Statistics Class 10: NCERT Maths Chapter 13

Statistics Class 10 NCERT Chapter 13 runs from page 170 to page 201 of the NCERT Mathematics textbook for the 2026-27 session. It extends the mean, median and mode you studied in Class IX from ungrouped data to grouped data, and the official chapter PDF is right below.

Download the official Statistics Class 10 NCERT Chapter 13 PDF from ncert.nic.in — the same file that appears in the printed book — so you can read or re-read any section while you work through the notes below.

Chapter 13 Statistics at a Glance

This is a compact, formula-heavy chapter. It teaches the mean, median and mode of grouped data, the cumulative frequency tables you build along the way, and the cumulative frequency curves called ogives (NCERT pp. 170–171).

The table below lists what the chapter file holds, so you know its size before you open it.

Every topic rests on the same machinery: data in class intervals, each represented by its class mark, with frequencies counted into the right class. Master that and the rest is substitution.

What Chapter 13 Statistics Teaches

This chapter takes the three measures of central tendency from Class IX and adapts them to grouped (class-interval) data, because real data is usually too large to handle observation by observation (NCERT p. 171).

The chapter moves in one direction: first the mean, then the mode, then the median, each time using the same kind of grouped frequency table.

One honest point: the introduction promises ogives, and the closing note tells you ogive construction needs continuous classes, but the worked body of the chapter concentrates on the three measures and on cumulative frequency tables (NCERT pp. 170–171, 201).

Mean of Grouped Data: Three Methods, One Answer

Grouped data has no individual scores, so each class is represented by its mid-point, called the class mark, and the mean is built from those marks and their frequencies (NCERT p. 173).

The class mark of an interval is the average of its two limits:

\[ \text{Class mark} = \frac{\text{Upper class limit} + \text{Lower class limit}}{2} \]

For the interval 10–25 the class mark is 17.5. The assumption underneath the whole section is that the observations in each class are centred at this mid-point — that is why the grouped mean is approximate, not exact (NCERT p. 174).

The mean formula

If \( x_i \) is the class mark of the \( i \)-th class and \( f_i \) its frequency, the mean of grouped data is \[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \]

Read it as: multiply each class mark by its frequency, add all those products, then divide by the total frequency (NCERT p. 172).

Direct method

The direct method applies the formula as it stands: build a column of \( f_i x_i \), sum it, and divide by \( \sum f_i \) (NCERT p. 174). It is quickest when the numbers are small.

Assumed mean method

When the class marks are large, choose one mark near the middle and call it \( a \), the assumed mean. Then work with the deviations \( d_i = x_i – a \), which are much smaller numbers (NCERT pp. 174–176).

\[ \bar{x} = a + \frac{\sum f_i d_i}{\sum f_i} \]

Why does this work? Subtracting \( a \) from every mark only shifts the data, so the mean shifts by the same amount — adding \( a \) back recovers the true mean (NCERT p. 175).

Step deviation method

If the deviations share a common factor — usually the class size — divide them by it first. Let \( u_i = \frac{x_i – a}{h} \), where \( h \) is the class size, and use (NCERT pp. 176–177):

\[ \bar{x} = a + h \left( \frac{\sum f_i u_i}{\sum f_i} \right) \]

This rescaling makes the table numbers tiny; multiplying by \( h \) at the end restores the correct mean.

Worked example: the same mean three ways

Heights of 30 students were grouped into five classes of size 10 cm. All three methods must give the same answer — here is the proof with original numbers.

Class marks are 135, 145, 155, 165, 175 and \( \sum f_i = 30 \).

How inexact is the grouped mean? The chapter’s marks data gives 59.3 as the exact mean of the ungrouped values, but 62 after the same data is grouped into classes of width 15 — the difference is entirely the mid-point assumption (NCERT p. 174).

The chapter’s Activity 1 makes the same point as the worked example: try any class mark as \( a \) and the mean stays the same — the mean does not depend on the choice of \( a \) (NCERT p. 176).

Mode of Grouped Data: The Modal Class Formula

For ungrouped data the mode is the value that occurs most often (NCERT p. 183). In grouped data you cannot see the mode directly — you can only locate the class with the maximum frequency, called the modal class, and the mode is a single value inside it (NCERT p. 184).

The formula places the mode inside the modal class:

\[ \text{Mode} = l + \left( \frac{f_1 – f_0}{2f_1 – f_0 – f_2} \right) \times h \]

where (NCERT p. 184):

• \( l \) = lower limit of the modal class;
• \( h \) = class size (all classes equal);
• \( f_1 \) = frequency of the modal class;
• \( f_0 \) = frequency of the class just before it;
• \( f_2 \) = frequency of the class just after it.

The ratio \( \frac{f_1 – f_0}{2f_1 – f_0 – f_2} \) asks how far the mode sits from the lower limit: if the class before is almost as full as the modal class, the mode sits near the lower end; if the class after is fuller, it moves up.

Worked example: mode of a distribution

Here is an original data set: a school library recorded the number of books issued per day over 45 days.

