These NCERT solutions for class 10 maths chapter 13 (Statistics) get you from a homework question to the exercise that answers it. The chapter has three exercises and 22 questions in NCERT Mathematics (Class 10, pp. 172–201): the mean in Exercise 13.1, the mode in Exercise 13.2, the median in Exercise 13.3.
Chapter 13 moves the mean, median and mode you met in Class IX from ungrouped lists to grouped frequency tables. That one change introduces class marks, cumulative frequency tables and ogives — and the real skill becomes picking the right formula and the right class inside the table.
This page tells you which exercise to open and which method it expects, so you reach the solution pages already knowing what to look for. From the Class 10 Maths hub you can reach every chapter’s directory; the Class 10 notes and the NCERT notes collection cover the rest of the syllabus.
| Exercise | What you get |
|---|---|
| Exercise 13.1 | Solved |
| Exercise 13.2 | Solved |
| Exercise 13.3 | Solved |
| Exercise | What you get |
|---|---|
| Exercise 13.1 | Solved |
| Exercise 13.2 | Solved |
| Exercise 13.3 | Solved |
Reference: NCERT textbooks (CBSE).
NCERT Solutions for Class 10 Maths Chapter 13: Exercise Map
The directory below lists every exercise in Chapter 13. Each row links to its step-by-step solutions; the notes under each exercise say what that exercise actually tests.
Exercise 13.1 — Mean of Grouped Data
Tests all three routes to the grouped mean — direct, assumed mean and step-deviation — and makes you justify the choice: two questions ask which method you used and why, and one works the formula backwards with the mean given and a frequency missing.
Later questions add decimal-width class intervals and unequal class sizes, where the method choice matters as much as the arithmetic.
Exercise 13.2 — Mode of Grouped Data
Drills the mode formula until locating the modal class is automatic. Two questions also demand the mean and a written comparison of the two measures, and one table hides zero-frequency classes just before the modal class — a test of whether you can still read off the preceding frequency \( f_0 \) correctly.
Exercise 13.3 — Median of Grouped Data
Builds the median through cumulative frequency tables. Two questions ask for all three measures together, one gives the median and asks for two missing frequencies, one starts from less-than cumulative data that must be rebuilt into class intervals, and one forces the conversion to continuous classes before the formula works at all.
The Methods of Chapter 13: Three Means, Mode and Median
Every formula in this chapter starts from the class mark, the midpoint chosen to represent a whole class:
\[ \text{Class mark} = \frac{\text{Upper class limit} + \text{Lower class limit}}{2} \]
The mean of grouped data has three equivalent routes:
\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \quad \text{(direct)}, \qquad \bar{x} = a + \frac{\sum f_i d_i}{\sum f_i} \quad \text{(assumed mean)}, \qquad \bar{x} = a + h\left( \frac{\sum f_i u_i}{\sum f_i} \right) \quad \text{(step deviation)} \]
where \( d_i = x_i – a \) and \( u_i = \frac{x_i – a}{h} \); all three give the same mean — the last two are simpler arithmetic, not different results.
The textbook’s rule for choosing (p. 180): use the direct method when the class marks \( x_i \) and frequencies \( f_i \) are small. When they are numerically large, use the assumed mean method or the step-deviation method. If the class sizes are unequal, step deviation still works when \( h \) is taken as a suitable divisor of all the deviations.
That grouped mean is approximate, not exact: the formula assumes the frequency of each class is centred on its class mark (pp. 173–77).
The mode needs the modal class — the class with the highest frequency — then uses:
\[ \text{Mode} = l + \left( \frac{f_1 – f_0}{2f_1 – f_0 – f_2} \right) \times h \]
where \( l \) is the lower limit of the modal class, \( h \) the class size, \( f_1 \) the modal class’s own frequency, \( f_0 \) the frequency of the class preceding it and \( f_2 \) the frequency of the class succeeding it (p. 184).
The median needs the median class first — the first class whose cumulative frequency is greater than (and nearest to) \( \frac{n}{2} \). Then:
\[ \text{Median} = l + \left( \frac{\frac{n}{2} – cf}{f} \right) \times h \]
where \( cf \) is the cumulative frequency of the class before the median class — the most misread symbol in the chapter (p. 193).
