

Write all key formulas on a revision sheet — recall speed matters in board exams
Show ALL working steps clearly — partial marks are awarded for method even if final answer is wrong
Practise past 5 years of CBSE board questions for this chapter
For proof questions: state the theorem/property used at each step by name
Time management: allocate time based on marks — 1 min per mark is a good rule

Where this shows up in the real world

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

Cricket averages

Batting averages (mean runs), strike rates, and economy rates are all statistical measures computed exactly as in this chapter.

School report cards

Class average, highest/lowest mark, and whether most students cluster around a particular score — all use mean, range, and mode.

Weather data

Mean temperature for a month, median rainfall, most common wind direction (mode) — these are daily applications of this chapter's tools.

Market research

Companies survey customers and build frequency tables to find the most popular product feature (mode) or average satisfaction score (mean).

Exam strategy

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

For direct computation questions (mean/median/mode), show each step — sum, count, and final answer. 3-mark questions expect at least 3 lines of working.
Always sort data BEFORE finding the median — and write 'Sorted data:' as the first step to earn the method mark.
For histogram questions, state whether class widths are equal or unequal, then decide whether to use frequency or frequency density on the y-axis.
Frequency polygon: join midpoints of each bar top AND include the points at zero frequency at the start and end of the distribution.
When asked to compare mean, median, and mode for a dataset, comment on outliers — 'Since there is an outlier at X, the median is a better measure than the mean.'

Going beyond the textbook

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

Weighted mean: if different groups have different sizes, the combined mean ≠ simple average of group means. Always use Σ(n·x̄)/Σn.
Median for grouped data (Class 10 preview): Median = l + ((n/2 − cf)/f) × h — start building intuition for what each symbol means.
Quartiles and interquartile range (IQR): Q₁ is the median of the lower half, Q₃ is the median of the upper half. IQR = Q₃ − Q₁ is a robust spread measure.
Mean deviation from median ≤ Mean deviation from mean — the median minimises mean absolute deviation (a beautiful result provable with basic inequalities).

Where else this chapter is tested

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

CBSE Class 9 Board6 marks in final exam. Chapter is compulsory.

NTSE (Stage 1)Mental ability section includes data interpretation based on these statistics fundamentals.

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Questions students ask

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

No. Pie charts were in Class 8. Class 9 statistics focuses on bar graphs, histograms and frequency polygons. Do not draw pie charts in Class 9 statistics answers.

Probability was removed from Class 9 in recent NCERT revisions. Class 9 ends with Statistics (Chapter 12). Probability returns in Class 10.

Class 10 Statistics extends mean/median/mode to GROUPED data with formulas: Mean = a + (Σf·d/Σf), Median = l + ((n/2 − cf)/f) × h, Mode = l + ((f₁-f₀)/(2f₁-f₀-f₂)) × h. The intuition you build here makes those formulas immediately understandable.

If two values tie for highest frequency, the dataset is BIMODAL (two modes). If three tie, it is multimodal. If all values appear equally often, there is NO mode.

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