Two chapter remarks to remember: the mode may be less than, equal to, or greater than the mean depending on the data (NCERT p. 186), and grouped data can be multimodal — the chapter keeps to single-mode problems only (NCERT p. 184).

The chapter works the same formula in Example 5 (family size) and Example 6 (marks), which is also a compare-and-interpret exercise: 52 is the modal marks while the mean marks is 62 (NCERT pp. 184–186).

Median of Grouped Data and Cumulative Frequency

The median is the middle observation. For grouped data you first add up frequencies as you go — that running total is the cumulative frequency — and then find the class that contains the middle value (NCERT pp. 188–190).

For ungrouped data, the rule from Class IX: arrange the values in ascending order; if \( n \) is odd the median is the \( \frac{n+1}{2} \)-th observation, and if \( n \) is even it is the average of the \( \frac{n}{2} \)-th and \( \left(\frac{n}{2}+1\right) \)-th observations (NCERT p. 188).

Cumulative frequency tables

Add a cumulative frequency column to the frequency table: each entry is the total of all frequencies up to and including that class. The last entry must equal \( n \), the total frequency (NCERT pp. 189–190).

Two standard forms are used (NCERT pp. 191–192):

• Less than type: for each upper limit, the number of observations below it — e.g. “less than 30” = \( 5 + 3 + 4 = 12 \).
• More than type: for each lower limit, the number of observations at or above it — e.g. “more than or equal to 30” = 41 in the chapter’s example.

Locating the median class

Compute \( \frac{n}{2} \). The median class is the class whose cumulative frequency is greater than and nearest to \( \frac{n}{2} \) (NCERT p. 193).

The median formula

\[ \text{Median} = l + \left( \frac{\frac{n}{2} – cf}{f} \right) \times h \]

Here \( l \) is the lower limit of the median class, \( f \) its frequency, \( h \) the class size, and \( cf \) is the cumulative frequency of the class preceding the median class — not the median class’s own cumulative frequency (NCERT p. 193). Taking \( cf \) from the wrong row is where students lose marks most often with this formula.

Worked example: median of marks

Marks out of 40 for 40 students were grouped as follows (original data).

Reading the median

A median of 28.5 in the chapter’s ungrouped example means about 50% of students scored less than 28.5 and 50% scored more (NCERT p. 190). The median always tells you the halfway point of the data, which is why it suits data with extreme values.

Missing frequencies

Example 8 shows a powerful trick: when one or two frequencies are unknown, the median itself gives you an equation. Here the median is 525, which fixes the median class at 500–600; substituting into the median formula gives \( x = 9 \), and the total frequency \( n = 100 \) then gives \( y = 15 \) (NCERT pp. 196–197). Exercise 13.3 Q2 asks exactly this kind of question.

Statistics Class 10 Formula Sheet

These five formulas are the whole chapter in miniature. Learn what each symbol means rather than just the expression, because the exercises test the symbols, not the formula.

Mean, Median or Mode: Which One to Use

The three measures usually give different numbers, so a question is really asking which kind of “average” fits the situation (NCERT pp. 197–198).

The chapter warns that the mean can mislead when one class is tiny and the rest are large — with frequencies like 2, 20, 25, 20, 21, 18, the mean does not reflect how the data behaves (NCERT p. 197).

Choosing the mean method: a decision rule

The chapter’s remark on method choice gives you this rule (NCERT pp. 179–180):

• Small \( x_i \) and \( f_i \) → use the direct method.
• Numerically large values → use the assumed mean method or step deviation.
• Unequal class sizes with large \( x_i \) → step deviation still works if you take \( h \) as a suitable common divisor of all \( d_i \).

Exercise 13.1 Q1 and Q5 ask you to name the method and justify it — the size of the class marks is the reason you give.

How to Read the Tables in This Chapter

Unlike most chapters, Chapter 13 has no figures — its learning objects are frequency tables and calculation tables, so “reading the diagram” means reading the table column by column.

Every mean calculation uses the same column progression:

• Class interval → class mark \( x_i \) → deviation \( d_i = x_i – a \) → \( u_i = \frac{d_i}{h} \).
• Next to those sit the frequency \( f_i \) and the product column \( f_i x_i \) or \( f_i u_i \).
• The bottom row is the total row, and its last entry feeds the formula: \( \sum f_i \), \( \sum f_i x_i \), \( \sum f_i d_i \), or \( \sum f_i u_i \).

You can follow this pattern in the chapter’s Tables 13.3, 13.4, 13.5 and 13.7 — each one simply adds one more column to the previous table (NCERT pp. 173–179).

Cumulative frequency tables work differently: each entry is a running total of frequencies up to that class, and the last entry must equal \( n \). In the less than type, entries are “below each upper limit”; in the more than type, “at or above each lower limit” (NCERT pp. 191–192).

Key Definitions in Statistics Class 10

These terms recur across all three sections, so it helps to see them in one place with the page where the book introduces each one.

Common Mistakes in Statistics Class 10

Most lost marks in this chapter come from six small misreadings of the table, not from the arithmetic.

Exam Notes for Statistics Class 10

The exercise questions are built to test method choice and interpretation, not just computation.