The three measures are tied by the empirical relationship \( 3 \times \text{Median} = \text{Mode} + 2 \times \text{Mean} \) (p. 198) — a quick check that three computed values sit together. Which measure is the right one depends on the question:
| Measure | What it answers | Best when | Fails when |
|---|---|---|---|
| Mean | The average of all observations | Comparing distributions; no extreme values | Extreme values pull it away from most of the data |
| Median | The middle-most value — 50% of data below, 50% above | A typical value is wanted and extremes exist | It ignores how large the other observations are |
| Mode | The most frequent value | The most popular item or common case is needed | Data may be multimodal or the mode far from the centre |
Common Mistakes on Chapter 13 Homework (and the Fix)
- The upper-limit convention. A value that falls exactly on an upper class limit is counted in the next class — a mark of 40 belongs to 40–55, not 25–40 (p. 173). One slip here shifts every cumulative frequency that follows.
- Skipping the class-mark column. Every mean method needs \( x_i \). Compute \( \frac{\text{upper} + \text{lower}}{2} \) for each class before you sum anything.
- Reading the wrong neighbours in the mode formula. Once the modal class is found, \( f_0 \) is the frequency of the class immediately before it and \( f_2 \) the frequency of the class immediately after it — swapping either changes the fractional term.
- Using the median class’s own cumulative frequency. In the median formula, \( cf \) belongs to the class before the median class. Substituting the median class’s own cumulative frequency turns the numerator negative and the median meaningless.
- Applying the mode, median or ogive to non-continuous classes. The Note to the Reader (p. 201) requires continuous classes first. If a table has gaps between consecutive classes, widen each class by half the gap at both ends before applying any formula or drawing an ogive.
- Building frequencies from less-than data by copying the totals. Each class frequency is the difference between successive cumulative totals (method of Example 7, p. 195) — subtract the previous total instead of reusing it.
How to Use These Solutions: Route from Homework to Exercise
Match the wording of your homework to the exercise that tests it:
- “Find the mean” or “calculate the average” → Exercise 13.1
- “Find the mode” or “modal …” → Exercise 13.2
- “Find the median” or “the median value” → Exercise 13.3
- “Find all three and compare / interpret” → Exercise 13.3 Question 1 (or Exercise 13.2 Question 1 when only the mean and mode are asked)
- “A frequency is missing” → Exercise 13.1 Question 3 when the mean is given; Exercise 13.3 Question 2 when the median is given
Work in the order 13.1 → 13.2 → 13.3. The median builds on cumulative frequency tables that use the same class intervals as the mean, so the first two exercises make the third mechanical. Note that this chapter covers grouped data only; the ungrouped mean, median and mode from Class IX are assumed background.
In the NCERT sequence, Statistics (Chapter 13) sits between Surface Areas and Volumes (Chapter 12) and Probability (Chapter 14) — the frequency-table fluency you build here is the same skill Chapter 14 expects. You can verify every exercise number and table against the official NCERT Class 10 Mathematics textbook — Chapter 13 PDF.
FAQs on Chapter 13 Statistics
Which exercise of Class 10 Maths Chapter 13 covers the mean of grouped data?
Exercise 13.1. Its nine questions cover the direct, assumed mean and step-deviation methods, and two of them ask you to say which method you chose and why.
Which exercise covers the mode in Chapter 13 Statistics?
Exercise 13.2. All six questions apply the mode formula, and three of them also ask for the mean so you can compare and interpret the two measures.
Which exercise covers the median in Chapter 13 Statistics?
Exercise 13.3. Its seven questions work through cumulative frequency tables, including one with missing frequencies and one that must be converted to continuous classes before the formula applies.
When should I use the direct method, the assumed mean method or the step-deviation method?
Use the direct method when the class marks and frequencies are small. When they are numerically large, switch to the assumed mean or step-deviation method; with unequal class sizes, step deviation still works by taking \( h \) as a suitable divisor of the deviations.
Why do I need continuous classes before finding the mode or median of grouped data?
The mode and median formulas assume the class intervals run without gaps, with the upper limit of one class equal to the lower limit of the next. If a table has gaps, adjust the boundaries first — widen each class by half the gap at each end — and only then apply the formula or draw the ogive.
What is the empirical relationship between mean, median and mode in Chapter 13?
Three times the median equals the mode plus two times the mean: \( 3 \times \text{Median} = \text{Mode} + 2 \times \text{Mean} \). Use it as a sanity check after computing all three measures.
Reference: NCERT textbooks (CBSE).