• Exercise 13.1 repeatedly asks “which method did you use and why” (Q1, Q5). Justify the choice using the remark on pp. 179–180: small values → direct; large values → assumed mean or step deviation. Q3 reverses the direction — you are given the mean and must find a missing frequency.
• Exercise 13.2 asks you to “compare and interpret” mode and mean (Q1, Q4). The same demand appears in Example 6, which contrasts 52 (modal marks) with 62 (mean marks) (NCERT pp. 185–186). A full answer states both numbers and says what each means in the context.
• Exercise 13.3 mixes median, mean and mode for one data set (Q1) and uses a given median to find missing frequencies (Q2) — the technique of Example 8 (NCERT pp. 196–197). Q4 carries the hint to convert classes to continuous form before using the median formula.

A full-marks answer names the modal or median class, writes the formula, substitutes with each symbol identified, and states the final value with its unit — followed by one line of interpretation whenever the question says “interpret” or “compare”.

Textbook contents and the examinable syllabus are not always identical — check the current official syllabus before deciding what to prioritise.

Revision Summary of Statistics Class 10

This rebuilds the chapter’s own Summary (NCERT pp. 200–201) in plainer words — five points cover the whole chapter.

1. Mean: the same mean comes from three methods — direct \( \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \), assumed mean \( \bar{x} = a + \frac{\sum f_i d_i}{\sum f_i} \), and step deviation \( \bar{x} = a + h \frac{\sum f_i u_i}{\sum f_i} \). All assume the frequency of a class is centred at its class mark.
2. Mode: locate the modal class, then \( \text{Mode} = l + \frac{f_1 – f_0}{2f_1 – f_0 – f_2} \times h \).
3. Cumulative frequency: the running total of frequencies up to and including a class; the last entry equals \( n \).
4. Median: find \( \frac{n}{2} \), pick the median class, then \( \text{Median} = l + \frac{\frac{n}{2} – cf}{f} \times h \), with \( cf \) from the class just before.
5. Note to the reader: mode, median and ogives all require continuous classes before the formulas apply; in an ogive the two axes need not share the same scale (NCERT p. 201).

Related Resources for Statistics Class 10

Keep the chapter in context: these pages sit around it in the book and in the notes library.

• Class 10 Mathematics notes — the book page covering every chapter, including Statistics.
• Surface Areas and Volumes (Chapter 12) — the chapter that comes just before Statistics in the book.
• Probability (Chapter 14) — the chapter that immediately follows Statistics.
• Class 10 hub — notes for every Class 10 subject.
• CBSE notes home — the full library from Class 6 to Class 12.

Sources and Data Verification

This page describes the NCERT Class 10 Mathematics textbook, Chapter 13 Statistics, as printed in the current NCERT edition — the official PDF carries the Reprint 2026-27 notice.

It covers this single chapter only, not the full CBSE curriculum.

The listing is maintained for the current academic session using the NCERT information available to us.

NCERT settles the textbook, its editions and the official PDF; CBSE settles the curriculum, syllabus and examinations.

Frequently Asked Questions about Statistics Class 10

Which method should I use to find the mean of grouped data: direct, assumed mean or step deviation?

Base the choice on the size of the numbers. If \( x_i \) and \( f_i \) are small, use the direct method. If the values are numerically large, use the assumed mean or step deviation method; step deviation is convenient when all deviations \( d_i \) share a common factor, normally the class size (NCERT pp. 179–180). All three give the same mean.

Why is the mean of grouped data different from the exact mean of the same data?

Because of the mid-point assumption. The book’s example gives 59.3 as the exact mean of 30 ungrouped marks but 62 once the same marks are grouped into classes of width 15 — grouping assumes every observation sits at its class mark (NCERT p. 174), so the grouped mean is approximate.

What is the difference between a modal class and the mode?

The modal class is the class interval with the maximum frequency; the mode is a single value inside it. In grouped data you cannot read the mode straight from the table, so you identify the modal class first and then locate the mode inside it using the formula (NCERT p. 184).

How do I identify the median class in a grouped frequency table?

Compute \( \frac{n}{2} \), where \( n \) is the total frequency. The median class is the class whose cumulative frequency is greater than and nearest to \( \frac{n}{2} \). In the chapter’s example \( n = 53 \) and \( \frac{n}{2} = 26.5 \), so the class 60–70 with cumulative frequency 29 is the median class (NCERT p. 193).

Do the mode and median formulas work when class intervals are not continuous?

No — both formulas, and ogive construction, assume continuous classes. Broken intervals such as 118–126, 127–135 must first be converted to 117.5–126.5, 126.5–135.5 (see the hint in Exercise 13.3 Q4 and the note on NCERT p. 201).

What does the empirical relation 3 Median = Mode + 2 Mean mean in a question?

It states that for the same data set, \( 3 \times \text{Median} = \text{Mode} + 2 \times \text{Mean} \) (NCERT p. 197). It is an empirical, observed relationship between the three measures — knowing any two lets you estimate the third, which is a quick check on your computed values.

Reference: NCERT Class 10 Mathematics textbook, chapter 13, official edition on ncert.nic.in.